Solar Energy Received on Flat-Plate Collectors Fixed on 2-Axis Trackers: Effect of Ground Albedo and Clouds

Abstract

This study investigates the performance of isotropic and anisotropic diffuse models to estimate the total solar energy received on flat-plate collectors fixed on dual-axis trackers. These estimations are applied at twelve sites selected in both hemispheres with different terrain and environmental conditions. The diffuse (or transposition) models used in this study are the isotropic Liu-Jordan (L&J), Koronakis (KOR), Badescu (BAD), and Tian (TIA), and the anisotropic Hay (HAY), Reindl (REI), Klucher (KLU), Skartveit and Olseth (S&O), and Steven and Unsworth (S&U). These models were chosen because of their simplicity in the calculations and minimum number of input values. The results show that a single transposition model is not efficient for all sites; therefore, the most appropriate models are selected for each site under all, clear, intermediate, and overcast conditions in skies. On the other hand, an increase in the ground albedo in the vicinity of the solar installation can increase the annual inclined solar availability on a two-axis tracker by at least 9% on average. Further, a linear dependence of the annual inclined solar energy on the variation of the ground albedo was found. Also, a linear relationship exists between the annual diffuse-fraction and cloud-modification factor values at the 12 sites.

1. Introduction

Systems equipped with tilted solar panels for exploiting the (renewable) energy of the sun have long been available in the market as commercial products. These systems consist of solar modules receiving the energy from the sun on flat-plate solar collectors that (i) operate at a fixed-tilt angle with southward orientation in the Northern Hemisphere or northward orientation in the Southern Hemisphere, (ii) continuously track the sun at a fixed-tilt angle rotating around a vertical axis, and (iii) continuously track the sun at a varying tilt angle fixed on a system with two axes, one vertical and one horizontal. Installations of mode I are known as fixed-tilt systems and are widely used because of the lower cost for the supporting frame [1]. Installations of mode II, also known as one- or single-axis systems, provide higher solar energy on the inclined surface, but have a slightly higher cost because of the need to maintain the moving parts [2]. Installations of mode III are considered the most effective and are known as two-, double-, or dual-axis systems [3]. Mode-I systems are also called static, while those of modes II and III are called dynamic.

Fixed-tilt solar systems are nowadays widely used as commercial products because of their simple design and low maintenance cost; because of their high commercial interest, they have received major attention from researchers for estimating the solar potential (or solar availability) at a specific site or region, e.g., [4,5,6]. Recently, attention has also been paid to dynamic mode-II solar systems because of relatively higher performance in terms of the solar energy received, e.g., [7]. The dynamic mode-III solar systems have been in use in the last 20 years or so because of higher performance than that of the other two types, e.g., [8,9]. In this context, several studies worldwide have been conducted and have shown that the solar energy captured by flat-plate collectors fixed on dual-axis systems is maximised in comparison to that received by fixed-angle or one-axis systems [3,10,11,12]. Speaking about these systems, much effort has been invested in improving their moving and electronic parts for the sun-tracking sensors [11,13]. Nevertheless, despite the development of technology, the performance of static or dynamic systems must be evaluated against solar radiation measurements on inclined surfaces. However, the presence of a very limited number of solar radiation stations worldwide has raised the necessity to develop and use suitable solar radiation models, e.g., [14,15,16], to estimate the (maximum) solar energy obtained on flat-plate solar collectors in any of the three operating modes. Nowadays, other methods (i) use a combination of ground-based solar data and modelling, e.g., [17], (ii) utilise solar data downloaded from international databases, e.g., [18,19], (iii) use available on-line facilities, e.g., [20], or (iv) even apply advanced statistical tools or fuzzy and neural networks models, e.g., [21].

Ref. [22] made a review of the existing one- and two-axis solar trackers equipped with solar panels; the researcher concluded that dual-axis trackers should be preferred in places with significant changes in the solar flux throughout the year, while single-axis systems must be used at places without significant changes in the solar flux throughout the year, i.e., around the equator. Ref. [23] gave a techno-economic and environmental assessment of the one- and two-axis systems for photovoltaic (PV) installations on building roofs; they came to the outcome that the single-axis systems are cost-efficient and provide a 23% improvement in the output power in comparison to that for a horizontal system, while the dual-axis systems have a higher improvement (32%), but they are not cost-efficient (probably due to their higher maintenance cost as said above). Another review paper by [13] concluded that double-axis tracking systems are preferred to be used at 16.67% of relevant installations, horizontal tracking at 16.67%, azimuth (single-axis) ones at 10%, and polar tracking at 4.44%. Ref. [24] examined the installation costs of the one- and two-axis tracking systems at 20 sites in the Northern Hemisphere (NH) and in the latitudinal band 20°–70°; they concluded that for a given site in the mentioned zone of latitudes, the PV-installation cost increases for the mode-III and mode-II systems, while this is lower for the mode-I ones. Ref. [25] made a comparative performance analysis study for four types of tracking systems at five sites in both hemispheres along the 25.5° E meridian; they came to the conclusion that the dual-axis systems can have a share up to 98% of the available annual solar energy of the site, the single-axis systems up to 94–95%, and the fixed-tilt systems up to 64–72% depending on the orientation of the inclined receiving surface. Another review paper by [10] suggests the superiority of the dual-axis systems to the single-axis ones. Ref. [26] has argued that solar air heaters (SAH) mounted on two-axis trackers provide 47.8% more energy efficiency, 20% more coefficient performance value, and 2.4 times more exergy efficiency in comparison to SAHs mounted on fixed-tilt systems. The higher performance of the two-axis systems in relation to that of the single-axis and fixed-tilt ones has also been confirmed by [27] for Saudi Arabia.

