Review and statistical analysis of different global solar radiation sunshine models

Renewable and Sustainable Energy Reviews 52 (2015) 1869–1880

Contents lists available at ScienceDirect

Renewable and Sustainable Energy Reviews journal homepage: www.elsevier.com/locate/rser

Review and statistical analysis of different global solar radiation sunshine models Milan Despotovic a,n, Vladimir Nedic a, Danijela Despotovic b, Slobodan Cvetanovic c a

Faculty of Engineering, University of Kragujevac, Kragujevac, Serbia Faculty of Economics, University of Kragujevac, Kragujevac, Serbia c Faculty of Economics, University of Nis, Nis, Serbia b

art ic l e i nf o

a b s t r a c t

Article history: Received 17 June 2015 Received in revised form 9 August 2015 Accepted 18 August 2015

For the optimal design and selection of solar energy conversion systems, as well as for other fields of interest, such as architecture, agriculture, hydrology and ecology, the knowledge of accurate global solar radiation data is extremely important. However, due to the cost and difficulty in solar radiation measurements these data are not easily available for many countries. Therefore many empirical models have been developed by various researchers to predict global solar radiation from readily available data. The number of developed models is relatively high, which makes it difficult to choose the most appropriate one for a particular purpose and site. There are several studies in which authors evaluate different models for specific location. However, there is no comprehensive study in which these models are evaluated in case of global use. The main objective of this study is to evaluate different solar radiation models on global scale, which might be helpful in the selection of the most appropriate and accurate model based on the available sunshine data. Using the radiation data corresponding to 924 sites throughout the world we conducted a detailed statistical analysis of 101 different solar radiation models that are available in literature. Ten statistical indicators were used to assess models performance. In addition, we introduced specific global performance indicator (GPI), by means of which all analyzed models are depicted with a single parameter and easily ranked. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Solar energy Global solar radiation Solar radiation models Model comparison

Contents 1. Introduction . . . . . . . . . . 2. Theoretical background . 3. Models used. . . . . . . . . . 4. Meteorological data . . . . 5. Statistical evaluation . . . 6. Results and discussion. . 7. Conclusion . . . . . . . . . . . Acknowledgment . . . . . . . . . References . . . . . . . . . . . . . . .

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1869 1870 1872 1872 1874 1877 1878 1879 1879

1. Introduction

Abbreviation: erMAX, maximum absolute relative error; MAE, mean absolute error (kWh/m2); MARE, mean absolute relative error; MBE, mean bias error (kWh/ 2 m ); RMSE, root mean squared error (kWh/m2); RMSRE, root mean squared relative error; RRMSE, relative root mean squared error (%) n Corresponding author. Tel.: þ 381 69 844 9679. E-mail address: [email protected] (M. Despotovic). http://dx.doi.org/10.1016/j.rser.2015.08.035 1364-0321/& 2015 Elsevier Ltd. All rights reserved.

Renewable energy sources such as solar energy have great potential to mitigate some negative environmental issues including climate change caused by intensive fossil fuel exploitation. With fast technological improvement and decreasing costs, solar energy will surely play a relevant share of future energy systems [1]. In the prediction, study, and design of solar energy systems,

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Nomenclature Greek symbols

δ ωss ϕ ψ

solar declination (○ ) sunset hour angle (○ ) latitude (○ ) longitude (○ )

Roman symbols n H i;c H i;m H m;avg

total number of observations ith calculated value (kWh/m2) ith measured value (kWh/m2) average of the measured value (kWh/m2)

data on solar radiation and its components at a particular location are very essential. The most critical input parameter employed in the design and prediction of the performance of a solar energy device at particular site is the global solar radiation on the horizontal surface [2]. Solar irradiation can be evaluated by processing images from satellites [3,4] or by on-ground measurements with pyranometers in meteorological stations. Satellite radiation measurements are not precise in estimating ground solar radiation since they require atmospheric models to estimate the solar radiation at ground level [5]. While processing satellite data provides less accurate values compared with ground measurements, it has the advantage of having much greater spatiotemporal coverage [6,7]. At locations outside the vicinity of a ground stations satellite-based methods generate more accurate solar radiation estimates than classical interpolation methods, which makes satellite approaches the best option for constructing accurate maps, especially over large areas [4]. On the other hand, although measurement of solar radiation is reliable, high cost of measuring equipment and the requirement of specific and periodic calibrations cause that solar radiation measurements are not easily available. This particularly pertains developing world, since the stations where solar irradiation is measured are mainly concentrated in developed countries [8,9]. For the designers and manufactures of solar equipment but also for other professionals, such as architects and agriculturists, it is very important to have methods to estimate the solar radiation based on readily available meteorological data. The most commonly used parameter in several empirical models which have been used to calculate solar radiation is sunshine duration because it can be easily and reliably measured, and data are widely available [10,11]. Other used parameters are air temperature [12,13], precipitation [14,15], relative humidity [16,17], cloudiness [18–20] and air pollution [21]. The first model for estimating monthly average daily global solar radiation (H) was proposed by Ångström [22], who derived a linear relationship between the ratio of average daily global radiation to the clear day radiation (H c ) at a given location and the ratio of average daily sunshine duration (S) to the maximum possible sunshine duration (S0 ). Prescott [23] modified Ångström relationship by replacing clear sky radiation with extraterrestrial radiation (H 0 ), which can be more readily determined. This replacement resulted in the formation of the Ångström–Prescott model, which has been the most widely used correlation for calculation of global solar radiation. Later other sunshine models appeared which are based on quadratic, cubic, logarithmic or exponential relationship [24]. The problem with these models is that they are location specific and require careful long-term measurements of sunshine duration and

