1. Introduction
Installations with tilted solar panels exploiting solar energy have long existed in the market. Solar flatplate panels are widely used to convert solar energy into electricity (e.g., PV installations). These systems consist of solar collectors receiving solar radiation on flatplate surface(s) that can operate in three different modes: (i) at a fixedtilt angle with southward orientation in the northern hemisphere or northward orientation in the southern hemisphere, (ii) at a fixedtilt angle rotated on a vertical axis (oneaxis or singleaxis) system that continuously follows the Sun, and (iii) at a varyingtilt angle fixed on a twoaxis (or dualaxis) system that continuously tracks the Sun. Installations of mode(i) are known as fixedtilt systems and are widely used because of their lower installation and maintenance costs. Installation of mode(ii) systems provide higher solar energy on the inclined surface but have a slightly higher cost because of the necessary maintenance of the moving parts. Installation of mode(iii) systems is considered the most effective because the solar rays are always normal to the receiving flatplate surface. These systems provide higher performance, but they are, though, associated with higher maintenance costs because of more moving parts. The first type of solar system is also called stationary or static, while the other two are named dynamic, because of their Suntracking ability. Recently, Kambezidis and Psiloglou [
1] examined the mode(i) static systems for the performance of fixedtilt flatplate solar collectors with southward orientation in Greece, but investigation of the solar energy potential across the country for mode(ii) and (iii) systems has never been made. The present work investigates the mode(iii) dynamic systems for the solar energy potential received on flatplate solar collectors for the first time in Greece.
Static solar systems are nowadays widely used in solar energy applications worldwide because of their simple construction and low maintenance cost. For this reason, they have received great attention from researchers (c.f., solar energy potential, solar availability) at a certain location or region, e.g., [
2,
3,
4,
5]. Another priority has been given to dynamic mode(ii) solar systems because of their relatively higher solar energy imprint, e.g., [
6]. As far as the dynamic mode(iii) solar systems are concerned, they have started being used in the last 20 years because of their higher performance compared to that of the other two types, e.g., [
7,
8]. Much effort has been invested, though, in improving both moving and electronic parts for the Suntracking sensors, e.g., [
3,
9], which are involved in the configuration of dynamic solar systems. Nevertheless, the performance of such systems must be evaluated against solar radiation measurements at firsthand, e.g., [
10]. However, the scarcity of solar radiation measuring stations worldwide has triggered the development of solar radiation modelling, e.g., [
11,
12,
13], to derive the optimum tilt angle and orientation for obtaining maximum solar energy on flatplate solar panels for static systems in both hemispheres. Other methods use a combination of groundbased solar data and modelling, e.g., [
14], or utilise solar data from international databases, e.g., [
15,
16].
Some studies, like the present work, have already been conducted in Greece. Tsalides and Thanailakis [
17] computed the optimum azimuth and tilt angles of PV arrays at 9 locations in Greece; they found that PV arrays having azimuth angles in the range ±30
^{o} (0
^{o} at south) receive about 40%  60% greater solar energy than that for tilt angles equal to the latitude of the sites. Koronakis [
18] found an optimum tilt angle of 25
^{o} toward the south for flatplate collectors and 30
^{o} for concentrated solar cells at Athens allyear round. Balouktsis et al. [
19] analysed the optimal tilt angle of PV installations at certain locations in Greece and found it to be around 25
^{o} to the south. Synodinou and Katsoulis [
20] estimated a tilt angle equal to the latitude of Athens for optimum solar energy harvesting at this location. Darhmaoui and Lahjouji [
21], by analysing the solar radiation databases of 35 sites around the Mediterranean, found the optimum tilt angles with south orientation; for Irakleio, Athens, and Mikra in Greece; these angles were estimated at 35.1
^{o}, 36.8
^{o}, and 38.7
^{o}, respectively. Kaldellis et al. [
22] found an optimum tilt angle for southoriented surfaces in Athens and central Greece of 15
^{o} during the summer. Jacobson and Jadhav [
23] have derived a review for the optimum tilt angles with south orientation in the northern hemisphere by using the PVWatts algorithm; for Athens, they estimated it at 29
^{o}. Raptis et al. [
24] estimated the optimum tilt angle for maximum energy reception on flatplate collectors with south orientation in Athens at 39
^{o}. Recently, Kambezidis and Psiloglou [
1] suggested a new methodology for estimating the optimum tilt angle for southoriented flatplate solar collectors in Greece; by applying the method, they found the optimum tilt angles in the range of 25
^{o}  30
^{o}, thus agreeing with the results of Koronakis, Balouktsis et al., and Jacobson and Jadhav. In 1996, the European Solar Radiation Atlas was derived [
25] and published in 2001 [
26]; it includes maps of the solar energy potential on horizontal and inclined surfaces over almost all of Europe, including Greece; the maps were derived from solar radiation databases across the continent covering the period 1981  1990 with a resolution of 10 km. Also, a Global Solar Atlas has been generated [
27] for almost all of the world, including Greece. These maps concern global solar horizontal irradiation, directnormal solar irradiation, and PV power potential. Calculations for these maps were made by using data in the periods 1994, 1999, 2007  2018 depending on the region. Moreover, a map of the solar potential over Greece on horizontal plane based on typical meteorological years (TMYs) was developed by Kambezidis et al. [
28]. Finally, a study about the future solar resource in Greece due to climate change has appeared in the literature [
29].
