Distribution of solar irradiance on inclined surfaces caused by moving clouds

Abstract

The distribution of solar irradiance in shadows of discrete (broken) clouds differs from the distribution calculated for inclined surfaces on the basis of traditional transposition models and changes fast. This phenomenon is studied in this paper. For calculations of dynamic distributions of irradiance on inclined surfaces, a formal point source of direct radiation near the real position of the sun is defined as the source of the “imaginable radiation.” This notion is used to create a one-dimensional (1D) simulation model, which allows the fast-changing distribution of irradiance to be calculated. In general, the coincidence of calculated and measured irradiance on inclined surfaces is good. The paper also shows how the current value of the diffuse component of solar radiation can be derived from measurements of total radiation in four differently tilted planes.

1 Introduction

Fast dynamic processes of solar radiation are caused by shadows of discrete, mainly cumulus, clouds: Cumulus humilis (Ch), Cumulus mediocris (Cm), and Cumulus fractus (Cf). Stochastic values of irradiance (on the horizontal plane) and its transients have been analyzed (Tomson 2010, 2014; Tomson and Hansen 2011), but moving clouds also involve different distribution laws on inclined surfaces, which are used in solar technology, that is, PV electricity generation and production of domestic hot water. Therefore, this phenomenon should have some importance and it is analyzed in the present study.

Regular solar radiation measurements and databases of solar irradiance have usually used recordings of beam radiation G _b , diffuse radiation G _d0 on the horizontal plane, and radiation reflected from the ground G _r (McArthur 2005). Here and below, solar radiation is expressed via its irradiance G. Global radiation on the horizontal plane G ₀ and total radiation on the plane of any optionally tilted plane G _T have to be found using recalculations.

Several transposition models have been developed for the recalculation. The fundamental law was proposed by Liu and Jordan (1963) and is still in use; significant development was achieved by Ineichen et al. (1990), who also considered radiation reflected from the ground. Overviews of such transposition models can be best found in books such as Perez et al. (2001), Muneer (2004), and Duffie and Beckman (2006). These models consider the static character of solar radiation using its averaged values and do not describe dynamic effects on inclined surfaces.

2 One-dimensional model of relative irradiance for the vertical profile along the sun’s azimuth

On observing the simultaneous irradiance in alternating shadows of clouds on differently tilted surfaces as shown in Fig. 1 (Tomson 2013), the non-uniform character of its distribution and frequent changes have to be underlined. The numbers in the legend of Fig. 1 are the “addresses” of the said distributions on the relative time-scale in seconds. They appear randomly. F is the averaged distribution for the whole packet.

Fig. 1

Relative solar irradiance on inclined surfaces, recorded in the shadows of Cumulus fractus clouds, which is a random function of relative time instants (presented by seconds)

To attempt to explain such behavior, a one-dimensional (1D) simulation model along the sun’s azimuth γ _s is proposed. The proposed model is an explication of the Liu and Jordan model suited for a study of dynamic processes. The proposed model requires some hypothetical assumptions: diffuse radiation and radiation reflected from the ground are considered to be isotropic; direct radiation is expected to be emitted from a single source although really it may be emitted from several sparse (thin) sites in non-uniform clouds.

Existing non-isotropy is considered via an imaginable point source of radiation with coordinates α _p ,γ _p somewhere near the actual position of the sun with the coordinates α _s ,γ _s . Radiation emitted from that source has irradiance G _p , which has the character of direct radiation with a corrected incident angle Θ _Tp (α _p ,γ _p ), Fig. 2, and magnitude, which can be calculated via the declared and measured initial data (Appendix).

Fig. 2

Layout of symbols used in the proposed 1D model. Definitions are shown in the nomenclature of symbols and numbers in brackets show the tilt angle of corresponding sensors (S)

In Fig. 2, the symbol Θ _z is the zenith angle, β is the tilt angle of the receiver plane, and S(0)… S(180) are sensors (pyranometers). The number in brackets shows their tilt angle.

