Solar Potential in Saudi Arabia for Inclined Flat-Plate Surfaces of Constant Tilt Tracking the Sun

Abstract

The objective of the present work is to investigate the optimally performing tilt angles in Saudi Arabia of solar panels that follow the daily motion of the Sun. To that end, the annual energy sums are estimated for surfaces with tilt angles in the range 5°–55° at 82 locations covering all Saudi Arabia. All calculations use a surface albedo of 0.2 and a near-real value, too. It is found that tilt angles of 40°, 45°, and 50°, respectively, are optimal for the three recently defined solar energy zones in Saudi Arabia. The variation of the energy sums in each energy zone on annual, seasonal and monthly basis is given for near-real ground albedos; the analysis provides regression equations for the energy sums as functions of time. A map of the annual global inclined solar energy for Saudi Arabia is derived and presented. The annual energy sums are found to vary between 2159 kWhm⁻²year⁻¹ and 4078 kWhm⁻²year⁻¹. Finally, a correction factor, introduced in a recent publication, is used; it is confirmed that the relationship between the correction factor and either the tilt angle or the ground-albedo ratio has a general application and it may constitute a nomogram.

1. Introduction

Installations with tilted solar collectors for exploiting the renewable energy of the Sun have long been available in the market as commercial products. Solar flat-plate panels are nowadays widely used for converting solar energy into electricity (PV installations) or hot water (solar heating systems). These stationary systems consist of solar panels that receive solar radiation (i) at a fixed tilt angle with a southward orientation in the northern hemisphere or a northward orientation in the southern hemisphere; (ii) at a fixed tilt angle following the motion of the Sun; and (iii) at a varying tilt angle following the motion of the Sun, a mechanism that ensures that the solar rays always remain normal to the receiving plane. The mode (i) is widely used because of its lower cost for the static supporting frame of the solar system. The mode (ii), also known as a one- or single-axis system, provides higher solar energy on the inclined surface mounted on a vertical rotating axis, but has a slightly higher cost in order to maintain the moving parts. The mode (iii) is considered the most effective and is known as a two- or double-axis system. It provides the best performance of the solar systems rotating with the aid of a vertical and a horizontal axis, but is associated with higher maintenance costs because of more mechanical moving parts. The first category of solar systems is also called stationary or static, while the other two are dynamic, because of their Sun-tracking ability. Farahat et al. [1] examined the mode (i) for the performance of flat-plate solar collectors in Saudi Arabia. This work is a continuation as it investigates the mode (ii) for the solar energy received on single-axis systems across the country.

Though static solar systems have received a lot of attention from researchers and solar energy developers, the dynamic ones (and especially the single-axis-mode collectors) have not been neglected either. Much effort has been invested at both the solar-energy-calculation level and the solar-system-development level to improve both the moving and the electronic parts for the Sun-tracking sensors [2,3]. The calculation of the optimum tilt angle and orientation for receiving maximum solar energy on solar flat-plate collectors has been the objective of various studies that use different strategies for obtaining solar radiation data: solar radiation modelling, e.g., [4,5]; combining with ground-based solar data, e.g., [6]; or utilising international data bases, e.g., [7,8]. Recently, a new method was presented by [9] for Greece and was applied in a recent study for Saudi Arabia [1]. The proposed method finds the maximum solar energy received by flat-plate collectors with southward (northward) orientation in the northern (southern) hemisphere. The idea behind this methodology is applied in the present study for solar collectors with constant tilt tracking the Sun.

An extensive review of the studies conducted for Saudi Arabia related to the present work can be found in [1].

From the above it is clear that no attempt has been made so far to construct a solar map for Saudi Arabia to show the solar potential on inclined flat planes that track the Sun. This gap is bridged in the present study, which includes three innovations. (i) For the first time, solar maps for Saudi Arabia showing maximum energy on optimally inclined flat surfaces tracking the Sun are derived. (ii) For the second time the three solar energy zones introduced by the same authors for Saudi Arabia [1] are also utilised here. (iii) The notion of the (ground-albedo) correction factor introduced in [1] is also used here, and the shape of the universal curves (representable as nomograms) of this parameter in relation to the tilt angle and the ground-albedo ratio is confirmed.

The structure of the paper is as follows. Section 2 describes the data collection and data analysis. Section 3 deploys the results of the study, while Section 4 presents the conclusions and main achievements of the work. Acknowledgements and References follow.

