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A study of zenith radiance in Pamplona under different sky conditions
Renewable Energy 35 (2010) 830–838
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Renewable Energy journal homepage: www.elsevier.com/locate/renene
A study of zenith radiance in Pamplona under different sky conditions J.L. Torres a, *, A. Garcı´a a, M. de Blas a, A. Gracia a, R. Illanes b a b
Department of Projects and Rural Engineering, Public University of Navarre, Campus Arrosadia, 31006 Pamplona, Navarre, Spain Department of Agricultural and Forestry Engineering, Polytechnic University of Madrid, Spain
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 December 2008 Accepted 27 August 2009 Available online 19 September 2009
Zenith radiance was measured in Pamplona (Spain) during sixteen months under different sky conditions. 5th degree polynomials that relate log(Lz) with solar elevation return the best correlations both when considering the entire dataset as well as when data are split into the five sky conditions considered. Besides, we have obtained simple relations, with high correlation coefficients and low Relative Root Mean Square Difference, to predict the values of the mean zenith radiance for a type of sky from the mean zenith radiance values of one or more of the remaining four types. Lastly, we obtained month–hour equal mean zenith radiance contours for each of the five sky types considered in the study as well as for all the skies as a whole. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Zenith radiance Sky types
1. Introduction Knowing the incident solar radiation and illumination over an inclined plane with different orientations is very important to help select suitable locations for solar collection systems, accurately estimate their energy production, and for the design of energyefficient buildings [1,2]. With the goal of improving the estimates of solar radiation and illumination and overcome the simplifications imposed by most commonly used models, for several years a significant number of research studies have focused on developing mathematical models of angular distribution of sky luminance and radiance [3–8]. With their use, it is possible to more accurately determine the existing solar illumination and radiation in complex terrains and urban environments, where we find simultaneously conditions of large variability in orientation and incline, and the presence of obstacles [9]. In the models proposed by Perez [7] and by Igawa [8], luminance or radiance (both magnitudes are directly related through the photopic curve) at any given point of the sky can be calculated from the luminance or radiance at the zenith. The interest in knowing the zenith luminance has also been stated in Refs. [10–12], in which zenith luminance in Madrid was studied under different sky conditions. In contrast with previous work, the physical variable studied here is the zenith radiance, not the zenith luminance. Besides, although the experimental data were collected in Pamplona (North of Spain), the fact that they have been arranged according to different types of sky allows extending the conclusions to other
* Corresponding author. Tel.: þ34 948 169175; fax: þ34 948 169148. E-mail address: [email protected] (J.L. Torres). 0960-1481/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2009.08.026
locations. We considered the following aspects: (1) Zenith radiance dependency on solar elevation under different types of sky, (2) the relationships, of a practical interest, that can be established among zenith radiance values under different sky conditions, and (3) the definition of curves of equal mean zenith radiance at each hour of the representative days of the twelve months of the year and under different sky conditions. 2. Experimental data We have acquired data of direct, global and diffuse irradiance, as well as zenith radiance every 10 min between April 2007 and July 2008, with a total of 21,537 valid measurements. For the irradiance measurements we have used two Kipp&Zonen CM11 pyranometers, one of them with a shadow ball attached, and a Kipp&Zonen CH 1 pyrheliometer, installed on a 2Ap 2-axis tracker/ positioner from the same manufacturer. For the zenith radiance, Lz (Wm2 sr1), we used a Sky Scanner EKO MS-321-LR. All the equipment was installed on a flat roof of one of the buildings of the Public University of Navarre in Pamplona (42.83 N, 1.6 W, 435 a.m.s.l). The sky line observed at the site can be seen in Fig. 1. The Sky Scanner has a moving head where the radiance sensor is mounted, that allows for radiance readings corresponding to the 145 positions of the celestial hemisphere recommended by CIE, the zenith position being one of them. The sensor has an aperture angle of 11 and readings have been corrected using a calibration factor that ensures that, on average, the diffuse irradiance Gd observed over a horizontal plane coincides with the value of the sum of the irradiances received from each of the 145 angular patches considered (excluded the one with the position of the sun in each moment) over the aforementioned plane (Eq. (1)).
J.L. Torres et al. / Renewable Energy 35 (2010) 830–838
and exponential (Eq. (4)) functional relations between Lz and as in order to find the best fit to the experimental data, and we have also analyzed polynomial relations between Lz and tan (as).
45 40
Elevation (º)
35
Lz ¼
30
n X
ai ais
ai are the coefficients of the different
i¼0
terms of the polynomial
25 20
logðLz Þ ¼
15
n X
(2)
ai ais
(3)
i¼0
10
Lz ¼ expðb þ c$as Þ
5 0
-180
-150
-120
-90
-60
East
-30
0
30
60
90
120
150
180
West
Azimut (º)
South
Fig. 1. Sky line of the measuring location.
Gd ¼
831
Z
L$ cos q$dU
(1)
2psr
L is the radiance from a generic patch of the celestial hemisphere (Wm2 sr1); q is the zenith angle of the normal direction to that generic patch; dU is the differential of solid angle from the horizontal to the patch (sr) Lz data have been divided in groups according to different types of sky. Several sky classifications can be found in the literature, and Ref. [13] includes a comparison among some of them. In our work we have used the classification proposed by Igawa et al. [8] that distinguishes among five types of sky: overcast, near overcast, intermediate, near clear and clear. The reasons to choose this classification are: (1) that all you need to do in order to determine the sky type is computing a simple index, the Sky Index, from data that are routinely registered in most measuring stations, and (2) that with five sky types it offers a higher sky differentiation than other classifications. Although we acknowledge that the CIE classification [14] considers a more diverse typology of up to 36 types of sky, we ruled it out in this article due to its complex selection process.
3. Zenith radiance data analysis 3.1. Dependence of zenith radiance on solar elevation for the dataset observed Fig. 2a represents Lz as a function of the solar elevation angle (as) on a logarithmic scale. We have studied polynomial Eqs. (2) and (3)
5th and 6th degree polynomials relating log(Lz) with as show very similar coefficients of determination (R) (0.7406 and 0.7408) and all their TStat coefficients (ai) have absolute values higher than 1. The 7th degree fit provides a practically insignificant improvement of R and some of the TStat of its coefficients have absolute values lower than one, meaning that we are unnecessarily increasing the polynomial degree. In order to have a parsimonious model, the 5th degree polynomial relation would be the most desirable.
log Lz ¼ 0:201189 þ 0:124783as 0:00601149 a2s þ0:000164154a3s 2:16223E 06a4s þ1:08837E 08a5s
b
500
100 50
10 5
1 0
10
20
30
40
50
Solar elevation in degrees
60
70
(5)
On the other hand, the exponential fit, also shown in Fig. 2b, results in a lower coefficient of determination (0.64) than the previously mentioned. However, in the experimental scatter plot, the trend shows a concave shape and the change in curvature for solar elevations under 15 that was observed by Soler and Gopinathan [10] in the case of luminance of cloudless skies, that was one of the reasons why these authors discarded the exponential model, cannot be detected with the naked eye. Given that in this article we are only using one predictor (as) to determine zenith radiance, in order to better establish the dependence of log(Lz) with respect to as, we have applied a moving average method, as Boland and Ridley [15] did for diffuse radiation. Besides, in the same way Soler and Gopinathan [10] did, we have obtained the mean zenith radiance values ðLz Þ in as five degree intervals. The plots corresponding to both methods are displayed in Fig. 3, both in logarithmic as well as in non-logarithmic scales, and Tables 1 and 2 show the results of the polynomial and exponential fit between log(Lz) and as, respectively. As it can be noticed, the 6th degree polynomial fit is better than the exponential, with the correlation coefficient reaching values over 0.99.