In view of the lack of a large number of stations measuring the solar radiation components worldwide (specifically solar flux on tilted surfaces), many of the studies mentioned above have resorted to solar radiation modelling. The usual approach is to convert the solar horizontal radiation values (hourly, daily, or monthly) to those on an inclined plane with a given tilt angle and orientation to the local horizon. These models are called transposition models, because they shift the solar horizontal radiation to an inclined one. Therefore, the total solar energy received on a sloped plane is, therefore, the sum of the direct, diffuse, and ground-reflected parts on the inclined surface. The transposition of the direct solar radiation values from its horizontal component to the inclined one is straightforward as it is simple geometry. The difficulty lies in the diffuse part, because the ground-reflected radiation can easily be estimated [28]. Such diffuse transposition models are extensive in the international literature (isotropic, anisotropic, decomposition). Some indicative studies dealing with estimating the total solar energy incident on an inclined surface are mentioned below.

Ref. [12] has applied some isotropic and anisotropic diffuse models to estimate the solar energy gain on fixed-tilt southward-oriented flat-plate collectors in Trinidad, the Caribbean Islands. Isotropic, anisotropic, and decomposition diffuse models were used by [29] to estimate the solar energy on inclined surfaces and compare the results. Similar to [29]’s work is the one by [4] for Brussels Uccle, Belgium. An isotropic and an anisotropic model were also utilised by [27] to estimate the annual solar energy potential in Saudi Arabia. These were also utilised by [30] for Greece. Ref. [31] estimated the global solar energy on tilted surfaces from horizontal radiation measurements at the Institute of Meteorology in Tunis, Tunisia, using their own diffuse radiation model. Ref. [32] has evaluated 26 transposition models and compared them; he also presented their accuracy by classifying them in four categories—cluster 1 (Perez), cluster 2 (Gueymard, Hay, Muneer, Reindl, Skartveit, Steven, Willmott), cluster 3 (Bugler, Liu-Jordan, Klucher), and cluster 4 (Badescu, Koronakis, Olmo, Temps, Tian)—and noting an overall diminishing accuracy from cluster-1 to cluster-4 models. Ref. [33] developed his own empirical diffuse model and evaluated it.

In another direction, Ref. [34] examined the performance of two types of SAHs, one type having 30° tilt to south with single-glass cover and a second type having 30° tilt to south with double-glass cover and found average thermal efficiencies of 18.2% and 26.4%, respectively. Further, Ref. [35] reviewed in detail the various design and operating parameters affecting the performance of SAHs in the three (static and dynamic) modes of operation and found that combining turbulators and fins can boost their efficiency; also, replacing water with nano fluids was shown to be beneficial to the thermal performance of SAHs.

From the above, one can conclude that no attempt has been made thus far for mode-III solar systems with the purpose of:

• evaluating isotropic and anisotropic diffuse models at various sites of the world with differing terrain and environmental features;
• choosing the most appropriate model(s) at the sites under various sky conditions;
• estimating the annual solar energy at the sites through the selected model(s);
• studying the effect of the ground-albedo value on the estimated total solar energy;
• examining the effect of clouds (cloudiness) on solar potential.

These gaps are bridged in the present work.

The structure of the paper is as follows. Section 2 describes the data collection and data processing. Section 3 deploys the results of the study and provides a discussion on the importance of the results obtained. Section 4 presents the conclusions and main achievements of the work. Acknowledgements and references follow.

2. Materials and Methods

Tο implement the goals of the study, 12 sites around the world were selected. The selection was based on the following criteria:

• different environmental characteristics;
• different terrain features;
• distribution across the continents.

Table 1. Selected sites for the scope of the present study.

#	Site’s Name (Abbreviation)	Country	φ (deg)	λ (deg)	z (m amsl)	Terrain Features (Topography)	Terrain Type	Period
1	Athens (ATH)	Greece	37.97 N	23.72 E	107	shrubs, trees (hilly)	II	2000
2	Boulder (BOU)	USA	40.05 N	105.01 W	1577	grass (flat)	I	1998
3	Carpentras (CAR)	France	44.08 N	5.06 E	100	cultivated land (hilly)	I	2018
4	De Aar (DAA)	South Africa	30.67 S	23.99 E	1287	sand (flat)	I	2017
5	Gandhinagar (GAN)	India	23.11 N	72.63 E	65	shrubs (flat)	II	2020
6	Ilorin (ILO)	Nigeria	8.53 N	4.57 E	350	shrubs (flat)	I	2003
7	Kishinev (KIS)	Moldova	47.00 N	28.82 E	205	grass (flat)	II	2020
8	Lerwick (LER)	UK	60.14 N	1.18 W	80	grass (hilly)	I	2003
9	Lindenberg (LIN)	Germany	52.21 N	14.12 E	125	cultivated land (hilly)	I	2018
10	Payerne (PAY)	Switzerland	46.82 N	6.94 E	491	cultivated land (hilly)	I	2013
11	Regina (REG)	Canada	50.21 N	104.71 W	578	cultivated land (flat)	I	2003
12	Solar Village (SOV)	Saudi Arabia	24.91 N	46.41 E	650	desert, sand (flat)	I	2002

φ and λ are the geographical latitudes and longitudes of the sites, respectively; φ is given in the NH (N) or the SH (S), and λ east (E) or west (W) of the Greenwich meridian. Both φ and λ values have been rounded to the 2nd decimal digit. In column 8, I denotes rural, and II denotes urban environment. The period of measurements is given in the last column. The selection of the sites and their solar radiation data were based on the BSRN (Baseline Surface Radiation Network), except for Athens (ATH); in this case, data from the Actinometric Station of the National Observatory of Athens not belonging to BSRN were used. The abbreviations of the sites (except for Athens) are those provided by the BSRN typology. A description of the BSRN operation can be found in [36].

deploys the selected sites in alphabetical order together with their geographical coordinates and environmental descriptions. Figure 1 shows the distribution of the 12 sites around the world.