2. Materials and Methods

2.1. Data Collection

For the implementation of the study hourly values of the following parameters were collected: direct, H_b (in Wm⁻²), and diffuse, H_d (in Wm⁻²), horizontal solar radiation. These values were downloaded from the PV—Geographical Information System (PV-GIS) tool [10] using the latest Surface Solar Radiation Data Set—Heliostat (SARAH) 2005–2016 data base (12 years) [11,12]. The PV-GIS provides solar radiation data for any location in Europe, Africa, Middle East including Saudi Arabia, central and southeast Asia and most parts of the Americas. Solar radiation in the PV-GIS platform is calculated from satellite observations and modelling [13,14].

A set of 82 sites in Saudi Arabia was selected to cover the whole territory of the country. Hourly values of H_b and H_d were downloaded from PV-GIS for all 82 sites;

Table 1. The 82 sites selected over Saudi Arabia to cover the whole territory of the country; φ is the geographical latitude, and λ the geographical longitude in the WGS84 geodetic system. This table is reproduction of Table 1 in [1].

#	Site	φ (Degrees N)	λ (Degrees E)
1	Dammam	26.42	50.09
2	Al Jubail	26.96	49.57
3	Ras Tanura	26.77	50.00
4	Abqaiq	25.92	49.67
5	Al Hofuf	25.38	49.59
6	Arar	30.96	41.06
7	Sakaka	29.88	40.10
8	Tabuk	28.38	36.57
9	Al Jawf	29.89	39.32
10	Riyadh	24.71	46.68
11	Al Qassim	26.21	43.48
12	Hafar Al Batin	28.38	45.96
13	Buraydah	26.36	43.98
14	Al Majma’ah	25.88	45.37
15	Hail	27.51	41.72
16	Jeddah	21.49	39.19
17	Jazan	16.89	42.57
18	Mecca	21.39	39.86
19	Medina	24.52	39.57
20	Taif	21.28	40.42
21	Yanbu	24.02	38.19
22	King Abdullah Economic City	22.45	39.13
23	Najran	17.57	44.23
24	Abha	18.25	42.51
25	Bisha	19.98	42.59
26	Al Sahmah	20.10	54.94
27	Thabhloten	19.83	53.90
28	Ardah	21.22	55.24
29	Shaybah	22.52	54.00
30	Al Kharkhir	18.87	51.13
31	Umm Al Melh	19.11	50.11
32	Ash Shalfa	21.87	49.71
33	Oroug Bani Maradh Wildlife	19.41	45.88
34	Wadi ad Dawasir	20.49	44.86
35	Al Badie Al Shamali	21.99	46.58
36	Howtat Bani Tamim	23.52	46.84
37	Al Duwadimi	24.50	44.39
38	Shaqra	25.23	45.24
39	Afif	24.02	42.95
40	New Muwayh	22.43	41.74
41	Mahd Al Thahab	23.49	40.85
42	Ar Rass	25.84	43.54
43	Uglat Asugour	25.85	42.15
44	Al Henakiyah	24.93	40.54
45	Ar Rawdah	26.81	41.68
46	Asbtar	26.96	40.28
47	Tayma	27.62	38.48
48	Al Khanafah Wildlife Sanctuary	28.81	38.92
49	Madain Saleh	26.92	38.04
50	Altubaiq Natural Reserve	29.51	37.23
51	Hazem Aljalamid	31.28	40.07
52	Turaif	31.68	38.69
53	Al Qurayyat	31.34	37.37
54	Harrat al Harrah Conservation	30.61	39.48
55	Al Uwayqilah	30.33	42.25
56	Rafha	29.63	43.49
57	Khafji	28.41	48.50
58	Unnamed 1	21.92	51.99
59	Unnamed 2	21.03	51.16
60	Unnamed 3	22.33	52.53
61	Unnamed 4	23.42	50.73
62	Unnamed 5	21.28	48.03
63	Unnamed 6	31.70	39.26
64	Unnamed 7	32.02	39.65
65	Unnmaed 8	31.02	42.00
66	Unnamed 9	30.63	41.31
67	Unnamed 10	29.78	42.68
68	Unnamed 11	28.68	47.49
69	Unnamed 12	28.41	47.97
70	Unnamed 13	28.05	47.53
71	Unnamed 14	27.97	47.88
72	Unnamed 15	27.15	48.98
73	Unnamed 16	27.21	48.56
74	Unnamed 19	27.15	48.02
75	Unnamed 18	27.66	48.52
76	Unnamed 19	24.74	48.95
77	Unnamed 20	28.34	35.17
78	Unnamed 21	26.27	36.67
79	Unnamed 22	21.89	43.06
80	Unnamed 23	18.76	47.54
81	Unnamed 24	21.38	53.28
82	Unnamed 25	19.24	52.79