Experimental zenith radiance
Experimental zenith radiance
a
(4)
400
300
200
100
0
0
10
20
30
40
50
60
70
Solar elevation in degrees
Fig. 2. (a) Experimental values of zenith radiance Lz (logarithmic scale) against solar elevation (as) for all sky types and 6th degree polynomial fit. (b) Experimental values of Lz against as for all sky types and exponential fit.
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J.L. Torres et al. / Renewable Energy 35 (2010) 830–838
Fig. 3. Moving average (window 100) of experimental zenith radiance as a function of solar elevation and experimental values ofLz calculated in 5 intervals of as a (points). (a) logarithmic scale (b) non-logarithmic scale.
3.2. Dependence of zenith radiance on solar elevation for the different types of sky considered
3.3. Relationships among the zenith radiance of different types of sky
All 21,537 data observed have been grouped into five different categories corresponding to each sky type. In principle, we can expect that the behavior of the zenith radiance will be influenced by the sky type, as it affects the angular distribution of the radiance on the celestial hemisphere. Fig. 4 includes the values of the zenith radiance observed in each of the sky types, versus solar elevation, on a logarithmic scale. As in Section 3.1., in order to determine the underlying trend we applied the moving average method and calculated the mean Lz values. The results are shown in Fig. 5a through e. In these figures we have represented also the curves resulting from the 5th degree polynomial fit from the regression analysis of the series represented in Fig. 5f. The latter makes it easier to compare the trends of the different types of sky. Overcast and clear skies are the ones with the lowest zenith radiance. Intermediate and near overcast skies show a very similar behavior in relation to zenith radiance and in fact the curves corresponding to their average values (see Fig. 5f) practically coincide. If we observe the trend lines, from overcast sky to clear sky, we can notice how the convex shape (with a slight change in curvature for high solar elevation values in the case of overcast skies) evolves, with the radius of curvature increasing to the point where for the clear skies there is an obvious change in curvature for as beyond 20 . Although we tried different degree polynomial fits, the 5th degree was the one that provided the best results. The values of the coefficients of the different terms (see Eq. (2) as well as the correlation coefficient for the different types of sky are displayed in Table 3. When the adjustment is made on the raw data observed (Table 3 data c), we notice an improvement in the polynomial model when the observations are grouped into different sky types, as the coefficients of determination (R) range from 0.76 for overcast skies to 0.87 for intermediate skies, and therefore always higher than the one referred to in Section 3.1 of 0.7406 for the sample that includes all the sky types together. If the adjustments are made on the filtered data using the moving average method or on the Lz values, the correlation coefficients are very similar to the ones obtained for the observations as a whole, proving that the 5th degree polynomial trend is kept in each of the sky types.
We have represented the Lz values obtained for each type of sky as a function of those obtained for each of the remaining skies. The trend lines that correspond to the best fit in each case are expressed analytically below, in Eqs. (6) through (25). As a sample, we have included in Fig. 6 the adjustments that behave best in each type of sky.
Lz ðover:Þ ¼ 4:08097 þ 0:461762 Lz ðn:over:Þ R2 ¼ 0:9829
(6)
Lz ðover:Þ ¼ 2:49529 þ 0:88892 Lz ðinterm:Þ 0:0048042 Lz 2 ðinterm:Þ þ1:1813 105 Lz 3 ðinterm:Þ
R2 ¼ 0:9865
(7)
Lz ðover:Þ ¼ 3:89162 þ 1:41726 Lz ðn:clearÞ 0:011126Lz 2 ðn:clearÞ þ3:26 105 Lz 3 ðn:clearÞ
R2 ¼ 0:9889
(8)
Lz ðover:Þ ¼ 8:68715 þ 3:04849 Lz ðclearÞ 0:0413323 Lz 2 ðclearÞ þ1:91215 104 Lz 3 ðclearÞ
R2 ¼ 0:9916
(9)
Lz ðn:over:Þ ¼ 7:20399 þ 2:12853 Lz ðover:Þ R2 ¼ 0:9829
(10)
Lz ðn:over:Þ ¼ 3:25335 þ 1:30181 Lz ðinterm:Þ 0:003242437 Lz 2 ðinterm:Þ þ4:13184 106 Lz 3 ðinterm:Þ R2 ¼ 0:9977
(11)
Lz ðn:overÞ ¼ 8:62744 þ 2:42374 Lz ðn:clearÞ 0:014748Lz 2 ðn:clearÞ þ3:6749 105 Lz 3 ðn:clearÞ R2 ¼ 0:9968
(12)
Lz ðn:over:Þ ¼ 20:4594 þ 5:68854Lz ðclearÞ 0:0658383Lz 2 ðclearÞ þ2:6724 104 Lz 3 ðclearÞ
R2 ¼ 0:9952
(13)
Table 1 Results of the 6th degree polynomial fit for the data filtered using a moving average method and for the mean values of as in 5 intervals.
Moving average Mean at every 5 as interval
a1
a2
a3
a4
a5
a6
a7
R2
0.16544 0.21238
0.14298 0.13029
8.442E-03 7.288E-03
2.981E-04 2.499E-04
5.670E-06 4.660E-06
5.430E-08 4.403E-08
2.044E-10 1.642E-10
0.9937 0.9994
J.L. Torres et al. / Renewable Energy 35 (2010) 830–838 Table 2 Results of the exponential model fit for the data filtered using a moving average method and for the mean values of as in 5 intervals.
þ2:3374 105 Lz 3 ðn:clearÞ R2 ¼ 0:9991
(16)
R2 ¼ 0.9747 R2 ¼ 0.9894
Lz ¼ exp(2.65722 þ 0.03487 as) Lz ¼ expð2:66982 þ 0:03449$as Þ
Moving Average Mean at every 5 interval of as
833
Lz ðinterm:Þ ¼ 16:2304 þ 4:79088Lz ðclearÞ 0:036003Lz 2 ðclearÞ þ1:0389 104 Lz 3 ðclearÞ R2 ¼ 0:9970
Lz ðinterm:Þ ¼ 7:81346 þ 0:053968Lz 2 ðover:Þ 2:97763 104 Lz 3 ðover:Þ R2 ¼ 0:9759
(14)
Lz ðn:clearÞ ¼ 21:2379 1:81158Lz ðover:Þ þ0:077747Lz 2 ðover:Þ
Lz ðinterm:Þ ¼ 5:81202 þ 0:57702Lz ðn:over:Þ
3:9554 104 Lz 3 ðover:Þ
þ0:0039Lz 2 ðn:over:Þ R2 ¼ 0:9961
R2 ¼ 0:9595
(15)
Lz ðinterm:Þ ¼ 6:14372 þ 2:06885Lz ðn:clearÞ
a
(19)
b Experimental zenith radiance
Experimental zenith radiance
(18)
Lz ðn:clearÞ ¼ 10:1977þ0:006184Lz 2 ðn:over:Þ R2 ¼ 0:9904
0:009094Lz 2 ðn:clearÞ
100 50
10 5
1
0
10
20
30
40
50
60
100 50
10 5
1 0
70
10
Solar elevation in degrees
20
30
40
50
60
70
60
70
Solar elevation in degrees
c
d Experimental zenith radiance
Experimental zenith radiance
(17)
100
10
1
0.1 0
10
20
30
40
50
60
100 50
10 5
1
0
70
10
20
30
40
50
Solar elevation in degrees
Solar elevation in degrees
Experimental zenith radiance
e
100 50
10 5
1 0
10
20
30
40
50
60
70
Solar elevation in degrees Fig. 4. Values of Lz (Wm2 sr1) as a function of as on a logarithmic scale and for different sky conditions: (a) Overcast. (b) Near overcast. (c) Intermediate. (d) Near clear. (e) Clear.