y~ j H h H0 Ic Nd R2 S S0 yij t-stat U95

median of scaled values of indicator j monthly average daily global radiation (Wh/m2) altitude (km) monthly average daily extraterrestrial radiation (Wh/ m2 ) solar constant ð ¼ 1367 W=m2 Þ the number of the day corresponding to a given date R squared monthly average daily sunshine duration (h) monthly average maximum possible daily sunshine duration (h) scaled value of indicator j for model i t-Statistic Uncertainty at 95% (kWh/m2)

global radiation [25]. On the other side, the number of such methods that have been developed for different locations is relatively high, which makes it difficult to choose the most appropriate correlation for a particular purpose and site [26]. There are several studies in which authors evaluate different models for specific location [27–30]. However, there is no comprehensive study in which these models are evaluated in case of global use. The main objective of this study is to evaluate different solar radiation models on global scale, which might be helpful in the selection of the most appropriate and accurate model based on the available meteorological data. Using the radiation data corresponding to 924 sites throughout the world we conducted a detailed statistical analysis of 101 different solar radiation models that are available in literature. Ten statistical indicators were used to assess models’ performance. In addition, we introduced specific global performance indicator (GPI), by means of which all analyzed models are depicted with a single parameter and easily ranked.

2. Theoretical background The monthly average daily extraterrestrial solar radiation on a horizontal surface is calculated from the following equation [31]:  24I c
H0 ¼ 1 þ 0:034 cos ð360N d =365Þ π   2πωss sin ϕ sin δ  cos ϕ cos δ sin ωss þ 360

ð1Þ

where I c is the solar constant ð ¼ 1367 W=m2 Þ, ϕ is the latitude of the site, N d is day of the year starting from January 1st (Table 1), and δ and ωss are the monthly mean daily solar declination and sunrise hour angle given, respectively, as [32]   284 þ Nd ð2Þ δ ¼ 23:45 sin 360 365 Table 1 Recommended average day for each month according to Klein [33]. Month

Date

Nd

Month

Date

Nd

January February March April May June

17 16 16 15 15 11

17 47 75 105 135 162

July August September October November December

17 16 15 15 14 10

198 228 258 288 318 344

M. Despotovic et al. / Renewable and Sustainable Energy Reviews 52 (2015) 1869–1880

Table 2 Different sunshine radiation models.

Table 2 (continued ) No.

No.

Author

1.

Page [34]

2.

Bahel et al. [35]

3.

Louche et al. [36] Gopinathan [37]

4.

5.

Zabara [38]

6.

Rietveld [39]

7.

Gopinathan [40]

8.

Akinoğlu and Ecevit [41]

9.

Glover and McCulloch [42] Hay [43]

10. 10. 12. 13. 14. 15. 16.

17.

18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

1871

Model   H S ¼ 0:23 þ 0:48 H0 S0    2  3 H S S S ¼ 0:16 þ 0:87 þ 0:34  0:61 H0 S0 S0 S0   H S ¼ 0:206 þ 0:546 H0 S0   S a ¼ 0:265 þ 0:07h þ 0:136 S0   S b ¼ 0:401 þ 0:108h þ 0:0325 S0    2  3 S S S  1:674 þ 2:680 a ¼ 0:395  1:247 S0 S0 S0    2  3 S S S b ¼ 0:395 þ 1:384 þ 2:055  3:249 S0 S0 S0    2 H S S ¼ 0:10 þ 1:02  0:44 H0 S0 S0   S a ¼  0:309 þ 0:539 cos ϕ  0:0693h þ 0:29 S  0 S b ¼ 1:527  1:027 cos ϕ þ 0:0926h  0:359 S0    2 H S S ¼ 0:145 þ 0:845  0:280 H0 S0 S0   H S ¼ 0:29 cos ϕ þ 0:52 H0 S0 

31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.