shows the names and provides the geographical coordinates of these sites, while Figure 1 is a map of Saudi Arabia showing their location. The criterion for selecting these sites is mentioned in [1].

2.2. Data Processing and Analysis

This section consists of five steps, which are briefly described as more details can be found in the companion paper [1].

Step 1: The downloaded hourly data from the PV-GIS platform were converted from the universal time coordinate (UTC) into local standard time (LST = UTC + 3 h).

Step 2: The hourly global horizontal radiation, H_g (in Wm⁻²), values were estimated as the sum H_g = H_b + H_d.

Step 3: Use of the XRONOS.bas algorithm was made to calculate the solar altitudes, γ, for all sites in LST; this code is based on the SUNAE routine introduced by Walraven [15], and all subsequent modifications [16,17].

Step 4: All radiation and solar geometry values were assigned to the nearest LST hour (i.e., values at hh:mm LST were assigned to hh:00 LST).

Step 5: All hourly solar radiation values were retained for subsequent analysis that (i) were greater than 0 Wm⁻², (ii) corresponded to γ ≥ 5°, and (iii) H_d ≤ H_g.

For estimating global solar irradiance on an inclined plane fixed on a one-vertical axis tracker, H_g,t,β (in Wm⁻²), there was adopted the isotropic model of Liu-Jordan (L-J) [18] (the subscript t stands for ‘tracking’, and β is the tilt angle of the inclined plane with respect to the local horizon, in degrees). The isotropic model was used to estimate the ground-reflected radiation from the surrounding surface, H_r,t,β (in Wm⁻²), received on the inclined flat-plate surface. The L-J model has proved to be as efficient as other more sophisticated models in providing the tilted total solar radiation in many parts of the world [19]. For a Sun-tracking surface mounted on a vertical axis the received total solar radiation is given by a slight modification of Equation (1) in [1]

H_g,t,β = H_b,t,β + H_d,t,β + H_r,t,β,

(1)

where the subscript t denotes tracking by the inclined surface. According to Liu-Jordan [18]

H_d,t,β = H_d·R_di,

(2)

H_r,t,β = H_g·R_r·ρ_g0, (or ρ_g)

(3)

R_di = (1 + cosβ)/2,

(4)

R_r = (1 − cosβ)/2,

(5)

H_b,tβ = H_b·cosθ/sinγ,

(6)

cosθ = sinβ·cosγ·cos(ψ − ψ′) + cosβ·sinγ.

(7)

where θ is the incidence angle (the angle formed by the normal to the inclined surface and the line joining the surface with the centre of the Sun), and ψ, ψ′ are the solar azimuths of the Sun and of the inclined plane, respectively; ψ = ψ′ in this case. The parameters R_di and R_r are called the isotropic sky-configuration and ground-inclined plane-configuration factors, respectively. In the L-J model the ground albedo usually takes the value of ρ_g0 = 0.2 (Equation (3)), which is considered as reference in solar radiation modelling, although it refers to grassland areas [19]. Also, close-to-reality ground-albedo values, ρ_g, were used in this work. To retrieve such values for the 82 sites, use of the Giovanni portal [20] was made (details in [1]). Annual mean ρ_g values were then computed and used to re-calculate H_g,t,β.

The tilt angle in the present study varied in the range 5°–55° in increments of 5° (i.e., 11 tilt angles). This range fully covers all latitudes of Saudi Arabia (from ≈18° N to 32° N). For every site and tilt angle, hourly values of H_g,t,β were estimated from Equation (1) for both ρ_g0 = 0.2 and ρ_g. From the hourly H_g,t,β values, annual, seasonal and monthly solar energy sums (in kWhm⁻²) under all-sky conditions were estimated for 82 sites, 11 tilt angles, and 2 ground-albedo values.