In clear-sky conditions α _p  → α _s , but in a general case, α _p  = α _s  ± Δα _p , where Δα _p varies randomly. This phenomenon is probably caused by invisible direct radiation from the sun (λ > 800 nm), which is able to shine through sparse (thin) sites in clouds and, to a lesser degree, by reflected radiation from the edges of clouds. Clouds and these sites are moving continuously and therefore G _p and α _p are variable. As Ch, Cm, and Cf clouds in the summer season appear mostly around noon (Tomson 2010), the value of α _s is mostly in the range of 30°–50° (at ∼60° N) and Δα _p has finite values. According to the definition, small fluctuations of γ _p around the sun’s azimuth Δγ _s  → 0 do not influence the value of G _pT . Possible fluctuations of G _pT due to fluctuations of the sun’s imaginable azimuth are lost on the background of instrumental, installation, and sampling errors. G ₀ is the global radiation on the horizontal plane comprising its components: direct radiation G _p0 and diffuse radiation G _d0, G _F is the (total) radiation on the illuminated vertical plane, G _B is the (total) radiation on the shaded vertical plane, and G _r is the pure reflected radiation on the reverse horizontal plane. Its relative value is equal to albedo, G ^* _r  = A. In the conditions of high latitudes (∼60° N), the sensors S(−90) and S(180) are always in shadow. Therefore, G _B has a diffuse origin, and according to the hypothesis proposed, it is a component of G _F too.

The imaginable (direct) component of radiation on the sensors S(90) and S(0) is a function of their incident angles if G _p  > 0 exists, and conversely, the value of G _p can be calculated from known values of G ₀ and G _F , as shown in the Appendix.

The analysis in the Appendix will be done in relative units marked with an asterisk, G ^* = G/G ₀, where G∈{G _B , G _F , G _r , and G _T }. Calculation of any real radiation regime is simple when the rates in relative units are known. A flow diagram of the proposed recalculation model is shown in the Appendix too.

In clear-sky conditions, α _p should be equal to α _s , but due to (instrumental and installation) errors, α _p  ≠ α _s and a finite angle shift Δα _p with a value of a few degrees always exists.

In dark overcast conditions with G ₀ ≈ 100 Wm⁻², G ^* _F  = G ^* _B , and therefore (according to Eq. (3) in the Appendix), tan[(1 − G ^* _d0) / (G ^* _F  − G ^* _B )] → ∝, which means that α _p  → 90°. In dark (overcast) conditions, according to the created 1D model, direct solar radiation is formally emitted from the zenith point, although its value degenerates to zero.

According to Eqs. (5) and (6) in the Appendix, direct radiation degenerates to zero also at azimuths perpendicular to the sun’s one (γ → ±90°), and therefore, these profiles correspond to profiles assessed by pure diffuse (and reflected) radiation.

The proposed 1D model is useful due to the possibility of describing dynamical processes on inclined surfaces (which is an advantage), but it requires especial measurements of the total radiation, and public actinometrical datasets cannot be used (which is a disadvantage).

3 Hardware used for observations

Figure 3 shows the portable measuring stand used in the study, which has five sensors, S(0) … S(180). In the summer season of 2014, over 30 profiles (diagrams of the distribution) of the relative irradiance on tilted planes G ^* _T were recorded with a sampling interval of 1 s. The azimuths of recording were fixed manually, and each particular recording lasted 10–20 s; also, the full “season” lasted ∼5 min. In case of reduced measurements, only azimuths of 0°, 180°, and ±22.5° were controlled. According to Tomson (2014), the correlation time of diffuse radiation in conditions of alternating clouds is of the order of 5–15 min, which means that significant errors cannot be made in the time-delay measuring G _B and G _F separately.

Fig. 3

The portable measuring stand used in the study. Tilt angles β∈{0…180°} of sensors (S) are fixed for each season. The ring with a sensor was rotated manually for different relative azimuths γ∈ {0, … 360°}

Observations were made mainly at the former and recultivated landfill hill of Tallinn city at 59.36° N, 24.65° E, which is covered by grass.