H S ¼ 0:280 þ 0:493 H0 S0   Raja and H S ¼ 0:388 cos ϕ þ 0:367 Twidell [44] H0 S  0 Raja and H S ¼ 0:388 cos ϕ þ 0:407 Twidell [44] H0 S0     Newland [45] H S S ¼ 0:34 þ 0:40 þ 0:17 H0 S0 S    0    Dogniaux and H S S ¼ 0:3702 þ 0:00506  0:00313 ϕ þ 0:32029 Lemoine [46] H0 S0 S0   Almorox and H S ¼ 0:2170 þ 0:5453 Hontoria [47] H0 S0   Srivastava H S ¼ 0:1382 þ 0:5564 and Pandey H0 S0 [48]    2 Srivastava S S a ¼  10:533 þ 27:18  17:222 and Pandey S0 S0 [48]    2 S S b ¼ 12:098  29:395 þ 18:676 S0 S0   Gopinathan H S ¼ 0:1961 þ 0:7212 and Soler [49] H0 S0    2 Almorox and H S S Hontoria [47] H0 ¼ 0:1840 þ 0:6792 S0  0:1228 S0    2  3 Almorox and H S S S  0:3657 Hontoria [47] H0 ¼ 0:230 þ 0:3809 S0 þ 0:4694 S0 S0   Almorox and H S ¼ 0:6902 þ 0:6142 Hontoria [47] H0 S0   Almorox and H S ¼  0:0271 þ 0:3096 Hontoria [47] H0 S0    2 Ögelman et al. H S S ¼ 0:195 þ 0:676  0:142 [50] H0 S0 S0    2  3 Bakirci et al. H S S S ¼ 0:6307  0:7251  0:4633 þ 1:2089 [51] H0 S0 S0 S0    2  3 Tarhan and H S S S ¼ 0:1520 þ 1:1334 þ 0:4516  1:1126 Sari [30] H0 S0 S0 S0     Tarhan and H S S 2 ¼ 0:1874 þ 0:8592  0:4764 Sari [30] H0 S0 S0   Aras et al. [52] H S ¼ 0:3078 þ 0:4166 H0 S0    2 Aras et al. [52] H S S ¼ 0:3398 þ 0:2868 þ 0:1187 H0 S0 S0      3 Aras et al. [52] H S S 2 S ¼ 0:4832  0:6161  1:0975 þ 1:8932 H0 S0 S0 S0   Ahmed and H S ¼ 0:324 þ 0:405 Ulfat [53] H0 S0

44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66.

Author

Model

   2 H S S ¼ 0:348 þ 0:320 þ 0:070 H0 S0 S0   Ulgen and H S ¼ 0:2671 þ 0:4754 Hepbasli [54] H 0 S0    2  3 Ulgen and H S S S  0:4834 Hepbasli [54] H 0 ¼ 0:2854 þ 0:2591 S0 þ 0:6171 S0 S0   Toğrul and H S ¼ 0:318 þ 0:449 Toğrul [55] H0 S   0 Toğrul and H S ¼ 0:2022ln þ 0:698 Toğrul [55] H0 S0    2 Toğrul and H S S ¼ 0:1541 þ 1:1714  0:705 Toğrul [55] H0 S0 S0    2  3 Toğrul and H S S S ¼ 0:1796 þ 0:9813  0:2657  0:2958 Toğrul [55] H0 S0 S0 S0 H Toğrul and ¼ 0:3396e0:8985ðS=S0 Þ Toğrul [55] H0  0:4146 Toğrul and H S ¼ 0:7316 Toğrul [55] H0 S0   Ulgen and H S ¼ 0:3092 cos ϕ þ 0:4931 Hepbasli [56] H 0 S0    2  3 Ulgen and H S S S  0:3708 Hepbasli [56] H 0 ¼ 0:2408 þ 0:3625 S0 þ 0:4597 S0 S0    2 Mecibah et al. H S S ¼ 0:57089 þ 0:01028  0:00005 [57] H0 S0 S0    2  3 Mecibah et al. H S S S þ 0:00028 ¼ 0:57211 þ 0:00901  0:00002 [57] H0 S0 S0 S0       2 Ertekin and H S S S 3 ¼  2:4275 þ 11:946 þ 7:9575  16:745 Yaldiz [28] H0 S0 S0 S0   Said et al. [58] H S ¼ 0:215 þ 0:527 H0 S0    2 Said et al. [58] H S S ¼ 0:1þ 0:874  0:255 H0 S0 S0    2 Aksoy [59] H S S ¼ 0:148 þ 0:668  0:079 H0 S0 S0   Tiris et al. [60] H S ¼ 0:2262 þ 0:418 H0 S  0 Veeran and H S ¼ 0:34þ 0:32 Kumar [61] H0 S  0 Veeran and H S ¼ 0:27þ 0:65 Kumar [61] H0 S  0 Lewis [62] H S ¼ 0:14þ 0:57 H0 S0    2  3 Lewis [62] H S S S ¼ 0:81 3:34  4:51 þ 7:38 H0 S0 S0 S0   Lewis [62] H S ¼ 0:12þ 0:58 þ 7:56  10  2 h H0 S0    2 Taşdemiroğlu H S S and Sever [18] H 0 ¼ 0:255 þ 0:014 S0 þ 0:001 S0    2  3 Samuel [63] H S S S ¼  0:14 þ 2:52 þ 2:24  3:71 H0 S0 S0 S0   Jain [64] H S ¼ 0:313 þ 0:474 H0 S  0 Jain [64] H S ¼ 0:307 þ 0:488 H0 S  0 Jain [64] H S ¼ 0:309 þ 0:599 H0 S  0 Raja and H S ¼ 0:335 þ 0:367 Twidell [44] H0 S  0 Luhanga and H S ¼ 0:241 þ 0:488 Andringa [65] H 0 S  0 Jain and Jain H S ¼ 0:240 þ 0:513 [66] H0 S  0 Khogali et al. H S ¼ 0:191 þ 0:571 [67] H0 S  0 Newland [45] H S ¼ 0:18þ 0:615 H0 S 0  Khogali et al. H S ¼ 0:297 þ 0:432 [67] H0 S  0 Khogali et al. H S ¼ 0:262 þ 0:454 [67] H0 S0   Alsaad [68] H S ¼ 0:174 þ 0:615 H0 S0 Ahmed and Ulfat [53]

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Table 2 (continued ) No. 67. 68. 69. 70.

Author Hinrichsen [69] Ahmad et al. [70] Jain [71]

74.