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J.L. Torres et al. / Renewable Energy 35 (2010) 830–838
Fig. 5. Moving average of experimental zenith radiance (jagged dark line), Lz (points) and 5th degree polynomial fit (smooth grey line) as a function of solar elevation for different sky types: (a) Overcast. (b) Near overcast. (c) Intermediate. (d) Near clear. (e) Clear. (f) Summary of the polynomial fit curves for the different types of sky: (a) Overcast. (b) Near overcast. (c) Intermediate.(d) Near clear. (e) Clear.
Table 3 Coefficients of the 5th degree polynomial fit and correlation coefficient for the different types of sky. Data
a1
a2
a3
a4
a5
a
0.11228 0.20722 0.03292
0.13879 0.12359 0.12850
0.00622 0.00536 0.00531
1.5811E-04 1.3596E-04 1.2709E-04
2.0236E-06 1.7607E-06 1.5559E-06
1.0080E-08 8.9102E-09 7.4955E-09
0.9935 0.9971 0.5784
0.44484 0.47445 0.24831
0.11024 0.10369 0.13073
0.00421 0.00366 0.00546
1.0178E-04 8.1389E-05 1.3803E-04
1.2980E-06 9.6520E-07 1.7825E-06
6.5456E-09 4.5824E-09 8.9242E-09
0.9979 0.9997 0.7132
0.32265 0.41732 0.23862
0.12727 0.10579 0.13041
0.00515 0.00354 0.00542
1.2274E-04 7.0444E-05 1.3114E-04
-1.4726E-06 -7.1104E-07 -1.5761E-06
6.9320E-09 2.8445E-09 7.3377E-09
0.9976 0.9995 0.7594
a b c
0.35364 0.45262 0.18797
0.09613 0.07450 0.11619
0.00346 0.00188 0.00487
7.6772E-05 2.5665E-05 1.1557E-04
8.4689E-07 9.2711E-08 1.2958E-06
3.6923E-09 -4.489E-10 5.5010E-09
0.9960 0.9988 0.7289
a
0.25585 0.37053 0.16324
0.10005 0.07908 0.10066
0.00508 0.00366 0.00521
1.4381E-04 9.9683E-05 1.5041E-04
1.9170E-06 1.2775E-06 2.0277E-06
9.7737E-09 6.2888E-09 1.0400E-08
0.9888 0.9986 0.6137
b c
Near Overcast (n.over.)
a b c
Intermediate (interm.)
a b c
Near clear (n.clear)
Clear (clear)
b c
a b c
Data filtered using a moving average. Lz Values. All data observed.
a6
R2
Sky types Overcast (over.)
80
Mean zenith radiance for near over cast skies
Mean zenith radiance for overcast skies
J.L. Torres et al. / Renewable Energy 35 (2010) 830–838
60
40
20
150
100
50
0
0 0
20
40
60
80
100
0
120
50
100
150
200
Mean zenith radiance for intermediate skies
Mean zenith radiance for clear skies
200
Mean zenith radiance for near clear skies
Mean zenith radiance for intermediate skies
835
150
100
50
150
100
0
50
0 0
50
100
150
0
Mean zenith radiance for near clear skies
20
40
60
80
100
120
Mean zenith radiance for clear skies
120
Mean zenith radiance for clear skies
100 80 60 40 20 0 0
50
100
150
Mean zenith radiance for near over cast skies Fig. 6. Lz values of one type of sky versus another and trend curves of the best fits.
Lz ðclearÞ ¼ 8:09769 þ 0:002126Lz 2 ðinterm:Þ R2 ¼ 0:9486
Lz ðn:clearÞ ¼ 2:14168 þ 0:50279Lz þ 0:00171Lz 2 ðinterm:Þ R2 ¼ 0:9981
(20)
þ2:1272 105 Lz 3 ðn:clearÞ
6:198 105 Lz 3 ðclearÞ
R2 ¼ 0:9912 (21)
Lz ðclearÞ ¼ 11:5714 0:582437Lz ðover:Þ þ0:02014Lz 2 ðover:Þ R2 ¼ 0:9763
(22)
Lz ðclearÞ ¼ 0:65462 þ 0:51731Lz ðn:over:Þ 0:006757Lz 2 ðn:over:Þ þ4:4455 105 Lz 3 ðn:over:Þ R2 ¼ 0:9986
Lz ðclearÞ ¼ 0:23577 þ 0:6283Lz ðn:clearÞ 0:004181Lz 2 ðn:clearÞ
Lz ðn:clearÞ ¼ 7:33078 þ 2:53207Lz ðclearÞ
R2 ¼ 0:9989
(24)
(23)
(25)
We can observe that correlation coefficients are high in all cases, and when we examine the relation between the Lz of two different sky types, this relation is linear in the case of overcast and near overcast skies, quadratic in the cases of clear and overcast, intermediate and near overcast, near clear and near overcast, near clear and intermediate, and clear and intermediate, and cubic in the 13 remaining possibilities. The comparison of the empirical values of Lz with the ones obtained applying Eq. (6) through (25) has allowed us to compute two other statistics of interest when assessing the accuracy of the models: the relative mean bias difference (MBDr) and the relative root mean square difference (RMSDr) expressed by Eqs. (26) and (27).
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J.L. Torres et al. / Renewable Energy 35 (2010) 830–838
Table 4 Percentual RMSDr of the adjustment between each two sky types. e.g., when we use the adjustment described by Eq (10), RMSDr is 7.8%. Overcast Overcast Near overcast Intermediate Near clear Clear
MBDr ¼
RMSDr ¼
Intermediate
7.1 7.8 10.7 15.9 13.35
1=yexp
Near overcast
6.34 2.87
4.32 7.78 3.22
X
yc yexp
3.45 10.75
Near clear
Clear
5.75 3.37 2.09
5.01 4.14 3.76 2.67
4. Month–hour distribution of mean zenith radiance for different sky conditions
8.13
. N
(26)
h X 2 i1=2 yc yexp =N 1=yexp $
the initial data used are Lz (clear skies) and 2.35% when we use a combination of those corresponding to the remaining types of sky. Lastly, the lowest RMSDr in the case of clear skies, 3.22%, is obtained when its Lz is calculated from those of the near overcast skies.