From the above, it is clear that an attempt has yet to be made to construct a solar map for Greece to show the solar energy potential on inclined flatplate surfaces that continuously track the Sun. This gap is bridged in the present study; for the first time, solar maps for Greece showing the energy on inclined flatplate surfaces tracking the Sun are derived.
The structure of the paper is the following.
Section 2 describes the data collection and data analysis.
Section 3 deploys the results of the study.
Section 4 provides a discussion, and
Section 5 presents the conclusions and main achievements of the work. Acknowledgements and References follow.
2. Materials and Methods
2.1. Data Collection
Hourly values of solar radiation were downloaded from the PV—Geographical Information System (PVGIS) tool [
30] using the Surface Solar Radiation Data SetHeliostat (SARAH) 2005  2016 database (12 years) [
31,
32]. The PVGIS platform provides solar radiation data through a userfriendly tool for almost any location in the world, including Greece. The methodology used for estimating solar radiation from satellites by the PVGIS tool is described in various works, e.g., [
33,
34].
In the present work, a set of 43 sites was arbitrarily chosen to cover the whole territory of Greece. The location of these sites has been adopted from a recent work on the solar radiation climate of Greece [
14].
Table 1 provides the names and geographical coordinates of the sites;
Figure 1 shows their location on the map of Greece.
TMYs for the above sites were downloaded from the PVGIS tool; these TMYs include hourly values of air temperature (in degrees C), relative humidity (in %), horizontal infrared radiation (in Wm^{2}), wind speed (in ms^{1}) and direction (in degrees), surface pressure (in Pa), global horizontal irradiance, H_{g} (in Wm^{2}), directnormal solar irradiance, H_{bn} (in Wm^{2}), and diffuse horizontal irradiance, H_{d} (in Wm^{2}). The latter three parameters were considered in this study. The TMYs were derived in the PVGIS platform from simulations from 2005 to 2016.
2.2. Data Processing and Analysis
To process the data used in this work, the following 5 steps were followed.
Step 1. The downloaded hourly data from the PVGIS website were transferred from universal time coordinate (UTC) into Greek local standard time (LST = UTC + 2 h). It must be mentioned that the PVGIS solar radiation values were provided at different UTC times for the 43 sites considered, e.g., at hh:48 or hh:09, where hh stands for any hour between 00 and 23.
Step 2. The routine SUNAE introduced by Walraven [
35] was used to derive the solar azimuths and elevations. However, the original SUNAE algorithm has been renamed to XRONOS (meaning time in Greek, X is pronounced CH) because of added modifications due to the right ascension and atmospheric refraction effects [
36,
37]. XRONOS ran for the geographical coordinates of the 43 sites in their TMYs to derive the solar altitudes,
γ, at all LST times calculated in step 1. Nevertheless, inconsistencies (gaps) in the solar azimuth angles, ψ, at both instances of sunrise and sunset were found during calculations in the XRONOS code. The discrepancy was overcome by implementing a modified XRONOS (mXRONOS) code in MatLab; a Fourier series approximation of the expression for ψ at the sunrise and sunset instances was derived and applied to all 43 sites. The mXRONOS algorithm is described in detail in an article recently published in the journal of Sun and Geosphere [
38].
Step 3. The hourly direct horizontal solar radiation, H_{b}, values were estimated at all sites by the expression H_{b} = H_{bn}·sinγ.
Step 4. All solar radiation and solar geometry values were assigned to the nearest LST hour (i.e., values at hh:48 LST or hh:09 LST were assigned to hh:00 LST). That was done to have all values in the database as integer hours.
Step 5. Only those hourly solar radiation values greater than 0 Wm^{−2} and corresponding to γ ≥ 5° (to avoid the cosine effect) were retained for further analysis. Also, the criterion of H_{d} ≤ H_{g} was required to be met at hourly level.
For estimating global solar irradiance on a flatplate solar collector fixed on a dualaxis system that continuously tracks the Sun, H
_{g,t} (in Wm
^{−2}), the isotropic model of LiuJordan (LJ) [
39], as well as the anisotropic model of Hay [
40,
41], was adopted (the subscript t stands for “tracking”). The isotropic and anisotropic models were used to estimate (i) the groundreflected radiation from the surrounding surface, H
_{r,t} (in Wm
^{−2}), and (ii) the diffuse inclined radiation, H
_{d,t} (in Wm
^{2}), received on the sloping flatplate surface. These models were adopted in the present study because of their simplicity and effectiveness in providing the tilted total solar radiation; a second reason for using both transposition models was to compare their results. The satisfactory performance of the LJ and Hay models has been verified by various studies, e.g., [
42,
43].
Figure 2 provides a schematic for a tilted surface receiving solar radiation. Deliberately, the tilted surface is not aligned along the direction of the Sun to show the various angles formed, i.e., the tilt angle of the surface, β, the solar altitude, γ, the incidence angle, θ (the angle between the normal to the surface and the direction toward the Sun), the solar azimuth, ψ, and the azimuth of the tilted plane, ψ’.