The following data were collected for all 12 sites: global horizontal irradiance, H_g (in Wm⁻²), and diffuse horizontal irradiance, H_d (in Wm⁻²). From those, the direct horizontal irradiance, H_b (in Wm⁻²), was calculated as H_g − H_d. The H_g and H_d data were downloaded from the BSRN network for the years indicated in Table 1, except for Athens, the data of which was part of the measuring solar radiation components at the Actinometric Station of the National Observatory of Athens (ASNOA), Greece. All data values are hourly averages. The BSRN network provides its data in UTC (Universal Time Coordinated), while the data from ASNOA are in LST (local standard time). Therefore, a transformation of all UTC data into LST ones for the 12 sites, except ATH, took place in the data pre-processing. During the same procedure, the hourly solar altitude, γ (in degrees), and solar azimuth, ψ (in degrees) values over the selected year for each station were calculated by using the XRONOS.bas code [37,38] (xronos means time in Greek with x being spelled as ch). XRONOS.bas has provided improvements to the original SUNAE code [39]. An improvement to the code was recently published to estimate ψ at the sunrise and sunset moments and overcoming the intrinsic discontinuities [40]. For the purpose of this work, the latest version of the XRONOS.bas code was adjusted to calculate the solar geometry parameters at the 12 sites halfway between two consecutive hours (i.e., 30′ after the hour). The following quality criteria were met:

• hourly values of H_g < H_d became H_g = H_d;
• solar radiation values corresponding to γ < 5° were rejected due to the cosine effect on the measuring pyranometers.

As shown in Table 1, single years were selected for analysis at each site instead of a period of years. That was performed on the rationale that no climatological analysis was intended to be conducted within the scope of this study. Instead, the goals posed in the end of the Introduction section could be well served by the analysis of 1 year’s data. For this reason, the year for each site was selected in the period 1999 to 2020 from the BSRN list of stations, with an additional restriction that the individual years cover the mentioned period as broadly as possible. This way, any weather peculiarities occurring over an extended area within a specific year would be avoided. Data for Athens were used from the ASNOA database, as the first author was a researcher at NOA until the end of 2019 and responsible scientist for the maintenance of the station for a long period until 2014. Also, the data of the KIS site, though part of the BSRN network, were provided by courtesy of the Institute of Atmospheric Sciences (AIP), Academy of Sciences, Moldova (Dr. A. Aculinin).

After processing the original data from all sites as mentioned above, some transposition (or diffuse) models were selected under the following criteria:

• be both isotropic and anisotropic;
• be simple in calculations;
• need the least input data;
• be used in the international literature.

These limitations guided us to choose the isotropic models of Liu and Jordan (L&J) [41], Koronakis (KOR) [42], Badescu (BAD) [43], and Tian et al. (TIA) [44], and the anisotropic models of Hay (HAY) [45], Reindl et al. (REI) [46], Klucher (KLU) [47], Skartveit and Olseth (S&O) [48], and Steven and Unsworth (S&U) [49]. Their expressions are given below.

For a surface mounted on a dual-axis solar tracker, the received total solar radiation is:

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

(1)

where H_g,t (in Wm⁻²) is the total solar radiation on the inclined surface (subscript t = tracking), H_b,t (in Wm⁻²) the direct solar radiation on the sloped plane, H_d,t (in Wm⁻²) the diffuse solar radiation on the tilted plane, and H_r,t (in Wm⁻²) the ground-reflected radiation on the sloped plane.

The solar radiation components in Equation (1) are given by the following analytical expressions:

H_r,t = H_g·R_r·ρ,

(2)

R_r = (1 − cosβ)/2 = (1 − sinγ)/2,

(3)

H_d,t = H_d·R_d,model, (model = any isotropic or anisotropic model),

(4)

R_b = max (cosθ/sinγ, 0); this calculation has no specific effect on the model’s calculations as it refers to momentary values of θ and γ and not to hourly averages,

(5)

H_b,t = H_b·cosθ/sinγ = H_b·cos0/sinβ = H_b/cosγ,

(6)

cosθ = sinβ·cosγ·cos(ψ − ψ’) + cosβ·sinγ,

(7)

H_ex = H₀·S·sinγ,

(8)

H₀ = 1361.1 Wm⁻²,

(9)

S = 1 + 0.033·cos(2·π·N/365),

(10)

where ρ is the ground albedo, R_r is the ground-inclined, and R_d is the sky-inclined configuration factors, respectively; β and θ are the tilt angle of the sloped surface in respect to the local horizon and the incidence angle (the angle between the sun and the normal to the inclined surface), respectively; for a dual-axis tracker β = 90° − γ, θ = 0° and ψ = ψ’ (because of the sun-tracking feature of the mode-III system), where ψ is the solar azimuth of the sun and ψ’ the azimuth of the inclined surface. S is the sun–earth distance correction factor, or else the earth’s orbit eccentricity. N is the day number of the year (N = 1 for 1 January, and N = 365 or 366 for 31 December in a non-leap or a leap year, respectively). Equations (2)–(8) are after [28] and (9)–(10) after [50] and [51], respectively. Near-real values for the ground albedo at each site were downloaded from the Giovanni portal [52] for pixels centred over each site (0.5° × 0.65° spatial resolution) in the year indicated in Table 1 for the individual site; monthly mean values of ρ were then estimated and used in Equation (2).