(27)
where yc is the value of the variable calculated with the equation; yexp is the empirical value of the variable; N is the number of observations; and yexp is the mean of yexp. The MBDr values obtained are very small in all cases, lower than 1010. The RMSDr values corresponding to all possible correlations between every two sky types are displayed in Table 4. Once the dependency between theLz of every two types of sky has been completely established, we can proceed to develop multiple correlations between the Lz of each type of sky and those of the remaining to determine if the combined use of different types of sky may improve the accuracy of the prediction of Lz . For each type of sky, we start by establishing the correlation with all the other types and analyzing the Tstat of the coefficients that correspond to each of the types of sky, reducing the number of types of sky involved in the correlation so that, while keeping the correlation coefficient, the number of types of sky involved is smaller. We have calculated a total of 10 multiple correlations using this procedure. The best are those summarized in Table 5. In this way, for example, theLz corresponding to clear skies can be obtained from those of three other types of sky, according to Eq. (28).
Using Eq. (5) and the equations implicitly considered in Table 3 (line b), the Lz values can be estimated as a function of as for every hour of the representative day of each month [16] and the contour lines of equal month–hour radiance can be displayed as in Fig. 7. The values of solar elevation that must be input in the equations are taken at the times shown on the left axis of the contour line graphs. Fig. 7 provides a quick estimate of the zenith radiance at any time of the representative days of the different months of the year, for each of the five types of sky, Fig. 7a through e., and for all sky types, Fig. 7f. 5. Conclusions From zenith radiance data measured in Pamplona every 10 min, we have been able to confirm that the dependence between zenith radiance and solar elevation can be expressed analytically by a 5th order polynomial between log(Lz) and as. This is the best type of fit in every case, that is, (a) when all the data are considered; (b) when the data are filtered using a moving average method; (c) when the analysis is done using mean Lz values, Lz , in 5 intervals of as; and (d) in each and every one of the different types of sky separately. In this regard, this conclusion agrees with that reached by Soler and Gopinathan [12] for the luminance in all types of sky as a whole and in the cloudless and cloudy skies separately in Madrid, which may lead to think that 5th degree polynomial fits can be extended to other locations. For the data obtained in Pamplona, we can draw the following conclusions with regards to obtaining Lz for a certain type of sky as a function of the values for other types of sky:
Lz ðclearÞ ¼ 0:89683 þ 1:2574Lz ðover:Þ þ 0:87147Lz ðn:over:Þ (28)
þ0:78868Lz ðn:clearÞ
Once again, the values of the MBDr computed are very small, below 1010 in all cases. The RMSDr values corresponding to the best multiple correlations between one type of sky and some of the others have been included in the last column of Table 5. The results show how the lowest RMSDr of Lz (overcast skies) is the one obtained when it is inferred from Lz (clear skies), 5.01%, or Lz (near overcast skies),Lz (near clear skies), and Lz (clear skies) together, 5.1%. The lowest RMSDr for Lz (near overcast skies) is 2.87% when derived from Lz (intermediate skies) and 2.5% when obtained from the combination of Lz (overcast skies), Lz (intermediate skies) and Lz (near clear skies). The lowest RMSDr for Lz (intermediate skies) is obtained when its values are estimated from a combination of Lz (near overcast skies) and Lz (near clear skies). In the case of Lz (near clear skies), the lowest RMSDr is close to 2.5%; 2.67% when
For overcast skies, Lz can be calculated with the highest accuracy from the Lz of clear skies (RMSDr ¼ 5.01%). For near overcast skies, Lz can be calculated with a slightly higher accuracy when a combination of the Lz of overcast, intermediate and near clear skies is used (RMSDr ¼ 2.5%) than when it is obtained only from values of intermediate skies (RMSDr ¼ 2.87%). However, the difference between RMSDr in both cases is very small, so it is recommended, due to its higher simplicity, to obtain Lz for near overcast skies from the Lz of intermediate skies. For intermediate skies, Lz can be calculated with the highest accuracy from the Lz of near overcast and near clear skies (RMSDr ¼ 1.37%). For near clear skies, Lz can be calculated with a similar accuracy from the Lz of clear skies alone (RMSDr ¼ 2.67%) or from those of all sky types except, of course, near clear skies
Table 5 Best multiple correlations of Lz for each type of sky. Lz (over.) Lz (over.) Lz (n.over.) Lz (interm.) Lz (n.clear) Lz (clear)
2.16322 0.95595 0.66064 0.97234 0.89683
Lz (n.over.)
Lz (interm.)
Lz (n.clear)
Lz (clear)
R2
RMSDr (%)
0.38001
1.532
0.31461 0.94074 0.63139
0.9912 0.9982 0.9996 0.9991 0.9834
5.10 2.50 1.37 2.35 11.17
0.58842 0.18555 0.34752 1.2574
0.57695 0.48761 0.87147
1.2679
0.23531 0.78868
J.L. Torres et al. / Renewable Energy 35 (2010) 830–838
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Fig. 7. Equal mean zenith radiance contours. (a) Overcast skies. (b) Near overcast skies. (c) Intermediate skies (d) Near clear skies. (e) Clear skies. (f) All skies. Radiance in Wm2 sr1.
838
J.L. Torres et al. / Renewable Energy 35 (2010) 830–838
(RMSDr ¼ 2.35%). Again, it is recommended to use the first option, which is easier and has nearly the same accuracy. For clear skies, Lz can be calculated from the values of near overcast skies with an RMSDr of 3.22%. Lz can still be calculated, even though there is no association between the zenith radiance values and the different sky types. In this case, Lz measurements should be available for a particular site together with global and diffuse irradiance values. Firstly, global and diffuse irradiance become the same in overcast skies, and Lz andLz can be easily determined for this type of sky. Secondly, Eqs. (10), (14), (18) and (22) allow estimating Lz for the rest of sky conditions as a function of Lz values corresponding to overcast skies. We have obtained contour lines that allow for a quick estimation of the zenith radiance at different times of the representative days of the twelve months of the year for each sky type. Acknowledgements This work has been performed as a part of Project ENE200764413/ALT, financed by the Spanish Government. References [1] Reinhart CF, Walkenhorst O. Validation of dynamic RADIANCE-based daylighting simulations for a test office with external blinds. Energy Buildings 2001;33(7):683–97. [2] Li DHW, Cheung GHW. Study of models for predicting the diffuse irradiance on inclined surfaces. Appl Energy 2005;81:170–86.