For a Suntracking surface the received total solar radiation is given by the following wellknown expression:
The solar radiation components in Equation (1) are calculated by the following analytical expressions:
where, in this case, θ = 0
^{o} and β = 90
^{o} – γ because the inclined surface is always normal to the solar rays (see
Figure 2); also, ψ = ψ’, because of the Suntracking feature of the mode(iii) system. R
_{d} and R
_{r} are the skyconfiguration and groundinclined planeconfiguration factors, respectively, S is the SunEarth distance correction factor, and N is the day number of the year (N = 1 for 1 January, and N = 365 for 31 December in a nonleap year or N = 366 in a leap year). In the LJ model the ground albedo usually takes the value of ρ
_{g0} = 0.2 (Equation (2)). Nevertheless, in the present study this value has been replaced with the nearreal groundalbedo one, ρ
_{g,} for all 43 sites. To retrieve the ρ
_{g} values for the 43 sites, use of the Giovanni portal [
48] was made; pixels of 0.5° × 0.625° spatial resolution were centered over each of the 43 sites for which monthly mean values of the ground albedo were downloaded in the period 2005  2016. Monthly mean ρ
_{g} values were then computed for all sites and were used to calculate H
_{g,t}.
To isolate those solar radiation values that corresponded to clearsky conditions only, use of the modified clearness index, k’
_{t}, was made as in [
49]. The significance of this modified index is that it does not depend on air mass [
50]. Its definition is the following:
where m is the optical air mass. Kambezidis and Psiloglou [
49] have defined the range for clear skies as 0.65 < k’
_{t} ≤ 1. This range has been used in the present study, while the allsky conditions are characterised by the full range of 0 < k’
_{t} ≤ 1. The atmospheric extinction index, k
_{e}, from [
52] was adopted; it is defined as k
_{e} = H
_{d}/H
_{b} [
53]. Its meaning is that it gives information about the percentage contribution of both the H
_{d} and H
_{b} solar radiation components to solar applications over an area and, more specifically, to PV installations. In other words, it denotes the significant fractional amount of each solar component in solar harvesting.
For every site, hourly values of H_{g,t} were estimated twice from Equation (1); the first time by using Equations (4a, 4b) for the LJ model and the second time by using Equations (4a – 4e) for the Hay model. From the hourly H_{g,t} values, annual, seasonal, and monthly solar energy sums (in kWhm^{−2}) under all and clearsky conditions were estimated for all sites. To implement all the above calculations, a MatLab code was developed, which included the routine mXRONOS.
4. Discussion
This section refers to the discussion of related results found by other researchers.
Hammad et al. [
63] compared the performance and cost between fixedtilt (static) and doubleaxis (dynamic) systems in Jordan. They found 31.29% more energy produced by the 2axis system in comparison with the static one, a figure quite comparable to our 31.2% (1.312 times) found in
Section 3.1. Further, the authors estimated the payback period to be 27.6 months and 34.9 months for the dynamic and static systems, respectively, with corresponding electricity costs of 0.080
$kWh
^{1} and 0.100
$kWh
^{1}.
Lazaroiu et al. [
64] found a 12%  20% increase in the energy produced by a dualaxis solar system in comparison to a fixedtilt one in Romania, quite lower than our 31.2%.
Michaelides et al. [
65] studied the performance of solar boilers for Athens, Greece, and Nicosia, Cyprus, by considering 1axis, seasonaltilt, and fixedtilt systems. They found that the solar fractions (the normalised difference between the hot water energy provided by the Sun and the auxiliary one supplied by electricity) are 81.4%, 76.2%, and 74.4% for Athens, and 87.6%, 81.6%, and 79.7% for Nicosia in the case of a singleaxis, a seasonaltilt, and a fixedtilt solar system, respectively.
As far as Saudi Arabia is concerned, Kambezidis et al. [
66] found that mode(iii) systems produce 4.22% more solar energy than mode(ii) ones, 28.81% more solar energy than mode(i) systems, and 37% in comparison to a flatplatereceiving surface on horizontal plane. Their result of 28.81% is close to our 31.2%.
A study for the USA by Drury et al. [
67] showed that mode(ii) tracking systems can increase power generation by 12%  25% in relation to fixedtilt ones, and mode(iii) tracking systems by 30%  45%; the latter finding agrees marginally with our 31.2%. These researchers estimated the installation cost at 0.25
$W
^{−1}, 0.82
$W
^{−1}, and 1.23
$W
^{−1} for fixedtilt, 1axis, and 2axis systems, respectively. In the same way, their operation and maintenance costs were estimated at 25
$kW
^{−1}year
^{−1}, 32
$kW
^{−1}year
^{−1}, and 37.5
$kW
^{−1}year
^{−1}, respectively.
Another study in Spain by Eke and Senturk [
68] concluded that a doubleaxis solar system may result in an increase in electricity by 30.7% compared to a fixedtilt one (a finding very close to our 31.2%).
Vaziri Rad et al. [
69] in a study about the technoeconomic and environmental features of different solartracking systems in Iran concluded that the dualaxis ones are the most efficient as they produce 32% more power on average compared to the fixedtilt mode (a figure quite comparable to our 31.2%).
From the above discussion, one can easily conclude that the additional solar energy gain on solar panels fixed on mode(iii) systems in comparison to mode(i) ones depends on the terrain (surface albedo) surrounding the site in question and not on the absolute values of solar radiation received at the location. This is confirmed by the comparable figures of 31.29% in Jordan, 30.7% in Spain, and 32% in Iran to our 31.2%. On the contrary, the diverging figures of 12%  20% in Romania, and 28.81% for Saudi Arabia may be attributed to the different landscape morphology in these cases to that of Greece. Further confirmation for this conclusion may be demonstrated by the wide range of solar energy gain within the USA (30%  45%) due to the high variety in the surface morphology (deserts, high mountains, coastal regions, plains); nevertheless, the range of solar energy gain includes 31.2% (equal to ours), implying that this result has been extracted for locations with similar terrain to the Greek territory.