Bahel et al. [72] Srivastava et al. [73] Hinrichsen [69] Maduekwe and Chendo [74] Rehman [75]

75.

Rehman [75]

76.

Ahmad and Ulfat [76] Jin et al. [77]

71. 72. 73.

77. 78. 79. 80. 81. 82.

83.

84.

85.

86. 87. 88. 89. 90.

91. 92. 93. 94.

95.

Table 2 (continued ) Model H H0 H H0 H H0 H H0 H H0 H H0 H H0

No. 

Author

Model



S S0   S ¼ 0:458 þ 0:175 S  0 S ¼ 0:177 þ 0:692 S  0 S ¼ 0:175 þ 0:552 S0   S ¼ 0:2006 þ 0:5313 S   0 S ¼ 0:23 þ 0:38 S  0 S ¼ 0:36 þ 0:34 S0



¼ 0:22 þ 0:42

  H S ¼ 0:3465 þ 0:352 H0 S0

  H S ¼ 0:3346 þ 0:558 cos ϕ þ 0:2 cos ψ þ 0:006hþ 0:3809 H0 S0

  H S ¼ 0:324 þ 0:405 H0 S0   H S ¼ 0:1332 þ 0:6471 H0 S0    2 Jin et al. [77] H S S ¼ 0:1404 þ 0:6126 þ 0:0351 H0 S0 S0      3 Jin et al. [77] H S S 2 S ¼ 0:1275 þ 0:7251 þ 0:1837  0:2299 H0 S0 S0 S0   Jin et al. [77] H S ¼ 0:0855 þ 0:002ϕ þ 0:03h þ 0:5654 H0 S0   Jin et al. [77] H S ¼ 2:1186  2:0014 cos ϕ þ 0:0304h þ 0:5622 H0 S0 H Jin et al. [77] ¼ 0:1094 þ 0:0014ϕ þ 0:0212h H0   S þ ð0:5176 þ 0:0012ϕ þ 0:015hÞ S0 H Jin et al. [77] ¼ 1:879  1:7516 cos ϕ þ 0:0205h H0   S þ ð1:0819  0:5409 cos ϕþ 0:0169hÞ S0 H Jin et al. [77] ¼ 0:0218 þ 0:0033ϕ þ 0:0443h þ H0   S ð0:9979  0:0092ϕ  0:0852hÞ S0  2 S ð þ ð 0:5579 þ 0:012ϕþ 0:1005hÞ S0 H Jin et al. [77] ¼ 4:251  4:1878 cos ϕ þ 0:0437h H0   S þ ð  10:5774 þ 11:4512 cos ϕ  0:0832hÞ S0  2 S þ ð12:7247  13:0994 cos ϕþ 0:1hÞ S0   Katiyar and H S ¼ 0:2281 þ 0:5093 Pandey [78] H0 S  0 Li et al. [79] H S ¼ 0:2223 þ 0:6529 H0 S   0 Yohanna et al. H S ¼ 0:17 þ 0:68 [80] H0 S0    2  3 Soler [81] H S S S ¼ 0:164 þ 0:6829  0:00094  1:0727 H0 S0 S0 S0    2 Maduekwe H S S ¼ 0:18 þ 1:16  0:91 and Chendo H0 S0 S0 [74]    2 Singh et al. H S S ¼ 0:7191  1:0673 þ 1:1448 [82] H0 S0 S0    2 Toğrul and H S S ¼ 0:21521 þ 0:62487  0:2205 Onat [83] H0 S0 S0    2 Toğrul et al. H S S ¼ 0:2816 þ 0:3713  0:0174 [84] H0 S0 S0   Toğrul et al. H S ¼ 0:195 þ 0:9191 [84] H0 S0  2  3 S S þ 0:6368  1:0739 S0 S0    2 Toğrul et al. H S S ¼ 0:2646 þ 0:2445 þ 0:982 [84] H0 S0 S0

 1:8662

S S0

3

 þ 1:0649

S S0

4

   2 H S S ¼ 0:348 þ 0:320 þ 0:07 H0 S S0  0 97. Toğrul et al. H S ¼  0:0344ln þ 0:1982 [84] H0 S0      S S þ  0:0201ln þ 0:4562 S0 S0    2  3 98. Teke et al. H S S S ¼  0:0631 þ 0:4813 þ 0:4122  0:764 [85] H0 S0 S0 S0   99. Yohanna et al. H S ¼ 0:17þ 0:68 [80] H0 S0   100. Rensheng H S ¼ 0:28 0:141 cos ϕ þ 0:026h þ 0:542 et al. [86] H0 S0      3 101. Rensheng H S S 2 S ¼ 0:15þ 1:145 þ 0:963  1:474 et al. [86] H0 S0 S0 S0 96.

Ahmad and Ulfat [76]

Fig. 1. Relationship between monthly mean clearness index and monthly sunshine fraction (WRDC data).