3. Results

3.1. 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°–55° (in 5° increments) by using both ground albedos, ρ_g0 and ρ_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. Maximum H_g,t,β annual sums for the 82 sites in Saudi Arabia for optimal tilt angles, opt. β, with reference albedo ρ_g0 and a near-real albedo ρ_g, under all-sky conditions for the period 2005–2016. The H_g values are rounded integers.

Site #	Hg,t,β/ρg0 (kWhm−2Year−1)/ Optimum β (Degrees)	Hg,t,β/ρg (kWhm−2Year−1)/ Optimum β (Degrees)	Hg,t,β/ρg (kWhm−2Year−1)/ Selected Optimum β (Degrees)
1	2826/45	2846/45	2846/45
2	2824/45	2873/45	2873/45
3	2785/45	2782/40	2782/45
4	2862/40	2925/45	2925/45
5	2872/40	2925/45	2925/45
6	2921/45	2993/50	2993/50
7	3015/45	3081/50	3081/50
8	3149/45	3173/45	3173/45
9	2963/45	3022/50	3022/50
10	2937/45	2991/45	2991/45
11	2915/45	2955/45	2955/45
12	2748/45	2804/45	2804/45
13	2897/45	2944/45	2944/45
14	2900/45	2953/45	2953/45
15	2991/45	3035/45	3035/45
16	2912/40	2918/40	2917/40
17	2794/40	2767/35	2767/40
18	2903/40	2909/40	2909/40
19	3035/45	3021/40	3021/40
20	2926/40	2931/45	2931/40
21	3058/45	3054/45	3053/40
22	2946/40	2940/40	2940/40
23	3111/40	3129/45	3128/40
24	2803/40	2803/40	2803/40
25	3078/40	3086/40	3086/40
26	3041/40	3123/45	3109/40
27	3039/40	3118/45	3103/40
28	4013/50	4115/55	4078/45
29	2974/45	3043/45	3043/45
30	3268/45	3360/50	3314/40
31	3069/40	3144/45	3132/40
32	3002/40	3083/45	3069/40
33	3062/40	3144/45	3131/40
34	3017/40	3070/45	3062/40
35	2999/40	3061/45	3050/40
36	2984/40	3046/45	3046/45
37	2976/45	3028/45	3028/45
38	2899/45	2956/45	2956/45
39	3001/45	3032/45	3025/40
40	3065/45	3097/45	3091/40
41	3041/45	3038/45	3037/40
42	2917/45	2967/45	2967/45
43	3005/45	3033/45	3033/45
44	3078/45	3082/45	3076/40
45	2984/45	3008/45	3008/45
46	3073/45	3111/45	3111/45
47	3149/45	3239/50	3228/45
48	3004/45	3060/50	3058/45
49	3118/45	3183/50	3181/45
50	3027/45	3058/50	3056/45
51	2926/45	3004/50	3004/50
52	3666/55	3731/55	3692/50
53	2332/50	2365/50	2365/50
54	2936/45	2980/50	2980/50
55	2891/45	2964/50	2964/50
56	2272/50	2325/55	2324/50
57	2721/45	2700/40	2696/45
58	3018/40	3100/45	3100/45
59	3029/40	3111/45	3097/40
60	3011/40	3091/45	3091/45
61	2961/40	3042/45	3042/45
62	3015/40	3079/45	3069/40
63	2841/45	2892/50	2892/50
64	2877/45	2926/50	2926/50
65	2845/45	2913/50	2913/50
66	2852/45	2922/50	2922/50
67	2908/45	2978/50	2978/50
68	2546/55	2608/55	2553/45
69	2760/45	2827/50	2827/45
70	2763/40	2833/45	2833/45
71	2764/45	2821/45	2821/45
72	2779/40	2859/50	2859/45
73	2817/45	2863/45	2863/45
74	2995/45	3082/50	3067/45
75	2883/45	2947/50	2945/45
76	2113/50	2171/50	2159/45
77	2914/40	3002/50	3002/45
78	3145/45	3138/45	3125/40
79	3114/45	3086/45	3086/40
80	3041/40	3085/45	3078/40
81	3084/40	3170/45	3170/45
82	2988/45	3076/50	3051/40