As far as the analytical expressions for the selected isotropic and anisotropic models are concerned, the differentiations are only related to the expression of the sky-inclined configuration factor, which is given by the following equations depending on the model:

R_d,L&J = (1 − cosβ)/2 = (1 − sinγ)/2,

(11)

R_d,KOR = (2 + cosβ)/3,

(12)

R_d,BAD = (3 + cos2β)/4,

(13)

R_d,TIA = 1 − β,

(14)

R_d,HAY = K_b·R_b + (1 − K_b)·R_d,L-J,

(15)

R_d,REI = (1 − K_b)·R_b + (1 + sin³(β/2)·√k_b)·R_d,L-J,

(16a)

K_b = H_b/H_ex,

(16b)

k_b = H_b/H_g,

(16c)

R_d,KLU = [(1 + cosβ)/2]·[1 + K·sin³(β/2)]·[1 + K·cos²θ·sin³(90 − γ)],

(17a)

K = 1 − k²_d and k_d = H_d/H_g (k_d = diffuse fraction),

(17b)

R_d,S&O = [(1 + cosβ)/2]·(1 − K_b − Ω) + Ω·cosβ + K_b·R_b,

(18a)

Ω = max (0, 0.3 − 2·K_b),

(18b)

R_d,S&U = 0.51·R_b + [(1 + cosβ)/2] − (1.74/1.26·π)·[sinβ − (π·β·cosβ/180) – π·sin²(β/2)].

(19)

Equations (11)–(15) are after [41,42,43,44,45], respectively, (16a,b) after [46], (17a,b) after [47], (18a,b) after [48], and Equation (19) after [49], and Equation (16c) after [53].

To classify the skies into clear, intermediate, and overcast, the methodology developed by [54] was applied here. This method makes use of the diffuse fraction and characterises the sky as clear when 0 < k_d ≤ 0.26, intermediate when 0.26 < k_d ≤ 0.78, and overcast for 0.78 < k_d ≤ 1. Those authors have demonstrated a worldwide applicability of their method, excluding the polar regions. Therefore, calculations of k_d were performed in the frame of the present work on an hourly basis from corresponding values of H_d and H_g.

The cloud-modification factor, CMF, is defined as the ratio of the actual global horizontal irradiance to the expected one under clear-sky conditions [55], i.e., CMF = H_g,AS/H_g,CS, where the subscripts AS and CS indicate all and clear skies, respectively. CMF determines the influence of clouds on solar radiation, and theoretically ranges between 0 (completely dark sky) and 1 (completely bright sky). Hourly values of H_g,CS (in Wm⁻²) were calculated from H_g,AS by applying the restriction of 0 < k_d ≤ 0.26 to the latter solar parameter. Then, the CMF = H_g,AS/H_g,CS was calculated for all H_g values meeting the clear-sky k_d limitation. That was performed for all sites by considering near-real albedo values in all calculations.

To investigate the effect of the ground-albedo value on the total inclined irradiation, Equation (2) was elaborated for the following albedo values: 0, 0.2, 0.5, 0.7, 1, and ρ_r (subscript r implies near-real albedo). These values were entered into the calculations with all 9 models and 12 sites; 54 (6 albedo values × 9 transposition models) databases of total inclined irradiation values were thus formed for each site.

Prior to any analysis of the databases, the selection of the transposition model(s) to be used was an imperative need. This means there should be a way to choose one of the 9 (isotropic and anisotropic) models at each site, which would be exclusively used in subsequent analyses. To perform this selection, the annual sums of the global inclined irradiations along their standard deviations were estimated through all 9 models at all 12 sites. For each site and each ground-albedo value (0, 0.2, 0.5, 0.7, 1, ρ_r), averages of the total inclined irradiation sums were estimated together with their standard deviations. Then, the model for which the annual total irradiation sum and standard deviation were closer to the averages for the 12 sites was chosen as the most appropriate transposition model for the specific site and ground albedo. This way, 48 (12 sites × 4 cloud conditions) different models were finally selected and are shown in

Table 2. Selected transposition models for all 12 sites and various sky conditions.

Site	Transposition Model
	L&J	KOR	BAD	TIA	HAY	REI	KLU	S&O	S&U
ATH					AS, CS, IS			OS
BOU					AS, CS	OS	IS
CAR					AS, CS, IS		OS
DAA		CS			AS		IS	OS
GAN					AS	CS		OS	IS
ILO	OS					CS		AS, IS
KIS					AS	OS	IS		CS
LER					AS, IS	CS	OS
LIN		OS			AS, CS, IS
PAY					AS, CS		IS	OS
REG					CS, IS		OS	AS
SOV					IS		AS	CS, OS

AS = all skies, CS = clear skies, IS = intermediate skies, OS = overcast skies. The distinction in sky conditions was made through the diffuse-fraction parameter, k_d.

. From this preliminary analysis, one can see that none of the selected models simultaneously serves all types of sky conditions (AS, CS, IS, OS). In the following, results will be presented for the AS and CS sky conditions only.

It should be mentioned here that all calculations in this work were performed by using MS Excel (v. 2406) spreadsheets.