ωss ¼ cos  1 ð  tan δ tan ϕÞ

ð3Þ

The monthly mean daily maximum possible sunshine duration is [32] 2 cos  1 ð  tan δ tan ϕÞ S0 ¼ 15

ð4Þ

3. Models used Different sunshine radiation models (101 in total) that were used for comparison are shown in Table 2.

4. Meteorological data Meteorological data used in this study for statistical evaluation of the models were taken from the World Radiation Data Centre (WRDC) [87]. WRDC serves as a central depository for solar radiation data collected from different places in the world, and provides a very large data set containing data on solar radiation, radiation balance and sunshine duration that covers the period 1955–2013. However, although these data are freely accessible through WRDC website, their collection is very troublesome since

M. Despotovic et al. / Renewable and Sustainable Energy Reviews 52 (2015) 1869–1880

Sunshine duration

1873

Global radiation 2500

2000 2000

Frequency

Frequency

1500

1000

1500

1000

500

500

0

0 0

10

20

30

40

50

60

0

10

20

Number of years

30

40

50

60

Number of years

Fig. 2. Histograms of sunshine duration and global radiation data.

Fig. 3. Locations that are not used for the study due to incomplete measurements.

Table 3 Sites that were discarded from analysis due to physically aberrant data. Country

City

Longitude

Latitude

Month

S0

H0

S/S0

H/H0

RUSSIA FINLAND FRANCE RUSSIA UAE ITALY

VERKHOYANSK SODANKYLA DUMONT D’URVILLE MIRNY SHARJAH INT.ARPT MODENA

133.4 26.6 140 93 55.5 11

67.6 67.4  66.7  66.6 25.3 44.7

12 12 6 6 1 11

0 0 1.327 1.551 10.598 9.391

0 0 6.421 10.290 6633.398 3817.978

Infinity Infinity 0.313 0.093 0.665 1.001

Infinity Infinity 4.093 2.653 1.082 0.312

only data for one year per site is possible to download in one pass. Subdirectory structure of the web database and separate data sets for sunshine duration and global radiation data additionally complicate the downloading of data. Therefore a Web crawler is written by means of which the needed data are fetched from the website, and only sites for which both radiation and sunshine duration exist were taken for analysis. These 138,674 pairs of

sunshine-radiation monthly readings are shown in Fig. 1. The same database comprising 72,984 readings from 677 cites during period from 1969 to 1993 was used by [88] in order to test the Suehrcke's equation [89]. In addition, Suehrcke et al. [25] analyzed recently 68,645 pairs of sunshine-radiation readings from 670 sites using the same WRDC database and came to the conclusion that the relationship between sunshine fraction and solar radiation is non-linear,

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Fig. 4. Locations used for the study.

which is contrary to the widely accepted linear Ångström– Prescott model. The period for which the data are available for different sites varies from one year to 65 years. Frequencies of number of years for which the sunshine duration and global radiation data are available are shown in histograms in Fig. 2. The monthly average values of daily global radiation and daily sunshine duration for specific site are found as arithmetic means of readings during the period for which these data are available for that site. In order to avoid seasonal bias, all sites that do not have measurements for all 12 months were discarded from analysis. These sites are shown in Fig. 3. In addition, from analysis were excluded sites that are partially or totally in polar night or have physically aberrant data (S=S0 4 1 or H=H 0 4 1, Table 3). In total, solar data corresponding to 924 sites throughout the world were used for this study. These locations are shown on a world map in Fig. 4.

5. Statistical evaluation In this study, 10 statistical quantitative indicators were used to evaluate different sunshine radiation models. These quantitative indicators are 1. MAE: mean absolute error. The MAE is the sum of absolute values of the errors divided by the number of observations. This quantity is often used in statistics as measure how close calculated values are to measured values. In [90] authors pointed out some advantages of MAE over the root mean squared error (RMSE) in dimensioned evaluations and inter-comparisons of average model performance error: MAE ¼

n   1X H i;m  H i;c  ni¼1

ð5Þ

2. RMSE: root mean squared error. The RMSE is a frequently used measure to compare forecasting errors of different models. The lower the RMSE value the better the predictive capability of a model in terms of its absolute deviation. However, presence of few large errors can result in greater value of RMSE: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 Xn

H i;m H i;c ð6Þ RMSE ¼ i¼1 n 3. MARE: mean absolute relative error. The MARE when expressed in percentages is also known as mean absolute percentage error (MAPE). This indicator is expressed as average absolute

value of relative differences between estimated and measured solar radiations.    n  1X H i;m  H i;c  MARE ¼ ð7Þ    H i;m  n i¼1

4. U95: uncertainty at 95%. Following Gueymard [91] and Behar et al. [92], this indicator is used in order to show more information about the model deviation. Uncertainty with 95% a confidence level is given as U95 ¼ 1:96ðSD2 þ RMSE2 Þ1=2

ð8Þ

where 1.96 is the coverage factor corresponding 95% confidence level, and SD is the standard deviation of the difference between the calculated and measured data. 5. RMSRE: root mean squared relative error vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u u1 Xn H i;m H i;c RMSRE ¼ t ð9Þ i¼1 n H i;m 6. RRMSE: relative root mean squared error. This indicator is calculated by dividing RMSE with average value of measured data. According to [93], model accuracy is considered excellent when RRMSE o10%, good if 10% o RRMSE o 20%, fair if 20% o RRMSE o 30%, and poor if RRMSE 4 30%: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pn