[3] Hooper FC, Brunger AP. A model for the angular distribution of sky radiance. J Sol Energy Eng Trans ASME 1980;102(3):196–202. [4] M Perraudeau. Luminance models. National Lighting Conference and Daylighting colloquium; 1988. p. 291–2. [5] Brunger AP, Hooper FC. Anisotropic sky radiance model based on narrow field of view measurements of shortwave radiance. Sol Energy 1993;51–1: 53–64. [6] Perez R, Michalsky J, Seals R. Modeling sky luminance angular distribution for real sky conditions: experimental evaluation of existing algorithms. J Illum Eng Soc 1992;21(2):84–92. [7] Perez R, Seals R, Michalsky J. All Weather Model for sky luminance distribution-Preliminary configuration and validation. Sol Energy 1993;50-3: 235–45. [8] Igawa N, Koga Y, Matsuzawa T, Nakamura H. Models of sky radiance distribution and sky luminance distribution. Sol Energy 2004;77-2:137–57. [9] Robinson D, Stone A. Solar radiation modeling in the urban context. Sol Energy 2004;77(3):295–309. [10] Soler A, Gopinathan KK. A study of zenith luminance on Madrid cloudless skies. Sol Energy 2000;69:403–11. [11] Soler A, Gopinathan KK, Robledo L, Ruiz E. Month–hour distributions of zenith luminance and diffuse iluminance in Madrid. Energy Convers Manage 2004;45:1141–52. [12] Soler A, Gopinathan KK. Relation between zenith luminances for cloudless, partly cloudy and overcast skies at Madrid. Energy Convers Manage 2004;45:2583–90. [13] Gracia AM, Torres JL, De Blas M, .Illanes R. Comparacio´n de las distintas clasificaciones de estados de cielo empleados en diferentes modelos de distribucio´n angular de radiancia o luminancia. In: Va´zquez M, Santos JM, editors. XIV Congreso Ibe´rico y IX Iberoamericano de energı´a solar. Vigo: Universidad de Vigo; 2008. [14] Cie S 011/E. . Spacial distribution of daylight-CIE standard general Sky. International Commission on Illumination CIE; 2003. [15] Boland J, Ridley B. Models of diffuse solar fraction. In: Badescu V, editor. Modeling solar radiation at the earth’s surface. Berlin: Springer; 2008. p. 193–218. [16] Duffie JA, Beckman WA. Solar engineering of thermal processes. 2nd ed. New York: Wiley Interscience; 1991.
Contents lists available at ScienceDirect
Renewable Energy journal homepage: www.elsevier.com/locate/renene
A study of zenith radiance in Pamplona under different sky conditions J.L. Torres a, *, A. Garcı´a a, M. de Blas a, A. Gracia a, R. Illanes b a b
Department of Projects and Rural Engineering, Public University of Navarre, Campus Arrosadia, 31006 Pamplona, Navarre, Spain Department of Agricultural and Forestry Engineering, Polytechnic University of Madrid, Spain
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 December 2008 Accepted 27 August 2009 Available online 19 September 2009
Zenith radiance was measured in Pamplona (Spain) during sixteen months under different sky conditions. 5th degree polynomials that relate log(Lz) with solar elevation return the best correlations both when considering the entire dataset as well as when data are split into the five sky conditions considered. Besides, we have obtained simple relations, with high correlation coefficients and low Relative Root Mean Square Difference, to predict the values of the mean zenith radiance for a type of sky from the mean zenith radiance values of one or more of the remaining four types. Lastly, we obtained month–hour equal mean zenith radiance contours for each of the five sky types considered in the study as well as for all the skies as a whole. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Zenith radiance Sky types
1. Introduction Knowing the incident solar radiation and illumination over an inclined plane with different orientations is very important to help select suitable locations for solar collection systems, accurately estimate their energy production, and for the design of energyefficient buildings [1,2]. With the goal of improving the estimates of solar radiation and illumination and overcome the simplifications imposed by most commonly used models, for several years a significant number of research studies have focused on developing mathematical models of angular distribution of sky luminance and radiance [3–8]. With their use, it is possible to more accurately determine the existing solar illumination and radiation in complex terrains and urban environments, where we find simultaneously conditions of large variability in orientation and incline, and the presence of obstacles [9]. In the models proposed by Perez [7] and by Igawa [8], luminance or radiance (both magnitudes are directly related through the photopic curve) at any given point of the sky can be calculated from the luminance or radiance at the zenith. The interest in knowing the zenith luminance has also been stated in Refs. [10–12], in which zenith luminance in Madrid was studied under different sky conditions. In contrast with previous work, the physical variable studied here is the zenith radiance, not the zenith luminance. Besides, although the experimental data were collected in Pamplona (North of Spain), the fact that they have been arranged according to different types of sky allows extending the conclusions to other
* Corresponding author. Tel.: þ34 948 169175; fax: þ34 948 169148. E-mail address: [email protected] (J.L. Torres). 0960-1481/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2009.08.026
locations. We considered the following aspects: (1) Zenith radiance dependency on solar elevation under different types of sky, (2) the relationships, of a practical interest, that can be established among zenith radiance values under different sky conditions, and (3) the definition of curves of equal mean zenith radiance at each hour of the representative days of the twelve months of the year and under different sky conditions. 2. Experimental data We have acquired data of direct, global and diffuse irradiance, as well as zenith radiance every 10 min between April 2007 and July 2008, with a total of 21,537 valid measurements. For the irradiance measurements we have used two Kipp&Zonen CM11 pyranometers, one of them with a shadow ball attached, and a Kipp&Zonen CH 1 pyrheliometer, installed on a 2Ap 2-axis tracker/ positioner from the same manufacturer. For the zenith radiance, Lz (Wm2 sr1), we used a Sky Scanner EKO MS-321-LR. All the equipment was installed on a flat roof of one of the buildings of the Public University of Navarre in Pamplona (42.83 N, 1.6 W, 435 a.m.s.l). The sky line observed at the site can be seen in Fig. 1. The Sky Scanner has a moving head where the radiance sensor is mounted, that allows for radiance readings corresponding to the 145 positions of the celestial hemisphere recommended by CIE, the zenith position being one of them. The sensor has an aperture angle of 11 and readings have been corrected using a calibration factor that ensures that, on average, the diffuse irradiance Gd observed over a horizontal plane coincides with the value of the sum of the irradiances received from each of the 145 angular patches considered (excluded the one with the position of the sun in each moment) over the aforementioned plane (Eq. (1)).
J.L. Torres et al. / Renewable Energy 35 (2010) 830–838
and exponential (Eq. (4)) functional relations between Lz and as in order to find the best fit to the experimental data, and we have also analyzed polynomial relations between Lz and tan (as).
45 40
Elevation (º)
35
Lz ¼
30
n X
ai ais
ai are the coefficients of the different
i¼0
terms of the polynomial
25 20
logðLz Þ ¼
15
n X
(2)
ai ais
(3)
i¼0
10
Lz ¼ expðb þ c$as Þ
5 0
-180
-150
-120
-90
-60
East
-30
0
30
60
90
120
150
180
West
Azimut (º)
South
Fig. 1. Sky line of the measuring location.
Gd ¼
831
Z
L$ cos q$dU
(1)
2psr
L is the radiance from a generic patch of the celestial hemisphere (Wm2 sr1); q is the zenith angle of the normal direction to that generic patch; dU is the differential of solid angle from the horizontal to the patch (sr) Lz data have been divided in groups according to different types of sky. Several sky classifications can be found in the literature, and Ref. [13] includes a comparison among some of them. In our work we have used the classification proposed by Igawa et al. [8] that distinguishes among five types of sky: overcast, near overcast, intermediate, near clear and clear. The reasons to choose this classification are: (1) that all you need to do in order to determine the sky type is computing a simple index, the Sky Index, from data that are routinely registered in most measuring stations, and (2) that with five sky types it offers a higher sky differentiation than other classifications. Although we acknowledge that the CIE classification [14] considers a more diverse typology of up to 36 types of sky, we ruled it out in this article due to its complex selection process.