3. Results

3.1. Estimations on Annual Basis

Figure 2 shows the effect of the ground albedo (ground reflectivity) on the total inclined irradiation received on a flat-plate collector mounted on a dual-axis tracker, worldwide under all-sky (AS) conditions. The inclined radiation was calculated by the selected transposition model for the site, as explained in the Methodology section. The ground albedo in the graph varies from 0 (total ground absorptivity, black-body-like) to 1 (total ground reflectivity, mirror-like). The average of the near-real ground-albedo values for all 12 sites was estimated at ≈0.19; the corresponding H_g,t,AS,0.19 values are also shown as data points next to those for H_g,t,AS,0.2. The best-fit curve to the data points for each site is linear, as expected from Equations (1) and (2), with the coefficient of determination, R², equal to 1. The differences calculated from the linear best-fit curves for the sites between ρ ≈ 0.19 (average ρ for all 12 sites) and ρ = 1 are 229.06 kWhm⁻² ATH, 215.26 kWhm⁻² BOU, 220.28 kWhm⁻² CAR, 266.17 kWhm⁻² DAA, 145.00 kWhm⁻² GAN, 152.86 kWhm⁻² ILO, 207.77 kWhm⁻² KIS, 153.92 kWhm⁻² LER, 208.03 kWhm⁻² LIN, 213.78 kWhm⁻² PAY, 239.04 kWhm⁻² REG, and 234.69 kWhm⁻² SOV. These differences divided by the annual solar energy of the site (2551.39 kWhm⁻² ATH, 2444.04 kWhm⁻² BOU, 2347.32 kWhm⁻² CAR, 3339.81 kWhm⁻² DAA, 3004.90 kWhm⁻² GAN, 1893.15 kWhm⁻² ILO, 2072.77 kWhm⁻² KIS, 1127.08 kWhm⁻² LER, 1908.44 kWhm⁻² LIN, 2113.71 kWhm⁻² PAY, 2565.96 kWhm⁻² REG, and 3530.17 kWhm⁻² SOV) correspond to approximately 9% ATH, 8.8% BOU, 9.4% CAR, 8.0% DAA, 4.8% GAN, 8.1% ILO, 10.0% KIS, 13.7% LER, 10.9% LIN, 10.1% PAY, 9.3% REG, and 6.6% SOV (average of 9.1% for all sites). In other words, if the albedo of the sites (artificially) increases towards 1, the sites should gain at least 9% in the energy received throughout the year. For this purpose, further investigation into transforming the vicinity of the installed solar system (e.g., a PV) to be as fully reflective as possible is needed in material technology. For a dual-axis tracker, the area to be covered by such a material has to be geometrically estimated for the best performance of the solar system throughout the year.

Turning now to the effect of clouds on the total solar irradiation for a mode-III system, Figure 3 shows the annual H_g,t,AS,ρg values as functions of the diffuse fraction, k_d,AS. Near-real ground-albedo, ρ_g, values were used in the estimations of H_g,t,AS,ρg. The dashed straight line is the best-fit linear curve to the data points with expression H_g,t,AS,ρg = −3494.45·k_d,AS + 4221.76 with R² = 0.75. This line was drawn to show how such a linear curve can adequately express the dependence of H_g,t,AS,ρg on k_d,AS. It is generally seen that the solar energy decreases with k_d, as expected; the greater the k_d,AS is (i.e., more cloudy weather), the lower the solar energy received under all-sky conditions.

Staying at the cloud effect on the received solar energies, the use of the cloud-modification factor, CMF, is introduced according to [55]. Figure 4 shows the variation in the total inclined solar irradiation, H_g,t,AS,ρg, for a dual-axis as a function of CMF. Near-real ground-albedo, ρ_g, values were used in the estimations of H_g,t,AS,ρg. The dashed straight line is the best-fit linear curve to the data points with expression H_g,t,AS,ρg = −3363.24·CMF − 357.06 with R² = 0.61. This line was drawn to show how adequately such a linear curve can express the interrelation of H_g,t,AS,ρg and CMF. Here, the dependence of H_g,t,AS,ρg on CMF is just opposite to its relation with k_d,AS. This should be expected because k_d = H_d/H_g and CMF = H_g/H_g,CS. As k_d increases, the sky is converted from clear to overcast, thus reducing H_g,t,AS,ρg. On the contrary, as CMF increases, the sky becomes more and more bright, thus increasing H_g,t,AS,ρg.

Figure 5 shows the dependence of the two sky-conditions factors, namely k_d,AS and CMF. The dashed straight line is the best-fit linear curve to the data points with expression k_d,AS = −2.68 · CMF − 1.08 and R² = 0.84. This line was drawn to show how adequately such a linear curve can express the dependence of k_d,AS on CMF. The trend of the data points is negative, i.e., CMF increases as k_d,AS is reduced. Such behaviour should be expected by the definitions of the two parameters. The greater the CMF value is, the brighter the sky becomes; a brighter sky implies an approach to a clear-sky situation, which compels k_d,AS to decrease.

Farahat et al. [1,2] and Kambezidis et al. [3] have defined the correction factor, CF, for the solar energy received on an inclined plane. Actually, CF is the ratio of the solar energy received on the inclined plane by taking into account a near-real ground albedo, ρ_g, to that under a constant albedo value, ρ_go, usually taken as 0.2. In other words, CF = H_g,t,AS,ρg/H_g,t,AS,0.2. Those researchers have shown that this factor is linearly dependent upon the ground-albedo ratio, ρ_r, defined as ρ_r = ρ_g/ρ_go = ρ_g/0.2. Further, these scientists showed that this linear relationship has a wide applicability throughout Saudi Arabia and claimed that it may be treated as a nomogram, i.e., it has a universal validity. To check whether this hypothesis is true worldwide, Figure 6 presents the CFs for the 12 sites as functions of their ρ_r = ρ_g/ρ_g0 = ρ_g/0.2; here, ρ_g takes the consecutive values of 0, 0.2, 0.5, 0.7, 1, as well as the near-real ρ_g value of each site, which, for simplicity, has been averaged over all near-real ρ_g of the sites, giving a value equal to ≈0.19 that corresponds to ρ_r ≈ 0.95. Therefore, the ρ_r ratios for all sites are 0, 0.95, 1, 2.5, 3.5, and 5. Indeed, the linear relationship between CF and ρ_r is found to apply at all 12 sites, thus demonstrating its universal applicability and verifying the nomogram hypothesis of [1,2,3]. All fitted lines have R² = 1. The physical interpretation of the correction factor is that one can calculate the solar energy on a plane inclined at β degrees to the local horizon and in an area with ground albedo ρ_g if the corresponding solar energy for ρ_go = 0.2 is known, and vice versa. It should be noted here that all data points in the graph coincide at CF = ρ_g = 1; this occurs since ρ_r = ρ_go/ρ_go = 1 and CF = H_g,t,AS,ρgo/H_g,t,AS,ρgo = 1.