1 i ¼ 1 H i;m  H i;c n  100 ð10Þ RRMSE ¼ Pn i ¼ 1 H i;m 7. MBE: mean bias error. This indicator expresses a tendency of model to underestimate (negative value) or overestimate (positive value) global radiation, while the MBE values closest to zero are desirable. The drawback of this test is that it does not show the correct performance when the model presents overestimated and underestimated values at the same time, since overestimation and underestimation values cancel each other. MBE ¼

n

 1X H  H i;c n i ¼ 1 i;m

ð11Þ

8. R2: coefficient of determination. This indicator is often used in statistics for estimating the performance of the models. It depicts the fraction of the calculated values that are the closest to the line of measurement data. While the ideal values of all other statistical indicators used in this study are 0, values of the coefficient of determination close to 1 indicate more efficient

Table 4 Statistical evaluation of different sunshine radiation models. RMSE

MARE

U95

RMSRE

RRMSE

R2

erMAX

MBE

t-stat

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

0.332 0.297 0.308 0.910 0.387 0.298 0.401 0.293 0.465 0.390 0.497 0.505 0.296 0.398 0.297 0.611 6.394 0.711 0.294 0.295 0.302 0.311 0.291 0.450 0.332 0.341 0.333 0.332 0.336 0.374 0.376 0.302 0.305 0.471 0.496 0.511 0.512 0.456 0.470 0.468 0.291 0.698 0.701 1.590 0.306 0.342 0.365 0.529 0.338 0.960 0.552 0.593 0.535 2.043 0.362

0.477 0.463 0.472 1.253 0.575 0.468 0.585 0.467 0.661 0.567 0.689 0.734 0.466 0.541 0.474 0.738 9.284 0.947 0.470 0.471 0.470 0.487 0.465 0.586 0.485 0.501 0.492 0.492 0.493 0.532 0.533 0.473 0.474 0.634 0.654 0.669 0.670 0.633 0.634 0.666 0.457 0.855 0.858 1.946 0.466 0.496 0.513 0.667 0.476 1.156 0.681 0.746 0.672 2.349 0.524

0.084 0.078 0.080 0.226 0.109 0.080 0.114 0.078 0.120 0.114 0.126 0.129 0.078 0.131 0.079 0.157 2.805 0.170 0.078 0.079 0.086 0.085 0.078 0.162 0.086 0.087 0.103 0.105 0.111 0.118 0.121 0.088 0.088 0.141 0.144 0.143 0.143 0.134 0.135 0.120 0.078 0.242 0.243 0.606 0.080 0.093 0.095 0.123 0.103 0.242 0.143 0.152 0.141 0.452 0.102

1.295 1.282 1.306 2.995 1.589 1.295 1.574 1.290 1.828 1.439 1.911 1.997 1.292 1.406 1.309 1.757 22.561 2.265 1.299 1.301 1.298 1.346 1.281 1.602 1.337 1.379 1.307 1.308 1.314 1.370 1.373 1.282 1.285 1.543 1.593 1.646 1.652 1.564 1.548 1.840 1.266 2.173 2.180 4.437 1.289 1.359 1.383 1.624 1.319 2.606 1.661 1.835 1.739 5.137 1.437

0.129 0.131 0.132 0.295 0.171 0.136 0.176 0.135 0.179 0.173 0.185 0.197 0.133 0.198 0.135 0.183 5.063 0.227 0.135 0.135 0.149 0.141 0.135 0.264 0.132 0.134 0.165 0.169 0.182 0.181 0.186 0.147 0.146 0.201 0.198 0.195 0.195 0.196 0.190 0.180 0.132 0.346 0.348 1.024 0.131 0.143 0.140 0.150 0.161 0.289 0.173 0.197 0.177 0.467 0.159

11.480 11.123 11.343 30.136 13.821 11.257 14.056 11.232 15.904 13.626 16.577 17.638 11.213 13.003 11.388 17.732 223.220 22.771 11.303 11.320 11.296 11.706 11.182 14.091 11.656 12.035 11.836 11.837 11.864 12.788 12.820 11.369 11.406 15.240 15.727 16.080 16.109 15.228 15.244 16.006 10.983 20.554 20.640 46.784 11.212 11.930 12.325 16.034 11.444 27.786 16.370 17.931 16.156 56.475 12.590

0.934 0.938 0.935 0.543 0.904 0.936 0.901 0.937 0.873 0.907 0.862 0.844 0.937 0.915 0.935 0.842  24.053 0.739 0.936 0.936 0.936 0.931 0.937 0.900 0.932 0.927 0.930 0.930 0.929 0.918 0.917 0.935 0.935 0.883 0.876 0.870 0.870 0.883 0.883 0.871 0.939 0.788 0.786  0.100 0.937 0.928 0.924 0.871 0.934 0.612 0.865 0.838 0.869  0.604 0.920

1.666 1.764 1.771 2.755 1.799 1.816 1.979 1.821 2.071 1.956 2.344 2.437 1.796 2.136 1.820 1.487 41.443 2.304 1.822 1.833 1.800 1.811 1.819 2.260 1.823 1.810 1.958 1.962 1.943 2.035 2.051 1.832 1.839 2.099 2.175 2.130 2.115 2.027 2.095 2.131 1.790 2.712 2.717 9.361 1.751 1.764 1.718 1.443 1.941 2.415 1.541 2.090 1.822 0.708 1.708