In the measurements made in 2012 (Fig. 1), other combinations of sensors S(0), S(22.5) … S(90) (over 15 recorded G ^* _T profiles) were used. Some of these results are used in the present study too. The sensors used during the whole study were photoelectrical pyranometers manufactured in Denmark (Soldata 2015), which have a transient time of microseconds and allow the dynamic behavior of radiation to be studied. Their total uncertainty is 3 %. Data were saved on a midi LOGGER GL-200 datalogger. The sampling step used was 45° (22.5° in 2012) in the vertical plane and 22.5° in the horizontal plane. The irradiance was studied in the conditions of clear sky, overcast sky, and discrete clouds. In the last case, irradiance in shadows and sunny windows between clouds were studied. Under clouds, irradiance was always recorded without visible (to the human eye) shadows of any object over the ground. The distributions of irradiance on inclined surfaces under clear-sky conditions and in sunny windows between clouds have no essential difference (Figs. 4 and 5).

Fig. 4

Distribution diagram of the relative irradiance on differently inclined surfaces. The diagram corresponds to a large clear-sky window at global irradiance of 588 Wm⁻² along the relative azimuth γ = 0°

Fig. 5

Distribution of measured and calculated relative irradiance on inclined surfaces in conditions of changing global radiation along the relative azimuth γ = 0° in a narrow window between moving discrete clouds

4 Distribution of irradiance in the vertical profile along the sun’s azimuth

The validity of the composed 1D simulation model was controlled via several recordings presented in this section. The model describes the correct principal character of every individual recording, and in the majority of cases, the coincidence with the actual recording is good. The reliability of the simulation is analyzed in the next section.

When evaluating these diagrams, it must be taken into account that the calculation step was 22.5° but the step of actual measurements was 45°, and its values at 22.5°, 67.5°… 157.5° were found as interpolated values.

The diagram shown in Fig. 4 has a clear maximum point G ^* _T (−45^ο) ≈ 1.65 and a concave character in the range 0 > β > −90°, where G ^* _T (−45^ο) ≈ 0.13. For these recordings, the coordinates of the sun are α _s  = 37.6° and γ _s  = 132.1°, and the calculated value of α _p is 35°. The difference between the azimuths α _s and α _p is probably caused by reflections from clouds (and/or instrumental and installation errors). The sector 127° > β > −53° is illuminated. In clear-sky conditions, the level of diffuse radiation is low, and therefore, G ^* _T(−45°) on the shadow side is low too. The derivative dG ^* _T /dβ has no certain sign and has both positive and negative values.

Figure 5 shows a diagram which demonstrates the dynamics of solar radiation due to the movement of discrete clouds.

First, the distribution of the irradiance in shadow was recorded. Two minutes later, the irradiance in sunshine was recorded. The characters of these two distribution diagrams differ distinctly, but diagrams recorded and calculated for the narrow sunny window are similar to that shown for a large window in Fig. 4. In this case, α _s  = 50.1° and azimuth γ _s  = 145.8°. Elevation angle of the imaginable point source depends on the illumination: in shadow α _p  = 42.6°, but in the sunny window, it is α _p  = 50.5° (≈α _s °).

Figure 6 shows the vertical profile under a dark overcast sky at G ₀ = 110 Wm⁻².

Fig. 6

Distribution of measured and calculated relative irradiance on inclined surfaces under thick cloud cover at a global irradiance of 110 Wm⁻² along the relative azimuth γ = 0°

Here, G ^* _T is monotonically decreasing while the tilt angle is increasing to both negative and positive tilt angles. In this case, the derivative is always negative: dG ^* _T /dβ < 0. Figure 6 corresponds to α _s  = 32.8°, γ _s  = 133.2°, and α _p  = 88°.

5 Reliability of the 1D model

The reliability analysis of the composed 1D simulation model is summarized in Table 1, where the first column shows the date of the measurement, the second column the sun’s elevation angle, the third column the shift in angle between the sun’s real and imaginable elevations, the fourth column the average global irradiance, and the next nine columns the difference between the measured M and calculated C values of G ^* _T (β) at corresponding tilt angles β.