The present work extends the universal nomogram of CF = f(ρ_r) by hypothesising that all the data points corresponding at the same ρ_r (except for ρ_r = 1) in Figure 6 are arrayed according to the absolute geographical latitude of the site (the term “absolute” ignores the minus sign for locations in the SH). In other words, an increase in the slope of each fitted line to the data points at the corresponding site comes from an increase in latitude. To demonstrate this assumption, let us consider the CF_ρr at ρ_r = 0, 0.95, 1, 2.5, 3.5, and 5 from Figure 6; the results are presented in Figure 7. The dashed lines are the best-fit quadratic curves to the data points, except for the curve for CF₁, which is fitted by a horizontal (dark green) line representing the threshold that separates the inverse dependence of CF_ρr on |φ|. It may be noted here that some data points at the same CF_ρr coincide (perhaps not discernible) since they correspond to sites of adjacent |φ| (namely KIS and PAY at approximately |φ| = 47°). The dark green line of CF₁ = 1 separates the graph into an upper section with increasing CF_ρr as |φ| increases and a lower one with CF_ρr decreasing as |φ| increases. The physical explanation for this effect is the following. In the lower portion, ρ_r < 1 because ρ_g < ρ_go = 0.2; therefore, H_g,t,AS,ρg < H_g,t,AS,ρgo. At higher latitudes, ρ_g increases and thus CF_ρr slightly decreases. The reverse explanation exists for deploying data points in the upper side of the graph. It is well observed, though, that there is a saturation band in all curves for |φ| < 30°. This result indicates sites located within the tropical zone (the zone between the Tropic of Cancer in the NH, φ ≈ 23.5° N, and the Tropic of Capricorn in the SH, φ ≈ 23.5° S); because of that, it is better to conclude that the saturation (flat variation of the curves in Figure 7) occurs for latitudes −23.5° ≤ φ ≤ 23.5°. The expressions for the best-fitted curves in Figure 7 are given in

Table 3. Regression expressions for the best-fitted curves to the annual data in Figure 7 and Figure 8.

Sky Conditions	Ground-Albedo Ratio, ρr	Regression Equation	R2
AS	0.00	CF0,AS = −8.3453 × 10−6·|φ|2 + 0.0004·|φ| + 0.9767	0.91
CS		CF0,CS = 6.8192 × 10−6·|φ|2 − 0.0007·|φ| + 0.7450	0.95
AS	0.95	CF0.95,AS = −4.1014 × 10−7·|φ|2 + 1.7972 × 10−5·|φ| + 0.9988	0.92
CS		CF0.95,CS = 1.2381 × 10−6·|φ|2 − 9.5143 × 10−5·|φ| + 0.9879	0.20
AS	1.00	CF1,AS = 1	1.00
CS		CF1,CS = 1	1.00
AS	2.50	CF2.5,AS = 1.2304 × 10−5·|φ|2 − 0.0005·|φ| + 1.0346	0.92
CS		CF2.5,CS = −1.4271 × 10−5·|φ|2 + 0.0013·|φ| + 1.3778	0.36
AS	3.50	CF3.5,AS = 2.0507 × 10−5·|φ|2 − 0.0009·|φ| + 1.0576	0.92
CS		CF3.5,CS = −2.4277 × 10−5·|φ|2 + 0.0022·|φ| + 1.6293	0.38
AS	5.00	CF5,AS = 3.2157 × 10−5·|φ|2 − 0.0014·|φ| + 1.0920	0.92
CS		CF5,CS = −3.9285 × 10−5·|φ|2 + 0.0036·|φ| + 2.0066	0.37

AS = all skies, CS = clear skies. The distinction in all- and clear-sky conditions was made by utilising the diffuse-fraction parameter, k_d. CF = correction factor; ρ_r = ρ_g/0.2; |φ| = absolute geographical latitude (in degrees).

.

An extension to Figure 7 is shown in Figure 8 for clear-sky conditions. The dashed lines are the best-fit quadratic curves to the data points, except for the case of CF₁, which is fitted by a horizontal (dark green) line representing the threshold that separates the inverse dependence of CF_ρr on |φ|. It may be noted here that some data points at the same CF_ρr coincide (perhaps not discernible) since they correspond to sites of adjacent |φ| (namely KIS and PAY at approximately |φ| = 47°). Here, the fitted curves to the data points have an opposite pattern to that in Figure 7. This is obvious since an increase in |φ| is associated with fewer clear-sky (CS) days and, therefore, lower CS solar radiation levels. The expressions for the best-fitted curves in Figure 8 are given in Table 3. Now, the coefficients of determination are much less than 0.90; this occurs for two reasons: (i) there is a greater scatter of the CS days throughout the year than the AS ones; the latter actually occur 365 days of the year, and (ii) the criterion of selecting CS days from the sites’ databases (0 < k_d ≤ 0.26) is not absolute, but some AS or IS days may be categorised as CS, too. Another important remark in this plot is that for ρ_r = 1, the corresponding CFs at any site should be and are equal to 1, because CF₁ = H_g,t,AS,ρgo/H_g,t,AS,ρgo = 1 according to the definition of CF earlier on.

3.2. Estimations on a Monthly Basis

One might wonder what the energy gain is in the case of using dual-axis trackers in the mentioned sites. For this reason, the solar energy gain, SEG, is defined as the ratio of the energy received on a flat surface fixed on the tracker to that on a horizontal plane, i.e., SEG = H_g,t/H_g.