5. Conclusions
The present study investigated the solar energy potential across Greece on flatplate solar panels that vary their tilt angle to receive solar radiation normally to their surfaces during the day. The main objective was to find the annual energy available in this configuration type under all and clearsky conditions. This was achieved by calculating the annual energy sum on flatplate surfaces with varying tilt angles that track the Sun across Greece; the solar availability on a horizontal plane was also included for reference purposes. For this reason, hourly solar radiation data in typical meteorological years derived from 2005 to 2016 were downloaded from the PVGIS platform for 43 sites of Greece. The energy received on the tilted surfaces was calculated for nearreal groundalbedo values downloaded from the Giovanni website.
The main result of the work was that the annual solar energy received by such (dynamic) mode(iii) systems varies between 2247 kWhm^{−2} and 2878 kWhm^{−2} for all skies and between 1806 kWhm^{−2} and 2617 kWhm^{−2} under clearsky conditions across Greece. These values have been calculated by using the HAY model. For the case of the LJ model, the above numbers become 2064 kWhm^{−2}  2709 kWhm^{−2} for all and 1743 kWhm^{−2}  2502 kWhm^{−2} for clearsky conditions. As reference, the corresponding values on the horizontal plane are 1726 kWhm^{−2} and 1474 kWhm^{−2}. It was found that flatplate solar panels mounted on a dualaxis tracking system provide 1.3 times higher energy than a fixedtilt (mode(i)) system in Greece. The distinction in clear skies was achieved by incorporating the modified clearness index, k’_{t}, in the calculations. In the rest of the analysis, only the HAY model was used by incorporating nearreal groundalbedo values, ρ_{g}.
The annual solar energy sums, and the monthly solar energy values averaged over all locations, and their corresponding TMYs were estimated under allsky conditions. A regression equation was provided as a bestfit curve to the monthly mean solar energy sums that can estimate the solar energy potential at any location in Greece with great accuracy (R^{2} > 0.98). This expression may prove especially useful to architects, civil engineers, solar energy engineers, and solar energy systems investors to assess the solar energy availability in Greece for solartracking flatplate solar systems throughout the year.
Seasonal solar energy sums were also calculated. They were averaged over all sites and their TMYs under allsky conditions. A new regression curve that best fits the mean values was estimated with absolute accuracy (R^{2} = 1). Maximum sums were found in the summer (527 kWm^{−2}) and minimum ones in the winter (382 kWm^{−2}), as expected.
Though unified curves have been presented for the monthly and seasonal solar energy yields in all of Greece numerically expressed in
Table 3, individual monthly and seasonal curves for all 43 sites were given in
Figure 6 and
Figure 8, respectively, for the interested scientist or engineer to see the individual solar energy yield variation.
Annual maps of H_{g,t,Hay/ρg} were derived from the annual mean solar energy sums of the 43 sites using the kriging geospatial interpolation method under all and clearsky conditions. In both cases, higher solar energy levels were found in southern Greece, a finding that may divide the country into two imaginary parts (northern and southern) at the latitude of φ ≈ 39^{o} N.
The atmospheric extinction index, k
_{e}, was also used in the present study introduced by [
52]. This index gives information about the contribution of the diffuse and direct solar radiation components in solar harvesting. A plot of the annual mean H
_{g,t,Hay/ρg} values versus k
_{e} showed a declining trend. Therefore, a map with annual mean k
_{e} values over Greece under allsky conditions revealed an almost reverse pattern to that for H
_{g,t,Hay/ρg}. Moreover, the intraannual variation of the monthly mean k
_{e} values as well as seasonal maps of the atmospheric extinction index over Greece were derived. A bestfit curve was produced for the intraannual variation. The seasonal k
_{e} maps showed patterns quite opposite to those for H
_{g,t,Hay/ρg}, at least for spring and summer.
A 3D graph of H_{g,t,Hay/ρg} versus φ and ρ_{g} presented a waveform pattern. That was attributed to the combination of the variation in both independent parameters (see Figures 14a and 16a). The intraannual variation of the ground albedo over Greece was also shown.
Figure 1.
Distribution of the 43 selected sites in Greece. The numbers in the circles refer to those in column 1 of
Table 1. This Figure is a reproduction of
Figure 1 in [
14].
Figure 1.
Distribution of the 43 selected sites in Greece. The numbers in the circles refer to those in column 1 of
Table 1. This Figure is a reproduction of
Figure 1 in [
14].
Figure 2.
Inclined surface (of a PV array) at a tilt angle β with arbitrary orientation. E, W, N, S denote East, West, North, and South, respectively. Also, the solar altitude, γ, the solar azimuth, ψ, the tilted surface’s azimuth, ψ’, and the incidence angle, θ, are shown.
Figure 2.
Inclined surface (of a PV array) at a tilt angle β with arbitrary orientation. E, W, N, S denote East, West, North, and South, respectively. Also, the solar altitude, γ, the solar azimuth, ψ, the tilted surface’s azimuth, ψ’, and the incidence angle, θ, are shown.
Figure 3.
Variation of the annual mean solar energy yield across the 43 sites in Greece on horizontal surface (green lines), and on flatplate surfaces fixed on mode(iii) dynamic (red lines) systems estimated by both diffuse transposition models LJ and Hay. The solid lines represent the variation of the annual yields under all skies, while the shortdashed ones under clearsky conditions. The horizontal straight lines show the average values all over the 43 sites and their TMYs.
Figure 3.