3. Zenith radiance data analysis 3.1. Dependence of zenith radiance on solar elevation for the dataset observed Fig. 2a represents Lz as a function of the solar elevation angle (as) on a logarithmic scale. We have studied polynomial Eqs. (2) and (3)
5th and 6th degree polynomials relating log(Lz) with as show very similar coefficients of determination (R) (0.7406 and 0.7408) and all their TStat coefficients (ai) have absolute values higher than 1. The 7th degree fit provides a practically insignificant improvement of R and some of the TStat of its coefficients have absolute values lower than one, meaning that we are unnecessarily increasing the polynomial degree. In order to have a parsimonious model, the 5th degree polynomial relation would be the most desirable.
log Lz ¼ 0:201189 þ 0:124783as 0:00601149 a2s þ0:000164154a3s 2:16223E 06a4s þ1:08837E 08a5s
b
500
100 50
10 5
1 0
10
20
30
40
50
Solar elevation in degrees
60
70
(5)
On the other hand, the exponential fit, also shown in Fig. 2b, results in a lower coefficient of determination (0.64) than the previously mentioned. However, in the experimental scatter plot, the trend shows a concave shape and the change in curvature for solar elevations under 15 that was observed by Soler and Gopinathan [10] in the case of luminance of cloudless skies, that was one of the reasons why these authors discarded the exponential model, cannot be detected with the naked eye. Given that in this article we are only using one predictor (as) to determine zenith radiance, in order to better establish the dependence of log(Lz) with respect to as, we have applied a moving average method, as Boland and Ridley [15] did for diffuse radiation. Besides, in the same way Soler and Gopinathan [10] did, we have obtained the mean zenith radiance values ðLz Þ in as five degree intervals. The plots corresponding to both methods are displayed in Fig. 3, both in logarithmic as well as in non-logarithmic scales, and Tables 1 and 2 show the results of the polynomial and exponential fit between log(Lz) and as, respectively. As it can be noticed, the 6th degree polynomial fit is better than the exponential, with the correlation coefficient reaching values over 0.99.
Experimental zenith radiance
Experimental zenith radiance
a
(4)
400
300
200
100
0
0
10
20
30
40
50
60
70
Solar elevation in degrees
Fig. 2. (a) Experimental values of zenith radiance Lz (logarithmic scale) against solar elevation (as) for all sky types and 6th degree polynomial fit. (b) Experimental values of Lz against as for all sky types and exponential fit.
832
J.L. Torres et al. / Renewable Energy 35 (2010) 830–838
Fig. 3. Moving average (window 100) of experimental zenith radiance as a function of solar elevation and experimental values ofLz calculated in 5 intervals of as a (points). (a) logarithmic scale (b) non-logarithmic scale.
3.2. Dependence of zenith radiance on solar elevation for the different types of sky considered
3.3. Relationships among the zenith radiance of different types of sky
All 21,537 data observed have been grouped into five different categories corresponding to each sky type. In principle, we can expect that the behavior of the zenith radiance will be influenced by the sky type, as it affects the angular distribution of the radiance on the celestial hemisphere. Fig. 4 includes the values of the zenith radiance observed in each of the sky types, versus solar elevation, on a logarithmic scale. As in Section 3.1., in order to determine the underlying trend we applied the moving average method and calculated the mean Lz values. The results are shown in Fig. 5a through e. In these figures we have represented also the curves resulting from the 5th degree polynomial fit from the regression analysis of the series represented in Fig. 5f. The latter makes it easier to compare the trends of the different types of sky. Overcast and clear skies are the ones with the lowest zenith radiance. Intermediate and near overcast skies show a very similar behavior in relation to zenith radiance and in fact the curves corresponding to their average values (see Fig. 5f) practically coincide. If we observe the trend lines, from overcast sky to clear sky, we can notice how the convex shape (with a slight change in curvature for high solar elevation values in the case of overcast skies) evolves, with the radius of curvature increasing to the point where for the clear skies there is an obvious change in curvature for as beyond 20 . Although we tried different degree polynomial fits, the 5th degree was the one that provided the best results. The values of the coefficients of the different terms (see Eq. (2) as well as the correlation coefficient for the different types of sky are displayed in Table 3. When the adjustment is made on the raw data observed (Table 3 data c), we notice an improvement in the polynomial model when the observations are grouped into different sky types, as the coefficients of determination (R) range from 0.76 for overcast skies to 0.87 for intermediate skies, and therefore always higher than the one referred to in Section 3.1 of 0.7406 for the sample that includes all the sky types together. If the adjustments are made on the filtered data using the moving average method or on the Lz values, the correlation coefficients are very similar to the ones obtained for the observations as a whole, proving that the 5th degree polynomial trend is kept in each of the sky types.
We have represented the Lz values obtained for each type of sky as a function of those obtained for each of the remaining skies. The trend lines that correspond to the best fit in each case are expressed analytically below, in Eqs. (6) through (25). As a sample, we have included in Fig. 6 the adjustments that behave best in each type of sky.
Lz ðover:Þ ¼ 4:08097 þ 0:461762 Lz ðn:over:Þ R2 ¼ 0:9829
(6)
Lz ðover:Þ ¼ 2:49529 þ 0:88892 Lz ðinterm:Þ 0:0048042 Lz 2 ðinterm:Þ þ1:1813 105 Lz 3 ðinterm:Þ
R2 ¼ 0:9865
(7)
Lz ðover:Þ ¼ 3:89162 þ 1:41726 Lz ðn:clearÞ 0:011126Lz 2 ðn:clearÞ þ3:26 105 Lz 3 ðn:clearÞ
R2 ¼ 0:9889
(8)
Lz ðover:Þ ¼ 8:68715 þ 3:04849 Lz ðclearÞ 0:0413323 Lz 2 ðclearÞ þ1:91215 104 Lz 3 ðclearÞ
R2 ¼ 0:9916
(9)
Lz ðn:over:Þ ¼ 7:20399 þ 2:12853 Lz ðover:Þ R2 ¼ 0:9829
(10)
Lz ðn:over:Þ ¼ 3:25335 þ 1:30181 Lz ðinterm:Þ 0:003242437 Lz 2 ðinterm:Þ þ4:13184 106 Lz 3 ðinterm:Þ R2 ¼ 0:9977
(11)
Lz ðn:overÞ ¼ 8:62744 þ 2:42374 Lz ðn:clearÞ 0:014748Lz 2 ðn:clearÞ þ3:6749 105 Lz 3 ðn:clearÞ R2 ¼ 0:9968
(12)
Lz ðn:over:Þ ¼ 20:4594 þ 5:68854Lz ðclearÞ 0:0658383Lz 2 ðclearÞ þ2:6724 104 Lz 3 ðclearÞ
R2 ¼ 0:9952
(13)
Table 1 Results of the 6th degree polynomial fit for the data filtered using a moving average method and for the mean values of as in 5 intervals.
Moving average Mean at every 5 as interval
a1
a2
a3
a4
a5
a6
a7
R2
0.16544 0.21238
0.14298 0.13029
8.442E-03 7.288E-03
2.981E-04 2.499E-04
5.670E-06 4.660E-06
5.430E-08 4.403E-08
2.044E-10 1.642E-10
0.9937 0.9994
J.L. Torres et al. / Renewable Energy 35 (2010) 830–838 Table 2 Results of the exponential model fit for the data filtered using a moving average method and for the mean values of as in 5 intervals.