0.139 0.011 0.034  0.898  0.062  0.044  0.197  0.056  0.071  0.322  0.015  0.196  0.013  0.265  0.051 0.533 6.317  0.677  0.053  0.053  0.055  0.049  0.074  0.138 0.068 0.079  0.200  0.198  0.193  0.278  0.280  0.138  0.143  0.429  0.442  0.436  0.433  0.407  0.424  0.074  0.019  0.482  0.487 1.565 0.048 0.107 0.167 0.450  0.023  0.950 0.457 0.486 0.340 2.041 0.105

32.161 2.591 7.713 108.246 11.478 10.047 37.627 12.669 11.362 72.597 2.344 29.202 2.981 59.318 11.382 110.174 97.751 107.756 11.882 11.878 12.331 10.642 16.975 25.488 14.968 16.741 46.891 46.160 44.838 64.633 64.905 32.215 33.220 96.700 96.508 90.354 89.154 88.441 94.883 11.771 4.392 71.867 72.514 142.448 10.866 23.174 36.342 96.341 5.096 152.249 95.439 90.347 61.833 184.778 21.482

1875

MAE

M. Despotovic et al. / Renewable and Sustainable Energy Reviews 52 (2015) 1869–1880

Model

1876

Table 4 (continued ) MAE

RMSE

MARE

U95

RMSRE

RRMSE

R2

erMAX

MBE

t-stat

56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101

0.531 0.545 1.040 0.348 0.295 0.294 0.318 0.325 0.322 0.294 0.327 0.562 0.507 0.481 0.407 0.341 0.942 0.394 0.364 0.904 0.374 0.397 0.401 0.401 0.860 2.667 0.878 2.693 0.930 2.713 0.298 0.615 0.415 2.705 0.480 0.464 0.368 0.406 0.398 0.396 0.376 0.414 3.858 0.415 0.401 0.324

0.694 0.709 1.216 0.496 0.454 0.469 0.483 0.513 0.485 0.449 0.507 0.699 0.642 0.711 0.545 0.490 1.112 0.540 0.510 1.325 0.532 0.551 0.556 0.556 0.988 3.417 1.006 3.449 1.075 3.438 0.459 0.826 0.636 3.311 0.664 0.602 0.517 0.553 0.546 0.543 0.533 0.556 4.224 0.636 0.537 0.474

0.154 0.156 0.270 0.111 0.079 0.081 0.082 0.084 0.099 0.082 0.085 0.130 0.173 0.117 0.103 0.086 0.211 0.129 0.118 0.230 0.118 0.107 0.108 0.108 0.225 0.830 0.224 0.821 0.230 0.847 0.079 0.154 0.102 0.573 0.129 0.163 0.089 0.100 0.096 0.096 0.121 0.101 0.930 0.102 0.097 0.082

1.655 1.686 2.694 1.321 1.257 1.288 1.337 1.415 1.295 1.245 1.404 1.683 1.668 1.815 1.424 1.327 2.506 1.397 1.347 3.506 1.370 1.492 1.506 1.509 2.220 8.052 2.250 8.126 2.394 8.071 1.271 1.988 1.675 7.492 1.839 1.667 1.371 1.443 1.431 1.420 1.373 1.438 8.941 1.675 1.375 1.294

0.212 0.212 0.314 0.174 0.132 0.139 0.134 0.141 0.159 0.135 0.140 0.156 0.254 0.179 0.142 0.132 0.227 0.195 0.182 0.331 0.181 0.153 0.153 0.153 0.248 1.065 0.243 1.037 0.246 1.124 0.131 0.209 0.165 0.622 0.176 0.274 0.129 0.138 0.134 0.133 0.186 0.138 0.931 0.165 0.131 0.129

16.696 17.055 29.242 11.931 10.910 11.273 11.618 12.324 11.659 10.800 12.185 16.815 15.446 17.092 13.100 11.771 26.742 12.985 12.269 31.855 12.788 13.236 13.360 13.377 23.745 82.157 24.197 82.917 25.843 82.652 11.034 19.861 15.298 79.599 15.960 14.482 12.419 13.288 13.139 13.065 12.820 13.363 101.567 15.298 12.911 11.387

0.860 0.854 0.570 0.928 0.940 0.936 0.932 0.924 0.932 0.941 0.925 0.858 0.880 0.853 0.914 0.930 0.640 0.915 0.924 0.490 0.918 0.912 0.910 0.910 0.717  2.394 0.706  2.457 0.664  2.435 0.939 0.802 0.882  2.186 0.872 0.895 0.922 0.911 0.913 0.914 0.917 0.910  4.187 0.882 0.916 0.935

2.125 2.119 2.430 2.018 1.744 1.822 1.783 1.877 1.924 1.749 1.849 1.420 2.366 2.118 1.646 1.697 1.202 2.117 2.057 2.604 2.035 1.764 1.765 1.761 1.408 4.576 1.370 4.102 1.259 7.959 1.753 2.201 2.045 1.295 2.099 2.333 1.659 1.668 1.690 1.654 2.051 1.616 1.161 2.045 1.580 1.688

 0.501  0.516  1.034  0.195 0.015  0.090 0.042  0.068  0.183  0.001  0.020 0.491  0.318  0.392 0.257 0.144 0.917  0.274  0.220 0.558  0.278 0.164 0.165 0.162 0.817  2.545 0.841  2.569 0.905  2.584 0.022  0.580  0.281 2.704  0.029 0.047 0.210 0.263 0.254 0.256  0.280 0.281 3.858  0.281 0.291 0.114