shows these maximum annual H_{g,t,opt. β} sums for all sites with the corresponding optimum tilts, opt. β (columns 2 and 3 for ρ_g0 and ρ_g, respectively). From this table, it is seen that the average H_{g,t,opt. β}/ρ_g0 (column 2) is 2745.5 kWhm⁻², while the average H_{g,t,opt. β}/ρ_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 ρ_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. β-derived values under the ρ_g calculations (H_{g,t,opt. β/ρg}) 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. β exists; to the contrary, there is a great mix of opt. β 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. β 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. β, 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. β = 40°, SEZ-B with selected opt. β = 45° and SEZ-C with selected opt. β = 50°). 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, ΔH_g,t,β/ρg, are defined as: ΔH_g,t,β/ρg = H_{g,t,opt. β/ρg} − H_{g,t,selected opt. β/ρg}. 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⁻²year⁻¹. The mean annual solar energy sums for all sites 1–43 and 44–82 are 3026.3 kWhm⁻²year⁻¹ and 2954.7 kWhm⁻²year⁻¹, respectively, while their corresponding mean differences are −4.4 kWhm⁻²year⁻¹ and −5.7 kWhm⁻²year⁻¹. This means that in the first group of sites, the absolute error derived from the manual selection of the optimum tilt angles is 0.15% (100 × 4.4/3026.3) while the error for the second group 0.19% (100 × 5.7/2954.7), both being quite insignificant. Therefore, the new distribution of the 82 sites in Figure 3 does not influence the solar energy yield across Saudi Arabia.

Contrary to the 3 SEZs adopted in the present work, Zell et al. [21] divided the country into five geographical areas in order to use and analyse the solar radiation data from [22]. Nevertheless, this division has of no practical value as it did not meet any solar radiation criteria.

Kaddoura et al. [7] estimated 12 optimal β for each of the 12 months of the year for Tabuk (#8 in Table 1), Al Jawf (#9), Riyadh (#10), Jeddah (#16), and Abha (#24). These β were derived from modelling and they, therefore, have a purely theoretical value.

In what follows, the notation H_g,t,β/ρg refers to the selected optimal β, i.e., to H_{g,t,selected opt. β/ρg}.

3.2. Monthly Energy Sums

The intra-annual variation of H_g,t,β/ρg in each specific SEZ region is shown in Figure 5. Figure 5a refers to all sites in SEZ-A with optimal β = 40°, Figure 5b to all sites in SEZ-B with optimal β = 45°, Figure 5c to the sites in SEZ-C with optimal β = 50°, and Figure 5d to all sites irrespective of SEZ and optimal β. The expressions for the lines that best fit the means and their coefficient of determination, R², are given in

Table 3. Regression equations for the best-fit curves to the monthly/seasonal mean H_g,t,β/ρg sums averaged over all respective sites in the period 2005–2016, together with their R² values; s is either month in the range 1–12 or season in the range 1–4 (1 = spring, 2 = summer, 3 = autumn, 4 = winter).

SEZ	Regression Equation	R2
A (months) A (seasons)	Hg,t,40/ρg = 0.0054 s6 − 0.218 s5 + 3.4241 s4 − 25.977 s3 + 95.627 s2 − 142.65 s +296.38 Hg,t,40/ρg = −2.0956 s3 − 0.5587 s2 + 10.341 s + 787.81	0.82 1
B (months) B (seasons)	Hg,t,45/ρg = 0.0049 s6 − 0.1809 s5 + 2.5754 s4 − 18.215 s3 + 65.134 s2 − 90.585 s +248.07 Hg,t,45/ρg = 44.915 s3 − 367.88 s2 + 870.06 s + 216.23	0.94 1
C (months) C (seasons)	Hg,t,50/ρg = 0.0051 s6 − 0.1867 s5 + 2.6857 s4 − 19.7280 s3 + 74.6630 s2 − 106.5200 s +235.9800 Hg,t,50/ρg = 74.807 s3 − 627.99 s2 + 1499.1 s − 174.91	0.98 1
A,B,C (months) A,B,C (seasons)	Hg,t,β/ρg = 0.0051 s6 − 0.1955 s5 + 2.9007 s4 − 21.259 s3 + 77.717 s2 − 112.08 s +263.4 Hg,t,β/ρg = 33.967 s3 − 286.87 s2 + 683.91 s + 344.56	0.94 1