Table 1 Reliability analysis of the composed 1D simulation model, which shows difference between calculated C and measured M values of relative irradiance on differently inclined surfaces, [%]

The results for three tilt angles (135°, 45°, and −45°) require comments. At 135°, the error has a relatively high value and the highest standard deviation among the results. This circumstance is the result of quantization: the edges of the illuminated sector are continuous variables, but the model uses a discrete sample interval (Δβ = 22.5° according to the analysis done), and a quantizing error is unavoidable. At 45°, the error is highest but the standard deviation has a moderate value. This circumstance refers to the systematic ∼5 % error of the model in calculating the maximum point of G ^* _T (β). At −45°, depending on the current elevation angle of sun, the sensor S(−90) may be in shadow or (lightly) illuminated, which results in an increased standard deviation.

For other optional points of G ^* _T (β), the error has a low value of ∼1 %. The difference between errors at +180° and −180° (less than 1 %) lies within the margin of instrumental errors.

Analysis of the calculated elevation of the imaginable point source should be of interest too. From the analysis of Table 1, a weak (0.7 % slope of the trend line) negative correlation of the difference Δα _p  = α _p  − α _s with the global radiation G ₀ can be found (Fig. 7). High values of G ₀ involve low values of Δα _p . In most cases, Δα _p  < 0, which means that the fictive source of imaginable radiation mostly lies below the sun’s real position. At a low value of global radiation, G ₀ < 200 Wm⁻², the standard deviation of Δα _p is high (over 25°), which is the result of instrumental and installation errors, which influence Δα _p significantly at low values of global radiation. The installation error is a result of the inadequacy of the sensors’ real positions with respect to their ideal positions. Installation of the portable stand on rugged grassland always results in a finite leveling error.

Fig. 7

Correlation diagram between angle shifts of the imaginable point source Δα _p versus global irradiance G ₀

6 Distribution of irradiance in space

The experimental distribution of the relative irradiance in space and the total irradiance on inclined planes G ^* _T (β γ) is shown in Figs. 8, 9, and 10. Here, γ is the azimuth of a vertical plane in which the irradiance is recorded relative to the solar azimuth γ _s .

Fig. 8

Spatial distribution diagram of the total irradiance on inclined planes G ^* _T (β,γ) in overcast conditions at a global irradiance of G ₀ = 110 Wm⁻² depending on direction and tilt angle

Fig. 9

Spatial distribution diagram of the total irradiance on inclined planes G ^* _T (β,γ) in conditions of a large sunny window between clouds at a global irradiance of 588 Wm⁻²

Fig. 10

Spatial distribution diagram of the total irradiance on inclined planes G ^* _T (β,γ) under sparse overcast clouds at global irradiance of ∼400 Wm⁻². The diagram is symmetric for the 0°/180° axis

Figure 8 shows the spatial distribution under overcast clouds. This distribution is invariant to the direction, and its profile along the sun’s azimuth γ _s is shown in Fig. 6. A similar distribution is valid for a shadow of a thick discrete cloud at the low value of G ₀ < 200 Wm⁻² too.

Figure 9 shows the distribution of G ^* _T (β, γ) in a large sunny window at G ₀ = 588 Wm⁻² between discrete clouds, which is symmetric for the 0°/180° axis. A similar form of the diagram is valid for any clear-sky recording. The direction γ = 0 coincides with the solar azimuth γ _s  = 132.1°, and its profile is shown in Fig. 4.

Figure 10 shows the distribution of G ^* _T (β γ) under sparse (thin uneven) overcast clouds at G ₀ ≈ 400 Wm⁻², which is symmetric for the 0°/180° axis but has a clear maximum due to sun (γ = 0°). Here, α _s  = 43.7°, γ _s  = 162°, and α _p  = 56°.

Similarly to Fig. 10, the diagrams are characteristic for recordings in shadows under most discrete cumulus (Ch, Cm, and Cf) clouds. Analysis of the recorded data shows that in 85 % of cases, the azimuth of imaginable radiation cannot be discriminated from the sun’s azimuth, while in 15 % of cases it may be different, but the deflection lies within the range of possible errors (including errors due to the sampling step of 22.5°). The noted quality of imaginable radiation motivated the usage of a 1D model to describe the vertical profile of solar radiation mainly along the sun’s azimuth, where a point source of imaginable radiation with a height angle α _p fluctuates up and down around the sun’s height angle α _s .