þ2:3374 105 Lz 3 ðn:clearÞ R2 ¼ 0:9991
(16)
R2 ¼ 0.9747 R2 ¼ 0.9894
Lz ¼ exp(2.65722 þ 0.03487 as) Lz ¼ expð2:66982 þ 0:03449$as Þ
Moving Average Mean at every 5 interval of as
833
Lz ðinterm:Þ ¼ 16:2304 þ 4:79088Lz ðclearÞ 0:036003Lz 2 ðclearÞ þ1:0389 104 Lz 3 ðclearÞ R2 ¼ 0:9970
Lz ðinterm:Þ ¼ 7:81346 þ 0:053968Lz 2 ðover:Þ 2:97763 104 Lz 3 ðover:Þ R2 ¼ 0:9759
(14)
Lz ðn:clearÞ ¼ 21:2379 1:81158Lz ðover:Þ þ0:077747Lz 2 ðover:Þ
Lz ðinterm:Þ ¼ 5:81202 þ 0:57702Lz ðn:over:Þ
3:9554 104 Lz 3 ðover:Þ
þ0:0039Lz 2 ðn:over:Þ R2 ¼ 0:9961
R2 ¼ 0:9595
(15)
Lz ðinterm:Þ ¼ 6:14372 þ 2:06885Lz ðn:clearÞ
a
(19)
b Experimental zenith radiance
Experimental zenith radiance
(18)
Lz ðn:clearÞ ¼ 10:1977þ0:006184Lz 2 ðn:over:Þ R2 ¼ 0:9904
0:009094Lz 2 ðn:clearÞ
100 50
10 5
1
0
10
20
30
40
50
60
100 50
10 5
1 0
70
10
Solar elevation in degrees
20
30
40
50
60
70
60
70
Solar elevation in degrees
c
d Experimental zenith radiance
Experimental zenith radiance
(17)
100
10
1
0.1 0
10
20
30
40
50
60
100 50
10 5
1
0
70
10
20
30
40
50
Solar elevation in degrees
Solar elevation in degrees
Experimental zenith radiance
e
100 50
10 5
1 0
10
20
30
40
50
60
70
Solar elevation in degrees Fig. 4. Values of Lz (Wm2 sr1) as a function of as on a logarithmic scale and for different sky conditions: (a) Overcast. (b) Near overcast. (c) Intermediate. (d) Near clear. (e) Clear.
834
J.L. Torres et al. / Renewable Energy 35 (2010) 830–838
Fig. 5. Moving average of experimental zenith radiance (jagged dark line), Lz (points) and 5th degree polynomial fit (smooth grey line) as a function of solar elevation for different sky types: (a) Overcast. (b) Near overcast. (c) Intermediate. (d) Near clear. (e) Clear. (f) Summary of the polynomial fit curves for the different types of sky: (a) Overcast. (b) Near overcast. (c) Intermediate.(d) Near clear. (e) Clear.
Table 3 Coefficients of the 5th degree polynomial fit and correlation coefficient for the different types of sky. Data
a1
a2
a3
a4
a5
a
0.11228 0.20722 0.03292
0.13879 0.12359 0.12850
0.00622 0.00536 0.00531
1.5811E-04 1.3596E-04 1.2709E-04
2.0236E-06 1.7607E-06 1.5559E-06
1.0080E-08 8.9102E-09 7.4955E-09
0.9935 0.9971 0.5784
0.44484 0.47445 0.24831
0.11024 0.10369 0.13073
0.00421 0.00366 0.00546
1.0178E-04 8.1389E-05 1.3803E-04
1.2980E-06 9.6520E-07 1.7825E-06
6.5456E-09 4.5824E-09 8.9242E-09
0.9979 0.9997 0.7132
0.32265 0.41732 0.23862
0.12727 0.10579 0.13041
0.00515 0.00354 0.00542
1.2274E-04 7.0444E-05 1.3114E-04
-1.4726E-06 -7.1104E-07 -1.5761E-06
6.9320E-09 2.8445E-09 7.3377E-09
0.9976 0.9995 0.7594
a b c
0.35364 0.45262 0.18797
0.09613 0.07450 0.11619
0.00346 0.00188 0.00487
7.6772E-05 2.5665E-05 1.1557E-04
8.4689E-07 9.2711E-08 1.2958E-06
3.6923E-09 -4.489E-10 5.5010E-09
0.9960 0.9988 0.7289
a
0.25585 0.37053 0.16324
0.10005 0.07908 0.10066
0.00508 0.00366 0.00521
1.4381E-04 9.9683E-05 1.5041E-04
1.9170E-06 1.2775E-06 2.0277E-06
9.7737E-09 6.2888E-09 1.0400E-08
0.9888 0.9986 0.6137
b c
Near Overcast (n.over.)
a b c
Intermediate (interm.)
a b c
Near clear (n.clear)
Clear (clear)
b c
a b c
Data filtered using a moving average. Lz Values. All data observed.
a6
R2
Sky types Overcast (over.)
80
Mean zenith radiance for near over cast skies
Mean zenith radiance for overcast skies
J.L. Torres et al. / Renewable Energy 35 (2010) 830–838
60
40
20
150
100
50
0
0 0
20
40
60
80
100
0
120
50
100
150
200
Mean zenith radiance for intermediate skies
Mean zenith radiance for clear skies
200
Mean zenith radiance for near clear skies
Mean zenith radiance for intermediate skies
835
150
100
50
150
100
0
50
0 0
50
100
150
0
Mean zenith radiance for near clear skies
20
40
60
80
100
120
Mean zenith radiance for clear skies
120
Mean zenith radiance for clear skies
100 80 60 40 20 0 0
50
100
150
Mean zenith radiance for near over cast skies Fig. 6. Lz values of one type of sky versus another and trend curves of the best fits.
Lz ðclearÞ ¼ 8:09769 þ 0:002126Lz 2 ðinterm:Þ R2 ¼ 0:9486
Lz ðn:clearÞ ¼ 2:14168 þ 0:50279Lz þ 0:00171Lz 2 ðinterm:Þ R2 ¼ 0:9981
(20)
þ2:1272 105 Lz 3 ðn:clearÞ
6:198 105 Lz 3 ðclearÞ
R2 ¼ 0:9912 (21)
Lz ðclearÞ ¼ 11:5714 0:582437Lz ðover:Þ þ0:02014Lz 2 ðover:Þ R2 ¼ 0:9763
(22)
Lz ðclearÞ ¼ 0:65462 þ 0:51731Lz ðn:over:Þ 0:006757Lz 2 ðn:over:Þ þ4:4455 105 Lz 3 ðn:over:Þ R2 ¼ 0:9986
Lz ðclearÞ ¼ 0:23577 þ 0:6283Lz ðn:clearÞ 0:004181Lz 2 ðn:clearÞ
Lz ðn:clearÞ ¼ 7:33078 þ 2:53207Lz ðclearÞ
R2 ¼ 0:9989
(24)
(23)
(25)
We can observe that correlation coefficients are high in all cases, and when we examine the relation between the Lz of two different sky types, this relation is linear in the case of overcast and near overcast skies, quadratic in the cases of clear and overcast, intermediate and near overcast, near clear and near overcast, near clear and intermediate, and clear and intermediate, and cubic in the 13 remaining possibilities. The comparison of the empirical values of Lz with the ones obtained applying Eq. (6) through (25) has allowed us to compute two other statistics of interest when assessing the accuracy of the models: the relative mean bias difference (MBDr) and the relative root mean square difference (RMSDr) expressed by Eqs. (26) and (27).