. It is seen that the R² statistic obtains high values; this allows a solar energy user or investor in Saudi Arabia to estimate the monthly energy production in any of the three SEZs in an accurate way by applying the regression equations. The graphs also contain curves for the mean ± 1 standard deviation (σ). It is apparent in all graphs (and especially in those that refer to SEZ-A and SEZ-B) that two secondary H_g,t,β maxima in March and October exist. These occur because of the variation of the solar elevation (or altitude, γ) throughout the year at a certain time during the day. To demonstrate this, Figure 6 shows the variation of H_g,t,40 during 2005–2016 at three sites in Saudi Arabia at high, mid and low latitudes, i.e., Arar (#6, 30.96 degN), Jeddah (#16, 21.49 degN), and Jazan (#17, 16.89 degN). Jazan has two energy maxima, one in April and another in October; Jeddah has two, one in March and another in October, while Arar shows a normal behaviour (maximum energy in July).

3.3. Seasonal Energy Sums

In solar energy installations the minimum and maximum possible energy received by them corresponds to the winter and summer months, respectively. Therefore, this section is devoted to analysing the solar energy totals during all seasons, i.e., spring (March-April-May), summer (June-July-August), autumn (September-October-November), and winter (December-January-February).

Figure 7 presents the total solar energy received on a flat surface in SEZ-A (Figure 7a), SEZ-B (Figure 7b), SEZ-C (Figure 7c), and all SEZs (Figure 7d) under all-sky conditions during spring, summer, autumn, and winter. The energy values are sums for each season averaged over the period 2005–2016 and for all sites belonging to the same SEZ (or all SEZs). Table 3 gives the regression equations for the curves that best fit the mean ones in each case. It is interesting to observe that the fits are ideal (R² = 1) in all cases.

3.4. Maps of Annual Energy Sums

Figure 8 shows the solar potential over Saudi Arabia in terms of the annual H_g and H_g,t,β/ρg sums. A gradual increase in the annual solar potential in the direction NE–SW for both horizontal and optimally-inclined flat planes is observed. Very similar patterns to those in the present study are given in the Solar Radiation Atlas for Saudi Arabia [23]. Farahat et al. [1] have come to similar conclusion as regards the maximum solar energy received by flat-plate collectors oriented to south with optimum inclination. They justified this observation by the latitude gradient of the sites and the variability in meteorology across the country from north to south [24].

3.5. Evaluation of the PV-GIS Tool

Various studies have presented validation results for the solar radiation PV-GIS-satellite-derived data by comparing them with ground-based solar radiation measurements from 30 BSRN (Baseline Solar radiation Network) stations [13,14,25]. The reported differences (%) in the form of 100 × (estimated values—measurements)/measurements were found to vary between −14% and +11% (the estimated values refer to the PV-GIS tool and the measurements come from the BSRN). Furthermore, Farahat et al. [1] demonstrated this comparison by taking monthly mean H_g values measured at the Actinometric Station of the National Observatory of Athens (ASNOA, 37.97° N, 23.72° E, 107 m above sea level) and corresponding values from the PV-GIS platform for the period 2005–2011. Figure 9 presents this comparison, which shows an excellent agreement (R² = 0.99). Nevertheless, the PV-GIS-estimated values seem to overestimate the measured H_g ones by +10%, a figure that is within the range in the above-mentioned studies, i.e., from −14% to +11%. Therefore, the PV-GIS data were accepted for use in the present study.

3.6. Correction Factor

Farahat et al. [1] introduced the notion of the correction factor, CF, which is defined as: CF = H_g,t,β/ρg/H_g,t,β/ρg0. Actually, CF is the ratio of the annual H_g,t,β sum at each site of the 82, calculated twice, once for ρ_g0 = 0.2 and a second time for ρ_g = actual value. The meaning of the CF is that it corrects the energy on an inclined surface under the influence of a ground albedo equal to 0.2 to that which is under the influence of the near-real ground-albedo value. Figure 10 presents the variation of CF as function of β for all 82 sites; the controlling parameter is the ratio ρ_r = ρ_g/ρ_g0. Note that the higher the ρ_r value is (i.e., for ρ_g > ρ₀), the more concave the best-fit curve is; in contrast, the lower the ρ_r value is (i.e., for ρ_g < ρ₀), the more convex the best-fit curve becomes. In the exceptional case of ρ_g = ρ₀ (as for site #24), CF = 1. All the data points at every β in Figure 10 correspond to the 82 sites. Figure 11 shows the distribution of the ρ_r values across the 82 sites.