At the beginning of Section 3.1, the extra energy to be gained by (artificially) increasing the albedo in the vicinity of the solar installation was discussed. Nevertheless, this technique needs more elaboration, and it is theoretical at the moment. However, the SEG defined above may be more realistic as it represents the benefit of harvesting solar energy by a solar collector fixed on a two-axis tracker rather than placing it horizontally. Figure 9 gives the monthly SEGs at the 12 sites; these gains were estimated as monthly ratios of the solar energy sums on the tracker to those on horizontal plane under AS conditions, i.e., SEG = ΣH_g,t,AS,ρg/ΣH_g,AS,ρg, where Σ = summation; moreover, the energies were calculated by the specific transposition model selected for each site (see Table 2) and by using near-real ground-albedo values. The red solid curve in Figure 9 represents the mean of all SEG values, while the vertical red bars denote ±1σ (standard deviations) around the mean curve; the blue dashed curve is the best-fit one to the mean (red) line expressed by SEG = 0.0223·t² − 0.2813·t + 2.1959 with R² = 0.95. The graph in Figure 9 is apocalyptic; the mean SEG ranges between 1.35·ΣH_g,AS,ρg to almost 2·ΣH_g,AS,ρg for all sites. Nevertheless, individual discrepancies from the mean curve exist due to the geographical location of the sites and the surrounding environments (terrain features). As expected, greater SEGs are achieved at higher latitudes because of “low suns”. On the contrary, low SEGs for the ILO site are apparent due to the “high-sun” effect throughout the year as ILO is close to the equator; because of that (ILO very close to the equator with φ = 8.53° N), solar radiation exerts two dips at the equinoxes (April, September); on the other hand, the months of October–December include some missing radiation values that result in lower-than-expected radiation levels and consequently SEGs < 1. According to the above rationale, higher gains are achieved at the higher latitude of REG.

3.3. Contour Plots

In Section 3.1, some of the results demonstrated the role of the ground albedo, ρ_g, of the sites in the annual solar energy, H_g,t,AS,ρg received on dual-axis trackers. To delve more into this effect, Figure 10 was prepared to show the variation in the annual H_g,t,AS,ρg sums as a function of the month and ρ_g for all sites. The x-axis in the plots denotes time (months), and the y-axis accommodates ρ_g (the near-real albedo). The contours represent values of H_g,t,AS,ρg. It should be noted here that the ρ_g value at any site does not remain constant throughout the year, but it varies slightly because of variations in the features of the surrounding terrain (changes in vegetation, periods with rain or snow). This is the reason for using monthly mean ρ_g values in all calculations for the sites, instead of annual averages. Moreover, it is interesting to observe that the time-albedo patterns of H_g,t,AS,ρg are not identical; there is a slight similarity, though, in the patterns for ATH (Figure 10a), CAR (Figure 10c), KIS (Figure 10g), LER (Figure 10h), LIN (Figure 10i), REG (Figure 10k), and SOV (Figure 10l). The common characteristic at these sites is a broad H_g,t,AS,ρg maximum during summer mainly, irrespective of any albedo changes. This outcome may be exploited by artificially increasing the albedo value in the winter months for extra gain in the received inclined solar energy. The rest of the sites have other patterns, which may be interpreted as an increase in the received inclined solar energy in the same way as for the other mentioned sites.

In the same way as with the time–albedo–solar energy contour plots, Figure 11 presents the time—cloud-modification factor—solar energy ones at the 12 sites. These plots show how the H_g,t,CS,ρg levels vary within the selected year in accordance with the state of cloudiness at the site; this cloudiness state is expressed by CMF = H_g/H_g,CS and denotes the fraction of a clear-sky solar radiation value, H_g,CS, necessary to make up the actual solar irradiance value, H_g. In other words, CMF is a measure of cloudiness; the lower the ratio, the higher the cloudiness; therefore, 0 < CMF < 1 with 0 for a completely dark sky and with 1 for a completely bright sky. Except for the DAA site that is located in SH, the patterns of the rest of the sites are very similar; they present higher H_g,t,CS,ρg levels at higher CMF values during summer or even extended to spring and autumn, depending on their geographical latitude. The pattern of DAA is reversed with maximum solar energy during wintertime. Nevertheless, maximum H_g,t,CS,ρg is achieved in the heart of the local summer.

Figure 12 provides the intro-annual variation of CMF for all 12 sites. The red solid curve represents the mean of all CMF values, while the vertical red bars denote ±1σ (standard deviation) around the mean curve; the blue dashed curve is the best-fit one to the mean line expressed by CMF = −0.0057·t² + 0.0775·t + 0.4220 with R² = 0.84. The blue curve can, therefore, inform any potential solar energy investor at the selected sites to consider this factor when dimensionalising solar installations, e.g., a PV.

3.4. Discussion

The solar energy received by flat-plate collectors at 12 sites fixed on dual-axis trackers in both hemispheres was investigated by adopting nine transposition models (four isotropic and five anisotropic) with the goal of estimating the inclined diffuse energies by using near-real ground albedos for the selected sites. The models were chosen for potential users based on their simplicity and easy access to the input values; these input parameters can easily be calculated by Equations (1)–(19). Upon selecting the most appropriate transposition model for each site, the effect of clouds and ground albedo on the inclined solar energies was examined. These issues were worthy of investigation, because no such study exists in the international literature so far to the authors’ knowledge.

The analysis of the databases of the 12 sites showed for the first time worldwide that a selected transposition model for a site under all-sky conditions does not necessarily work as efficiently under clear-, intermediate-, or overcast-sky conditions as in all-sky ones. Nevertheless, the HAY model seemed to be more appropriate for 9 out of the 12 sites (75%) under all-sky conditions. This means that solar energy scientists, investors, or entrepreneurs should take this fact into account in the case that a solar system (e.g., a PV) is to be installed at one of the 12 sites or generally at any site.