Variation of the annual mean solar energy yield across the 43 sites in Greece on horizontal surface (green lines), and on flatplate surfaces fixed on mode(iii) dynamic (red lines) systems estimated by both diffuse transposition models LJ and Hay. The solid lines represent the variation of the annual yields under all skies, while the shortdashed ones under clearsky conditions. The horizontal straight lines show the average values all over the 43 sites and their TMYs.
Figure 4.
Variation of the annual mean solar energy yield, H_{g,t,Hay}, versus the geographical latitude, φ, across the 43 sites in Greece on flatplate solar collectors fixed on mode(iii) solar tracker under all (black circles), and clear (red circles) sky conditions over their TMYs. The black horizontal lines (solid for all and dashed for clear skies) show the annual averages. The arrows (black for all and red for clear skies) denote the ±1σ from the mean.
Figure 4.
Variation of the annual mean solar energy yield, H_{g,t,Hay}, versus the geographical latitude, φ, across the 43 sites in Greece on flatplate solar collectors fixed on mode(iii) solar tracker under all (black circles), and clear (red circles) sky conditions over their TMYs. The black horizontal lines (solid for all and dashed for clear skies) show the annual averages. The arrows (black for all and red for clear skies) denote the ±1σ from the mean.
Figure 7.
Intraannual variation of H_{g,t,Hay/ρg} under (a) all, and (b) clearsky conditions, averaged over all sites in Greece and over each month in their TMYs. The black solid line represents the average monthly H_{g,t,Hay/ρg} sums. The red lines correspond to the mean H_{g,t,Hay/ρg} + 1σ curves, and the blue lines to the mean H_{g,t,Hay/ρg} ‒ 1σ ones. The green lines refer to the bestfit curves to the mean H_{g,t,Hay/ρg} ones. The grey lines denote the 95% confidence interval.
Figure 7.
Intraannual variation of H_{g,t,Hay/ρg} under (a) all, and (b) clearsky conditions, averaged over all sites in Greece and over each month in their TMYs. The black solid line represents the average monthly H_{g,t,Hay/ρg} sums. The red lines correspond to the mean H_{g,t,Hay/ρg} + 1σ curves, and the blue lines to the mean H_{g,t,Hay/ρg} ‒ 1σ ones. The green lines refer to the bestfit curves to the mean H_{g,t,Hay/ρg} ones. The grey lines denote the 95% confidence interval.
Figure 8.
Seasonal mean H
_{g,t,Hay/ρg} variation under
(a) all, and
(b) clearsky conditions for all 43 sites in Greece. The seasonal values are sums of the hourly solar radiation ones for each site. The numbers in the legend correspond to the sites shown in column 1,
Table 1. The numbers 1  4 in the
xaxis refer to the seasons in the sequence 1 = spring to 4 = winter.
Figure 8.
Seasonal mean H
_{g,t,Hay/ρg} variation under
(a) all, and
(b) clearsky conditions for all 43 sites in Greece. The seasonal values are sums of the hourly solar radiation ones for each site. The numbers in the legend correspond to the sites shown in column 1,
Table 1. The numbers 1  4 in the
xaxis refer to the seasons in the sequence 1 = spring to 4 = winter.
Figure 9.
Seasonal variation of H_{g,t,Hay/ρg} in Greece. The black lines represent the seasonal means. The red dashed lines refer to the mean H_{g,t,Hay/ρg} + 1σ curves, and the blue dashed ones to the mean H_{g,t,Hay/ρg} ‒ 1σ curves, under (a) all, and (b) clearsky conditions. The H_{g,t,Hay/ρg} values have been averaged over all 43 sites, and over each season in their TMYs. The green lines refer to the bestfit curves to the mean ones. The numbers 1 – 4 in the xaxis refer to the seasons in the sequence 1 = spring to 4 = winter.
Figure 9.
Seasonal variation of H_{g,t,Hay/ρg} in Greece. The black lines represent the seasonal means. The red dashed lines refer to the mean H_{g,t,Hay/ρg} + 1σ curves, and the blue dashed ones to the mean H_{g,t,Hay/ρg} ‒ 1σ curves, under (a) all, and (b) clearsky conditions. The H_{g,t,Hay/ρg} values have been averaged over all 43 sites, and over each season in their TMYs. The green lines refer to the bestfit curves to the mean ones. The numbers 1 – 4 in the xaxis refer to the seasons in the sequence 1 = spring to 4 = winter.
Figure 10.
Distribution of the annual H_{g,t,Hay/ρg} sums across Greece, averaged over their TMYs; (a) all, and (b) clearsky conditions. The kriging geospatial interpolation method has been used to draw the isolines from the available 43 values.
Figure 10.
Distribution of the annual H_{g,t,Hay/ρg} sums across Greece, averaged over their TMYs; (a) all, and (b) clearsky conditions. The kriging geospatial interpolation method has been used to draw the isolines from the available 43 values.
Figure 11.
Interdependence of the annual mean H
_{g,t,Hay/ρg} and H
_{g,t,Hay/ρg} values for
(a) all, and
(b) clearsky conditions. The data points are averages over the TMY of each site. The linear fits to the data points have the following expressions:
(a) H
_{g,t,Hay/ρg} = 0.9436H
_{g,t,Hay/ρg} + 298.4800 with R
^{2} = 0.9848, and
(b) H
_{g,t,Hay/ρg} = 1.0017H
_{g,t,Hay/ρg} + 81.5810 with R
^{2} = 0.9860. The distant data points on the bestfit green dotted lines correspond to Kastellorizo (site #15 in
Table 1 and
Table 2, and
Figure 1).