A diagram of CF vs. ρ_r is shown in Figure 12, where a linear relationship exists along all sites at the same β. The controlling parameter in this case is β; as β increases, so does the slope of the linear fit to the data points. The data points along each line correspond to the 82 sites.

Beside the above, a more detailed analysis was performed for CF. In this analysis, the average values of CF = H_g,t,β/ρg/H_g,t,β/ρg0 were calculated for all βs in the range 5°–55° and all sites belonging to the same SEZ, as well as all sites irrespective of SEZ. The results are shown in Figure 13. It is clearly seen that all mean CF values have an increasing trend with increasing β, because a flat-plate tilted surface receives more reflected radiation from its surroundings as its tilt angle increases. Moreover, the standard deviation, σ, of the mean CF increases with β, because H_g,t,β increases with increasing β, a fact that produces larger dispersion of the annual solar energy across all sites. In contrast, σ becomes smaller in the transition from SEZ-A to SEZ-C sites, a result that comes from the combination of the multitude of sites in each SEZ and the dispersion of the individual H_g,t,β values in the SEZ as shown by R² in Table 3 (decreasing dispersion by increasing R² from SEZ-A to SEZ-C). Note that the majority of the sites lies in SEZ-B (28 sites in SEZ-A, 40 in SEZ-B, and 14 in SEZ-C).

Table 4. Regression equations for the best-fit curves to the CF–β data points averaged over all respective sites in the period 2005–2016, together with their R² values, for SEZ-A, SEZ-B, SEZ-C, and all SEZs.

β (SEZ)	Regression Equation	R2
40° (A)	CF = 8.0863 × 10−9 β3 + 4.1615 × 10−6 β2 + 3.9379 × 10−5 β + 0.9998	1
45° (B)	CF = 1.1413 × 10−8 β3 + 7.0911 × 10−6 β2 + 7.0532 × 10−5 β + 0.9997	1
50° (C)	CF = 2.0347 × 10−8 β3 + 6.4319 × 10−6 β2 + 9.3106 × 10−5 β + 0.9996	1
40°,45°,50° (A,B,C)	CF = 1.1802 × 10−8 β3 + 5.9782 × 10−6 β2 + 6.4260 × 10−5 β + 0.9997	1

gives the regression equations for the curves that best fit the data points in each SEZ and all SEZs, too. These regression relationships have the same shape as those of the sites in Figure 10. This occurs because few sites have ρ_r < 1, and, therefore, the average of the CF across all sites in the same SEZ, or all SEZs, gives a relationship resembling a site with ρ_r > 1.

From the above it is confirmed that the curves of CF vs. β (Figure 10) and CF vs. ρ_r (Figure 12) are universal and may be represented graphically as nomograms to allow any one of those variables to be calculated from the other two; this was demonstrated for the first time worldwide by [1].

4. Conclusions and Discussion

The present study investigated solar availability across Saudi Arabia on flat-plate solar panels that track the Sun along its daily path in the sky. The main objective was to find the optimum tilt angles of solar panels that produce maximum annual energy in this configuration under all-sky conditions. This was achieved by calculating the annual energy sum on flat-plate surfaces with tilt angles tracking the Sun in the range 5°–55° with an increment of 5° at 82 sites across Saudi Arabia; the solar availability on a horizontal plane was also included for reference purposes. The calculations of the energy received on the tilted surfaces were performed for a ground albedo equal to 0.2 (as a reference value) and also for a near-real ground albedo.