As stated in Section 1, there are three main configurations for installing a flat-plate solar collector: the fixed-tilt mode-I (static), the fixed-tilt one-axis mode-II (dynamic), and the variable tilt dual-axis mode-III (dynamic). Among the three, the dual-axis solar systems are more expensive than single-axis or fixed-tilt ones, because of their higher cost of maintaining their moving parts. This, of course, cannot be a barrier to a solar investor as he/she has to examine the cost-efficiency issue of the installation on an annual basis by taking into account the climatology of the area of interest. Nevertheless, there has been a general attitude nowadays towards using dual-axis solar systems because such systems may provide extra solar energy as high as 41.34% compared to that by a fixed-tilt one [9]. Ref. [24] concluded that a dual-axis solar tracker is more economically preferable when the size of the installation becomes bigger. Ref. [3], in a study on the efficiency of all three modes of operation in Saudi Arabia, concluded that a mode-III system delivers 23.6% more solar energy on an annual basis than a mode-I one and 4.2% more in comparison to a mode-II system. Also, Ref. [34] examined the performance of two types (A, B) of SAHs, both having 30° tilt to south with type A including a single-glass cover and type B a double-glass cover; they found average thermal efficiencies of 18.2% and 26.4%, respectively. Further, Ref. [35] presented in detail the various design and operating parameters affecting the performance of SAHs in the three modes of operation and found that combining turbulators and fins can boost their efficiency; they also demonstrated that replacing water with nano fluids can be beneficial to the thermal performance of SAHs.

Despite the above results, the present study could not provide a performance comparison among the three modes of operation, because it only considered dual-axis trackers. Nevertheless, the study revealed for the first time worldwide the effect of the ground albedo on the solar energy received by flat-plate collectors fixed on dual-axis tracking systems at the sites of interest.

Another goal of the present research was to investigate the influence of the ground albedo on the solar energy received by mode-III systems. It was, therefore, demonstrated that by increasing (artificially) the ground albedo towards 1, a solar energy gain of at least 9% is achieved in comparison to that delivered by the system without ground-albedo intervention. This is a very encouraging outcome, as it may trigger the relevant industries to design and produce such materials for ground use.

Another important outcome from the present work was the confirmation of the universal applicability of the (correction factor–albedo ratio) nomogram introduced by [1]. Indeed, the present research demonstrated the validity of the nomogram at all 12 sites examined. The value of this relationship is that anyone can easily estimate the inclined solar energy received at any of the 12 sites under any ground-albedo value in the range 0–1.

A third aim of the present study was to investigate the influence of clouds on the solar potential at the 12 sites by using the cloud-modification factor. That was performed by plotting the total annual solar energies at the sites versus the cloud-modification factor (Figure 4) and by fitting the data points by a linear curve. This curve can serve as an estimator of the solar energy received at any of the 12 sites under AS conditions; it should be reminded here that the cloud-modification factor varies between 0 and 1, i.e., between a completely dark sky and a fully bright one. Ref. [56] used this factor to show the influence of clouds on fixed-tilt solar systems at 25 cities around the world; they constituted a relationship between the geographical latitude of the sites, φ, the optimum tilt angle, β_opt, at each of them, and the corresponding cloud-modification factor, thus deriving a quadratic regression expression of the form φ − β_opt = f(cloud-modification factor).

Furthermore, the present work constructed contour plots of the monthly inclined solar energy sums versus the ground albedo or the cloud-modification factor at the 12 sites under all-sky conditions and showed the effect of either factor on the received solar energy during the selected year for the site.

An additional remark should refer to future research on this subject. This implies the application of the derived methodology to many other sites in the world in order to testify the conclusions drawn in the present study. A key point in such a work is the selection of the most appropriate transposition model for each site based upon real solar radiation measurements or statistics in their absence; then, a categorisation of the selected models according to the environmental features of the site’s area may provide a valuable tool for interested solar entrepreneurs. In this line, Table 2 of the present study showed that the HAY model has a wide applicability as 9 of the 12 sites used it for inclined solar energy estimations under all-sky conditions. Finally, a further investigation is required to show whether there is a relationship between the flat portions of the fitted curves to the CMF-|φ| data points in Figure 7 with −23.5° < φ < +23.5° and the same solar declination range (−23.5° < δ < +23.5°).

4. Conclusions

The present study set the following aims:

• to evaluate the performance of isotropic and anisotropic diffuse models at 12 sites of the world with differing terrain and environmental features;
• to choose the most appropriate model(s) at these sites under various sky conditions;
• to estimate the total solar energy at the sites through the selected model(s);
• to study the effect of ground albedo of the sites on the estimated total solar energy;
• to investigate the effect of clouds (cloudiness) at the sites on the potential solar availability.

All items were successfully dealt with. For the purpose of the study, BSRN solar radiation data from 11 sites in the world were used in single years, alongside data from ASNOA (Athens). Nine transposition models were adopted for the estimation of the diffuse irradiance on solar collectors fixed on two-axis trackers at the 12 sites. The most appropriate transposition model for each site was selected upon a criterion.

The main conclusions of the study are, therefore, the following.

• A single transposition model is not efficient for all sites in the world as such models are site-specific more or less.
• On the other hand, a selected model for all-sky conditions may not be representative for clear-, intermediate-, and overcast-sky conditions (see Table 2).
• In all sites, an (artificial) increase in the ground reflectivity in the vicinity of the solar installation may increase the inclined solar availability by at least 9% on average.
• There is a linear dependence of annual H_g,t values on ρ_g under all-sky conditions.
• There is also a linear dependence of annual H_g,t values on k_d at the 12 sites under all-sky conditions.
• Similar linear dependence exists between annual H_g,t values and CMF under all-sky conditions.
• A linear relationship occurs between annual k_d values and CMF at the 12 sites under all-sky conditions.
• A plot of the annual CF values against ρ_r at all sites under all-sky conditions forms a bundle of linear lines, each line for every site, all passing through the point CF = ρ_r = 1.
• A plot of the annual CF values against |φ| at all sites under all- and clear-sky conditions forms a bundle of quadratic lines, each line for every site.
• A plot of the monthly SEG for all sites under all-sky conditions showed quadratic dependence on time (month).
• Similar behaviour was shown between the monthly CMF values and time (month) for all sites under all-sky conditions.
• Contour plots of the monthly H_g,t values against time (month) and ρ_g or CMF for the 12 sites indicated the dominating patterns of H_g,t as functions of ρ_g and CMF at the sites under all-sky conditions.