Figure 11.
Interdependence of the annual mean H
_{g,t,Hay/ρg} and H
_{g,t,Hay/ρg} values for
(a) all, and
(b) clearsky conditions. The data points are averages over the TMY of each site. The linear fits to the data points have the following expressions:
(a) H
_{g,t,Hay/ρg} = 0.9436H
_{g,t,Hay/ρg} + 298.4800 with R
^{2} = 0.9848, and
(b) H
_{g,t,Hay/ρg} = 1.0017H
_{g,t,Hay/ρg} + 81.5810 with R
^{2} = 0.9860. The distant data points on the bestfit green dotted lines correspond to Kastellorizo (site #15 in
Table 1 and
Table 2, and
Figure 1).
Figure 12.
(a) Scatter plot of the annual mean datapoint values of (H
_{g,t,Hay/ρg}, k
_{e}) over Greece under allsky conditions, and averaged over their TMYs. The green dashed line is a linear fit to the data points with equation H
_{g,t,Hay/ρg}, = ‒4256.9347k
_{e} + 4224.0925 and R
^{2} = 0.2148.
(b) Map of the annual mean k
_{e} values under allsky conditions across Greece and averaged over their TMYs. The kriging geospatial interpolation method has been used to draw the isolines from the available 43 values. The distant data point at H
_{g,t,Hay/ρg}, ≈2900 kWhm
^{2} in
(a) corresponds to Kastellorizo (site #15 in
Table 1 and
Table 2, and in
Figure 1).
Figure 12.
(a) Scatter plot of the annual mean datapoint values of (H
_{g,t,Hay/ρg}, k
_{e}) over Greece under allsky conditions, and averaged over their TMYs. The green dashed line is a linear fit to the data points with equation H
_{g,t,Hay/ρg}, = ‒4256.9347k
_{e} + 4224.0925 and R
^{2} = 0.2148.
(b) Map of the annual mean k
_{e} values under allsky conditions across Greece and averaged over their TMYs. The kriging geospatial interpolation method has been used to draw the isolines from the available 43 values. The distant data point at H
_{g,t,Hay/ρg}, ≈2900 kWhm
^{2} in
(a) corresponds to Kastellorizo (site #15 in
Table 1 and
Table 2, and in
Figure 1).
Figure 13.
Intraannual variation of k_{e} over Greece under allsky conditions. The values are averages over all 43 sites and in their TMYs. The black line presents the mean k_{e} variation; the red line is the mean k_{e} + 1σ curve; the blue line is the mean k_{e} ‒ 1σ curve; the green line shows the nonlinear fitted curve to the mean k_{e} one with equation k_{e} = ‒1.1927t^{6} ‒ 823.7400t^{5} ‒ 26324.0000t^{4} ‒ 3x10^{6}t^{3}‒ 4x10^{7}t^{2} ‒ 18x10^{8}t ‒ 4x10^{9} and R^{2} = 0.9908; t is month (1 for January, 12 for December).
Figure 13.
Intraannual variation of k_{e} over Greece under allsky conditions. The values are averages over all 43 sites and in their TMYs. The black line presents the mean k_{e} variation; the red line is the mean k_{e} + 1σ curve; the blue line is the mean k_{e} ‒ 1σ curve; the green line shows the nonlinear fitted curve to the mean k_{e} one with equation k_{e} = ‒1.1927t^{6} ‒ 823.7400t^{5} ‒ 26324.0000t^{4} ‒ 3x10^{6}t^{3}‒ 4x10^{7}t^{2} ‒ 18x10^{8}t ‒ 4x10^{9} and R^{2} = 0.9908; t is month (1 for January, 12 for December).
Figure 14.
Maps of the atmospheric extinction index, k_{e}, over Greece under allsky conditions, for (a) spring, (b) summer, (c) autumn, and (d) winter. The k_{e} values are seasonal averages over their TMYs. The kriging geospatial interpolation method has been used to draw the isolines from the available 43 values.
Figure 14.
Maps of the atmospheric extinction index, k_{e}, over Greece under allsky conditions, for (a) spring, (b) summer, (c) autumn, and (d) winter. The k_{e} values are seasonal averages over their TMYs. The kriging geospatial interpolation method has been used to draw the isolines from the available 43 values.
Figure 15.
(a) Scatter plot of the annual mean H
_{g,t,Hay/ρg} values as function of
(a) the altitude, z (m amsl), and
(b) the nearreal ground albedo, ρ
_{g}, at the 43 sites in Greece under allsky conditions, and averaged over their TMYs. The vertical black dashed line in
(a) shows the altitude of z = 25 m amsl; the green dotted line in
(b) is the bestfit curve to the (H
_{g,t,Hay/ρg}, ρ
_{g}) data points with equation H
_{g,t,Hay/ρg} = 4x10
^{12}ρ
_{g}^{6} ‒ 3x10
^{12}ρ
_{g}^{5} + 1x10
^{12}ρ
_{g}^{4} ‒ 2x10
^{11}ρ
_{g}^{3} + 2x10
^{10}ρ
_{g}^{2} ‒ 9x10
^{8}ρ
_{g} + 2x10
^{7} and R
^{2} = 0.2224 at a 95% confidence interval. The distant data point of H
_{g,t,Hay/ρg} ≈ 2900 kWhm
^{2} in both graphs corresponds to Kastellorizo (site #15 in
Table 1 and
Table 2, and in
Figure 1).