The first outcome of the work was that the optimum tilt angles for flat-plate solar collectors mounted on a vertical axis that track the Sun over Saudi Arabia are 40°, 45°, and 50°. The second finding of the study was that these three optimum tilt angles can group the 82 sites into the three (solar energy) zones (SEZ) defined in [1], i.e., SEZ-A with 40°, SEZ-B with 45°, and SEZ-C with 50°. The third conclusion was related to the variation of the annual maximum solar energy in each SEZ, i.e., 2767–3314 kWhm⁻²year⁻¹ (average 3044.9 kWhm⁻²year⁻¹ in SEZ-A), 2159–4078 kWhm⁻²year⁻¹ (average 2975.6 kWhm⁻²year⁻¹ in SEZ-B), 2324–3692 kWhm⁻²year⁻¹ (average 2932.5 kWhm⁻²year⁻¹ in SEZ-C), and 2159–4078 kWhm⁻²year⁻¹ (average 2991.9 kWhm⁻²year⁻¹ in all SEZs). In relation to fixed-tilt solar systems reported in [1], the one-vertical axis solar systems in Saudi Arabia show an average increase in the maximum annual solar energy yield of +21.8% for SEZ-A, +22.6% for SEZ-B, +31.2% for SEZ-C, and +23.5% for all SEZs. Beside the annual energy sums, monthly solar energy values averaged over all locations belonging to the same SEZ as well as to all SEZs were estimated under all-sky conditions. Regression equations were provided as best-fit curves to the monthly mean energy sums that estimate the solar energy potential per SEZ (and all SEZs) with great accuracy (R² ≥ 0.82). These expressions may prove very useful to architects, civil engineers, solar energy engineers, and solar-energy-system investors in order to assess the solar energy availability in Saudi Arabia for Sun-tracking flat-plate solar collectors throughout the year.

Seasonal solar energy sums were also calculated. They were averaged over all sites in the same SEZ as well as over all sites (all SEZs), under all-sky conditions. For every case, regression curves that best fit the mean values were estimated with absolute accuracy (R² = 1). Maximum sums were found in the summer (836.6 kWm⁻²), and minimum ones in the winter (664.1 kWm⁻²), as expected. The corresponding figures for the maximum energy received on a flat-plate solar collector in Saudi Arabia having an optimum tilt angle towards south are 659.9 kWm⁻² and 532.3 kWm⁻², respectively for summer and winter [1]. Therefore, the single-axis solar systems provide a high increase in the maximum seasonal energy yield in comparison to the static ones of +26.8% and +24.8%, respectively.

The correction factor, CF, introduced in [1], was used in this work, too. A graph of CF as a function of the tilt angle (in the range 5°–55°) showed exponential growth for sites having ratios ρ_r = ρ_g/ρ₀ > 1, or exponential decay in the cases of ρ_r < 1. Such curves are assumed to be universal (i.e., representable as nomograms), since similar graphs in [1] had the same shape. Nevertheless, this universality remains to be confirmed at other locations in the world with different climate and terrain characteristics. A graph of CF as a function of ρ_r was prepared for different values of the tilt angle in the range 5°–55°. Best-fit curves to the data points were estimated and were found to be linear, with decreasing slopes in proportion to decreasing tilt angles. These curves are also assumed to be nomograms, since similar graphs in [1] had the same shape. Nevertheless, this kind of nomogram has to be confirmed at other sites worldwide with different climate and terrain characteristics.

Three innovations appeared in the present study: (i) For the first time, solar maps for Saudi Arabia of the maximum energy received on optimally-inclined flat surfaces mounted on a system with a vertical axis that rotates and tracks the Sun were derived; (ii) for the first time, the three energy zones identified in [1] were used here and the same sites as in [1] were accommodated in the same zones; (iii) universal curves (nomograms) of CF in relation to β and ρ_r were derived for this case, too.

As far as the utilisation of the results presented here is concerned, this can be summarisd as follows. The solar industry has now a new rule for the inclined supporting frames mounted on single-axis solar systems; the inclined frame must have an angle of 40°, 45°, or 50° to the local horizon, depending on the SEZ to be installed. The same utilisation guidelines given by [1] for the static solar systems apply to this case, too.

This last paragraph of the work is devoted to the performance efficiency and operation cost of a static (fixed-tilt) and a single-axis solar PV system. Talavera et al. [26] examined the levelised cost of electricity (LCOE, in €kWh⁻¹) for the two systems at five sites (two in Spain, one in Saudi Arabia, one in Brazil, and one in the USA) and found it varying according to site and type of the PV system. Overall, the LCOE for the single-axis PV system is a little bit higher than that for a fixed-tilt one. Nevertheless, the difference is not big and varies as +4.4%, 0%, 0%, +8.6%, and +1% for the five sites, respectively. Another study [27] concluded that the amount of electricity generation by a single-axis PV system in the USA is about 12–25% higher than that for a fixed-tilt one. A study for Jordan [28] showed that a dual-axis PV system generates 31.29% more annual electricity than a fixed-tilt one. For Europe, it has been found [29] that a single-axis PV system produces an annual energy yield of 30% higher in southern, and 25% in central Europe and up to 50% in northern Scandinavia.