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J.L. Torres et al. / Renewable Energy 35 (2010) 830–838
Table 4 Percentual RMSDr of the adjustment between each two sky types. e.g., when we use the adjustment described by Eq (10), RMSDr is 7.8%. Overcast Overcast Near overcast Intermediate Near clear Clear
MBDr ¼
RMSDr ¼
Intermediate
7.1 7.8 10.7 15.9 13.35
1=yexp
Near overcast
6.34 2.87
4.32 7.78 3.22
X
yc yexp
3.45 10.75
Near clear
Clear
5.75 3.37 2.09
5.01 4.14 3.76 2.67
4. Month–hour distribution of mean zenith radiance for different sky conditions
8.13
. N
(26)
h X 2 i1=2 yc yexp =N 1=yexp $
the initial data used are Lz (clear skies) and 2.35% when we use a combination of those corresponding to the remaining types of sky. Lastly, the lowest RMSDr in the case of clear skies, 3.22%, is obtained when its Lz is calculated from those of the near overcast skies.
(27)
where yc is the value of the variable calculated with the equation; yexp is the empirical value of the variable; N is the number of observations; and yexp is the mean of yexp. The MBDr values obtained are very small in all cases, lower than 1010. The RMSDr values corresponding to all possible correlations between every two sky types are displayed in Table 4. Once the dependency between theLz of every two types of sky has been completely established, we can proceed to develop multiple correlations between the Lz of each type of sky and those of the remaining to determine if the combined use of different types of sky may improve the accuracy of the prediction of Lz . For each type of sky, we start by establishing the correlation with all the other types and analyzing the Tstat of the coefficients that correspond to each of the types of sky, reducing the number of types of sky involved in the correlation so that, while keeping the correlation coefficient, the number of types of sky involved is smaller. We have calculated a total of 10 multiple correlations using this procedure. The best are those summarized in Table 5. In this way, for example, theLz corresponding to clear skies can be obtained from those of three other types of sky, according to Eq. (28).
Using Eq. (5) and the equations implicitly considered in Table 3 (line b), the Lz values can be estimated as a function of as for every hour of the representative day of each month [16] and the contour lines of equal month–hour radiance can be displayed as in Fig. 7. The values of solar elevation that must be input in the equations are taken at the times shown on the left axis of the contour line graphs. Fig. 7 provides a quick estimate of the zenith radiance at any time of the representative days of the different months of the year, for each of the five types of sky, Fig. 7a through e., and for all sky types, Fig. 7f. 5. Conclusions From zenith radiance data measured in Pamplona every 10 min, we have been able to confirm that the dependence between zenith radiance and solar elevation can be expressed analytically by a 5th order polynomial between log(Lz) and as. This is the best type of fit in every case, that is, (a) when all the data are considered; (b) when the data are filtered using a moving average method; (c) when the analysis is done using mean Lz values, Lz , in 5 intervals of as; and (d) in each and every one of the different types of sky separately. In this regard, this conclusion agrees with that reached by Soler and Gopinathan [12] for the luminance in all types of sky as a whole and in the cloudless and cloudy skies separately in Madrid, which may lead to think that 5th degree polynomial fits can be extended to other locations. For the data obtained in Pamplona, we can draw the following conclusions with regards to obtaining Lz for a certain type of sky as a function of the values for other types of sky:
Lz ðclearÞ ¼ 0:89683 þ 1:2574Lz ðover:Þ þ 0:87147Lz ðn:over:Þ (28)
þ0:78868Lz ðn:clearÞ
Once again, the values of the MBDr computed are very small, below 1010 in all cases. The RMSDr values corresponding to the best multiple correlations between one type of sky and some of the others have been included in the last column of Table 5. The results show how the lowest RMSDr of Lz (overcast skies) is the one obtained when it is inferred from Lz (clear skies), 5.01%, or Lz (near overcast skies),Lz (near clear skies), and Lz (clear skies) together, 5.1%. The lowest RMSDr for Lz (near overcast skies) is 2.87% when derived from Lz (intermediate skies) and 2.5% when obtained from the combination of Lz (overcast skies), Lz (intermediate skies) and Lz (near clear skies). The lowest RMSDr for Lz (intermediate skies) is obtained when its values are estimated from a combination of Lz (near overcast skies) and Lz (near clear skies). In the case of Lz (near clear skies), the lowest RMSDr is close to 2.5%; 2.67% when
For overcast skies, Lz can be calculated with the highest accuracy from the Lz of clear skies (RMSDr ¼ 5.01%). For near overcast skies, Lz can be calculated with a slightly higher accuracy when a combination of the Lz of overcast, intermediate and near clear skies is used (RMSDr ¼ 2.5%) than when it is obtained only from values of intermediate skies (RMSDr ¼ 2.87%). However, the difference between RMSDr in both cases is very small, so it is recommended, due to its higher simplicity, to obtain Lz for near overcast skies from the Lz of intermediate skies. For intermediate skies, Lz can be calculated with the highest accuracy from the Lz of near overcast and near clear skies (RMSDr ¼ 1.37%). For near clear skies, Lz can be calculated with a similar accuracy from the Lz of clear skies alone (RMSDr ¼ 2.67%) or from those of all sky types except, of course, near clear skies
Table 5 Best multiple correlations of Lz for each type of sky. Lz (over.) Lz (over.) Lz (n.over.) Lz (interm.) Lz (n.clear) Lz (clear)
2.16322 0.95595 0.66064 0.97234 0.89683
Lz (n.over.)
Lz (interm.)
Lz (n.clear)
Lz (clear)
R2
RMSDr (%)
0.38001
1.532
0.31461 0.94074 0.63139
0.9912 0.9982 0.9996 0.9991 0.9834
5.10 2.50 1.37 2.35 11.17
0.58842 0.18555 0.34752 1.2574
0.57695 0.48761 0.87147
1.2679
0.23531 0.78868
J.L. Torres et al. / Renewable Energy 35 (2010) 830–838
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Fig. 7. Equal mean zenith radiance contours. (a) Overcast skies. (b) Near overcast skies. (c) Intermediate skies (d) Near clear skies. (e) Clear skies. (f) All skies. Radiance in Wm2 sr1.
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J.L. Torres et al. / Renewable Energy 35 (2010) 830–838
(RMSDr ¼ 2.35%). Again, it is recommended to use the first option, which is easier and has nearly the same accuracy. For clear skies, Lz can be calculated from the values of near overcast skies with an RMSDr of 3.22%. Lz can still be calculated, even though there is no association between the zenith radiance values and the different sky types. In this case, Lz measurements should be available for a particular site together with global and diffuse irradiance values. Firstly, global and diffuse irradiance become the same in overcast skies, and Lz andLz can be easily determined for this type of sky. Secondly, Eqs. (10), (14), (18) and (22) allow estimating Lz for the rest of sky conditions as a function of Lz values corresponding to overcast skies. We have obtained contour lines that allow for a quick estimation of the zenith radiance at different times of the representative days of the twelve months of the year for each sky type. Acknowledgements This work has been performed as a part of Project ENE200764413/ALT, financed by the Spanish Government. References [1] Reinhart CF, Walkenhorst O. Validation of dynamic RADIANCE-based daylighting simulations for a test office with external blinds. Energy Buildings 2001;33(7):683–97. [2] Li DHW, Cheung GHW. Study of models for predicting the diffuse irradiance on inclined surfaces. Appl Energy 2005;81:170–86.
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