Figure 15.
(a) Scatter plot of the annual mean H
_{g,t,Hay/ρg} values as function of
(a) the altitude, z (m amsl), and
(b) the nearreal ground albedo, ρ
_{g}, at the 43 sites in Greece under allsky conditions, and averaged over their TMYs. The vertical black dashed line in
(a) shows the altitude of z = 25 m amsl; the green dotted line in
(b) is the bestfit curve to the (H
_{g,t,Hay/ρg}, ρ
_{g}) data points with equation H
_{g,t,Hay/ρg} = 4x10
^{12}ρ
_{g}^{6} ‒ 3x10
^{12}ρ
_{g}^{5} + 1x10
^{12}ρ
_{g}^{4} ‒ 2x10
^{11}ρ
_{g}^{3} + 2x10
^{10}ρ
_{g}^{2} ‒ 9x10
^{8}ρ
_{g} + 2x10
^{7} and R
^{2} = 0.2224 at a 95% confidence interval. The distant data point of H
_{g,t,Hay/ρg} ≈ 2900 kWhm
^{2} in both graphs corresponds to Kastellorizo (site #15 in
Table 1 and
Table 2, and in
Figure 1).
Figure 16.
Maps of the global solar irradiation, H_{g,t,Hay/ρg}, over Greece under allsky conditions, for (a) spring, (b) summer, (c) autumn, and (d) winter. All values are seasonal averages over their TMYs. The kriging geospatial interpolation method has been used to draw the isolines from the available 43 values.
Figure 16.
Maps of the global solar irradiation, H_{g,t,Hay/ρg}, over Greece under allsky conditions, for (a) spring, (b) summer, (c) autumn, and (d) winter. All values are seasonal averages over their TMYs. The kriging geospatial interpolation method has been used to draw the isolines from the available 43 values.
Figure 17.
(a) 3D plot of H_{g,t,Hay/ρg} versus φ and ρ_{g}; (b) scatter plot of ρ_{g} versus φ. In both graphs the H_{g,t,Hay/ρg} and ρ_{g} values are annual averages for each site in its TMY under allsky conditions.
Figure 17.
(a) 3D plot of H_{g,t,Hay/ρg} versus φ and ρ_{g}; (b) scatter plot of ρ_{g} versus φ. In both graphs the H_{g,t,Hay/ρg} and ρ_{g} values are annual averages for each site in its TMY under allsky conditions.
Figure 18.
Intraannual variation of the nearreal ground albedo, ρ_{g}; the monthly values are averages over all 43 sites and their TMYs under allsky conditions. The black line shows the mean ρ_{g} curve, the red and blue lines the mean ρ_{g} + 1σ and mean ρ_{g} ‒ 1σ curves, respectively; the green dashed line represents the best fit curve to the mean ρ_{g} one with equation ρ_{g} = ‒1.1927t^{6} ‒ 823.7400t^{5} ‒ 2.6324 x10^{4}t^{4} ‒ 3x10^{6}t^{3} ‒ 4x10^{7}t^{2} ‒ 18x10^{8}ρ_{g} ‒ 2x10^{9} with R^{2} = 0.9846 at a 95% confidence interval; t is month (1 = January, …, 12 = December).
Figure 18.
Intraannual variation of the nearreal ground albedo, ρ_{g}; the monthly values are averages over all 43 sites and their TMYs under allsky conditions. The black line shows the mean ρ_{g} curve, the red and blue lines the mean ρ_{g} + 1σ and mean ρ_{g} ‒ 1σ curves, respectively; the green dashed line represents the best fit curve to the mean ρ_{g} one with equation ρ_{g} = ‒1.1927t^{6} ‒ 823.7400t^{5} ‒ 2.6324 x10^{4}t^{4} ‒ 3x10^{6}t^{3} ‒ 4x10^{7}t^{2} ‒ 18x10^{8}ρ_{g} ‒ 2x10^{9} with R^{2} = 0.9846 at a 95% confidence interval; t is month (1 = January, …, 12 = December).
Table 1.
The 43 sites selected over Greece to cover the whole area of the country. This Table is a reproduction of
Table 1 in [
14]. φ = geographical latitude, and λ = geographical longitude (both in the WGS84 geodetic system); z = altitude; N = North of equator; E = East of Greenwich meridian; amsl = above mean sea level.
Table 1.
The 43 sites selected over Greece to cover the whole area of the country. This Table is a reproduction of
Table 1 in [
14]. φ = geographical latitude, and λ = geographical longitude (both in the WGS84 geodetic system); z = altitude; N = North of equator; E = East of Greenwich meridian; amsl = above mean sea level.