7 Usage of 1D simulation model

Figure 11 shows a small packet of distribution diagrams artificially generated with the 1D simulation model using randomly generated height angles of an imaginable radiation source in the range of 30°< α _p  < 60°.

Fig. 11

Artificial distribution of solar irradiance on inclined surfaces generated at a random elevation angle of the imaginable point radiation in the range of 30^ο < α _p  < 60^ο

The numbers in the legend show values of α _p , which are presented in order of their appearance. The bold average line “aver” coincides with the case α _p  = 44.2°.

Comparison of Fig. 11 with Fig. 1 shows the lack of principal differences, and in such a way, the natural phenomenon can be reproduced artificially. The artificial G ^* _T may be useful for the précised (most exact) calculations of solar energy if the statistics of sunny windows versus cloud shadows are known.

8 Summary

The present study highlights the dynamic behavior of the solar radiation mainly in conditions of moving discrete clouds. Shadows of clouds quantify the value of irradiance and involve its different distribution on inclined surfaces. To describe such dynamic behavior, an imaginable but effective notion of a “point source of direct radiation” can be used. For the calculation of this distribution, a 1D calculation model has been proposed which shows a good coincidence with measured recordings. It is also shown how the current value of the diffuse component of solar radiation can be derived from measurements of global radiation and total radiation on three differently tilted planes.

Appendix

One-dimensional simulation model, described by flow diagram below is created using relative units marked with an asterisk G ^* = G/G ₀, where G∈{G ₀, G _B , G _F , G _r , and G _T }. All symbols are declared in the nomenclature. Building the model for the vertical profile along the sun’s azimuth, a simple relation is valid Θ _Tp  = π/2 − (α _p  + β). According to the isotropic sky model (Quashning and Hanitch 1998),

$$ {G^{*}}_B=\mathrm{A}\times \left(1\hbox{--} \cos \beta \right)/2+{G_d}_0\times \left(1+ \cos \beta \right)/2 $$

(1)

Considering that at β = −90°, its cosine is zero, cos β = 0, which results in

$$ {G}^{*}{{}_d}_0=2\times {G^{*}}_B\hbox{--} A $$

(2)

Imaginable (direct) component of radiation on the sensors S(90) and S(0) can be calculated from G _p according to their incident angles if G _p  > 0 exists. Also backward – value of G _p can be calculated from known values of G ₀ and G _F , what is shown below.

Flow diagram of calculations of the proposed 1D simulation model.

The diffuse component on the illuminated vertical plane is equal to that in shadow G ^* _dF  = G ^* _dB  = G ^* _B . The relative total radiation on the illuminated vertical plane is G* _F  = G* _B  + G* _p  × cos Θ _Tp and that on the horizontal plane is G ^* _p0 = G ^* _p  × sinα _p = = 1 − G ^* _d0, from which the following equation can be expressed:

$$ {a}_p=\mathrm{atan}\left[\left(1\hbox{--} {G}^{*}{{}_d}_0\right)/\left({G^{*}}_F\hbox{--} {G^{*}}_B\right)\right] $$

(3)

and

$$ {G^{*}}_p=\left(1\hbox{--} {G}^{*}{{}_d}_0\right)/ \sin {\alpha}_p $$

(4)

Imaginable component on an optionally declined plane can be calculated

$$ {G^{*}}_{pT}={G^{*}}_p\times \cos \left(\beta +{\alpha}_p-\pi /2\right) $$

(5)

Total radiation (irradiance) on an optionally declined plane is sum of diffuse ( 1 ) and imaginable (5) components

$$ {G^{*}}_T={G^{*}}_{dT}+{G^{*}}_{pT} $$

(6)

If the relative irradiance on a tilted plane different from the sun’s azimuth G ^* _T(γ) has to be found, then Eq. (6) should be used:

$$ {G^{*}}_{pT}\left(\gamma \right) = {G^{*}}_{pT} \times \cos \gamma $$

(7)

Using Eq. (5), G ^* _pT  ≥ 0 always has positive (or zero) values.