Site number 
Site name / Region / z (m amsl) 
λ (^{o} E) 
φ (^{o} N) 
1 
Agrinio/Western Greece/25 
21.383 
38.617 
2 
Alexandroupoli/Eastern Macedonia and Thrace/3.5 
25.933 
40.850 
3 
Anchialos/Thessaly/15.3 
22.800 
39.067 
4 
Andravida/Western Greece/15.1 
21.283 
37.917 
5 
Araxos/Western Greece/11.7 
21.417 
38.133 
6 
Arta/Epirus/96 
20.988 
39.158 
7 
Chios/Northern Aegean/4 
26.150 
38.350 
8 
Didymoteicho/Eastern Macedonia and Thrace/27 
26.496 
41.348 
9 
Edessa/Western Macedonia/321 
22.044 
40.802 
10 
Elliniko/Attica/15 
23.750 
37.900 
11 
Ioannina/Epirus/484 
20.817 
39.700 
12 
Irakleio/Crete/39.3 (also written as Heraklion) 
25.183 
35.333 
13 
Kalamata/Peloponnese/11.1 
22.000 
37.067 
14 
Kastelli/Crete/335 
25.333 
35.120 
15 
Kastellorizo/Southern Aegean/134 
29.576 
36.142 
16 
Kastoria/Western Macedonia/660.9 
21.283 
40.450 
17 
Kerkyra/Ionian Islands/4 (also known as Corfu) 
19.917 
39.617 
18 
Komotini/Eastern Macedonia and Thrace/44 
25.407 
41.122 
19 
Kozani/Western Macedonia/625 
21.783 
40.283 
20 
Kythira/Attica/166.8 
23.017 
36.133 
21 
Lamia/Sterea Ellada/17.4 
22.400 
38.850 
22 
Larissa/Thessaly/73.6 
22.450 
39.650 
23 
Lesvos/Northern Aegean/4.8 
26.600 
39.067 
24 
Limnos/Northern Aegean/4.6 
25.233 
39.917 
25 
Methoni/Peloponnese/52.4 
21.700 
36.833 
26 
Mikra/Central Macedonia/4.8 
22.967 
40.517 
27 
Milos/Southern Aegean/5 
24.475 
36.697 
28 
Naxos/Southern Aegean/9.8 
25.533 
37.100 
29 
Orestiada/Eastern Macedonia and Thrace/41 
26.531 
41.501 
30 
Rodos/Southern Aegean/11.5 (also written as Rhodes) 
28.117 
36.400 
31 
Samos/Northern Aegean/7.3 
26.917 
37.700 
32 
Serres/Central Macedonia/34.5 
23.567 
41.083 
33 
Siteia/Crete/115.6 
26.100 
35.120 
34 
Skyros/Sterea Ellada/17.9 
24.550 
38.900 
35 
Souda/Crete/140 
21.117 
35.550 
36 
Spata/Attica/67 
23.917 
37.967 
37 
Tanagra/Sterea Ellada/139 
23.550 
38.317 
38 
Thira/Southern Aegean/36.5 
25.433 
36.417 
39 
Thiva/Sterea Ellada/189 
23.320 
38.322 
40 
Trikala/Thessaly/114 
21.768 
39.556 
41 
Tripoli/Peloponnese/652 
22.400 
37.533 
42 
Xanthi/Eastern Macedonia and Thrace/83 
24.886 
41.130 
43 
Zakynthos/Ionian Islands/7.9 (also known as Zante) 
20.900 
37.783 
Table 2.
Annual solar energy sums for the 43 sites in Greece for flatplate solar collectors mounted on mode(iii) dynamic systems, H_{g,t,Hay/ρg} under all and clearsky conditions within their TMYs. The H_{g} values are rounded integers in kWhm^{−2}.
Table 2.
Annual solar energy sums for the 43 sites in Greece for flatplate solar collectors mounted on mode(iii) dynamic systems, H_{g,t,Hay/ρg} under all and clearsky conditions within their TMYs. The H_{g} values are rounded integers in kWhm^{−2}.
Site number 
H_{g,t,Hay/ρg,all skies}

H_{g,t,Hay/ρg,clear skies}

1 
2505 
2141 
2 
2305 
1906 
3 
2406 
2027 
4 
2515 
2171 
5 
2554 
2202 
6 
2548 
2228 
7 
2379 
2032 
8 
2272 
1856 
9 
2415 
2039 
10 
2504 
2181 
11 
2269 
1806 
12 
2528 
2177 
13 
2526 
2175 
14 
2558 
2211 
15 
2878 
2617 
16 
2388 
1963 
17 
2330 
1927 
18 
2640 
2311 
19 
2588 
2130 
20 
2571 
2235 
21 
2425 
2070 
22 
2336 
1941 
23 
2488 
2194 
24 
2422 
2094 
25 
2473 
2131 
26 
2278 
1921 
27 
2641 
2288 
28 
2514 
2182 
29 
2266 
1868 
30 
2583 
2274 
31 
2486 
2141 
32 
2299 
1916 
33 
2552 
2203 
34 
2247 
1831 
35 
2553 
2207 
36 
2502 
2177 
37 
2438 
2075 
38 
2525 
2191 
39 
2567 
2227 
40 
2425 
2093 
41 
2623 
2280 
42 
2419 
2031 
43 
2506 
2177 
Sum 
106245 
90848 
Average 
2471 
2113 
Standard deviation (σ) 
127 
157 
Average + 1σ 
2598 
2270 
Average ‒ 1σ 
2344 
1956 
Table 3.
Regression equations for the bestfit curves to the monthly and seasonal mean H_{g,t,Hay/ρg} sums averaged over all 43 sites in Greece and over their TMYs, together with their R^{2} values; t is either month in the range 1  12 (1 = January,…,12 = December) or season in the range 1  4 (1 = spring, …, 4 = winter). The regression equations are given for all and clearsky conditions. R^{2} is the coefficient of determination.
Table 3.
Regression equations for the bestfit curves to the monthly and seasonal mean H_{g,t,Hay/ρg} sums averaged over all 43 sites in Greece and over their TMYs, together with their R^{2} values; t is either month in the range 1  12 (1 = January,…,12 = December) or season in the range 1  4 (1 = spring, …, 4 = winter). The regression equations are given for all and clearsky conditions. R^{2} is the coefficient of determination.