109.913 111.808 170.014 45.026 3.436 20.618 9.242 14.093 42.888 0.144 4.098 103.827 59.897 69.529 56.286 32.488 153.162 62.094 50.284 48.911 64.633 32.782 32.761 31.944 155.260 117.545 160.244 117.532 164.217 120.023 5.145 103.711 51.870 149.059 4.609 8.209 46.878 56.993 55.207 56.278 64.905 61.844 235.937 51.870 68.027 25.993

M. Despotovic et al. / Renewable and Sustainable Energy Reviews 52 (2015) 1869–1880

Model

M. Despotovic et al. / Renewable and Sustainable Energy Reviews 52 (2015) 1869–1880 4

models:

Pn 2

1877

R ¼ 1P

i¼1

n i¼1



H i;m  H i;c

2

H i;m  H m;avg

2

9. erMAX: maximum absolute relative error ! H  H i;c j erMAX ¼ max j i;m H i;m

3.51

ð12Þ

ð13Þ

10. t-stat: t-Statistic. This indicator, which had a long history of popular usage, was first proposed by Stone [94] to be used in conjunction with RMSE and MBE for more complete evaluation of solar radiation estimation models. Later on t-statistics became a widely accepted test for validating whether or not the calculated values of solar radiation are not significantly different from their measured counterparts [92,91,95,10,29]. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn  1ÞMBE2 tstat ¼ ð14Þ RMSE2  MBE2

3

2.76

2

1.7 1.32

0

1.245

1.04

1

0.291

MAE

0.449

RMSE

0.941

0.27

0.348

0.319

0.078

0.129

0.108

MARE

U95

RMSRE RRMSE

1.202

1.03

0.490 0.001

R^2

erMAX

0.001

|MBE| (t-stat)*0.01

Fig. 5. Boxplot diagram of calculated statistical indicators for investigated models except those that have negative R2. Whiskers represent minimum and maximum of all of the data.

6. Results and discussion Results of statistical analysis of different sunshine radiation models are given in Table 4. As it has been already said in the previous section, the lower the absolute values of all statistical indicators except the R2, the more accurate the estimate. The R2 value around 1 indicates that there is a perfect linear relationship between the estimated and measured values whereas R2 around zero shows that there is no linear relationship. However, although R2 ranges from 0 to 1, it could be seen that models 17, 44, 54, 81, 83, 85, 89, and 98 have negative R2, which is possible due to computational definition of coefficient of determination. But, these negative values indicate that corresponding models do not predict well, that is they fit worse than a horizontal line. This is also visible from other statistical indicators for these models which are abnormally high. By far the worst statistical indicators show model 17. Since the authors in original study [48] did not provide any appraisal of that model, it is not possible to realize whether such bad performance is only on global scale, particularly because model 16 that comes from the same study shows very different indicators. It is interesting that models 77–85, which originate from the same source [77], can be clustered into two groups according to performance on the global scale. The first group consists of models 81, 83 and 85 that have negative R2 value, and the second group comprises other six models. Statistical indicators for these models within each group are quite even whereas the models show very different performance between groups. On the other hand, statistical indicators for these nine models in the original study are far more alike. The calculated statistical indicators for rest of the models are in a wide range (Fig. 5). Based on the mean absolute error, the most accurate model is model 23 with MAE ¼ 0.291 kWh/m2. The lowest RMSE (RMSE ¼ 0.449 kWh/m2) has model 65. The same model is found as the most accurate based on U95 ¼ 1.245 kWh/ m2, RRMSE ¼ 10.8%, R2 ¼ 0.941, MBE ¼  0.001 kWh/m2, and tstat ¼ 0.144. Based on indicator MARE, the lowest value has model 2 (MARE ¼ 0.78). Model 101 shows the best performance according to indicator RMSRE (0.129) and the lowest maximum absolute relative error erMAX has the model 54 (erMAX ¼ 0.708). At first sight it could be concluded that model 65 has the best overall performance of all investigated models. However, it is very difficult to compare or rank large number of different models in such way. One way to overcome this issue is to use a global performance indicator (GPI) that represents multiplication of all used statistical indicators [92]. We introduced different global

Fig. 6. Physical interpretation of the idea of computing GPI.

performance indicators, which we believe is more appropriate and derived on more realistic basis. The idea for such GPI is presented in Fig. 6. If the value of some statistical indicator of a model is less than median, then the farther the value from median value of all other models, the more accurate the model compared to other models. If the value of statistical indicator is higher than median, then the farther the value from median, the less accurate the model compared to other models. In addition, in order to avoid predominant influence of any particular indicator, we scaled values of all indicators so that minimum value of every indicator is 0 and maximum value of every indicator is 1. Thus, the GPI of model i is defined as GPIi ¼

10 X

αj ðy~ j  yij Þ

ð15Þ

j¼1

where y~ j is median of scaled values of indicator j, yij is the scaled value of indicator j for model i, αj equals  1 for j ¼ 7 (R2), and equals 1 for all other indicators. The more the accuracy of the model the higher the value of the GPI. There are several advantages of GPI defined in such way over the GPI calculated as product of individual indicators. For instance, if the value of one indicator is 0 then the product of all indicators will automatically be 0, regardless of the values of other indicators that could be rather high. In addition, the method of computing GPI in the way presented in this study enables that models with negative R2 could also be compared amongst themselves, which is not possible otherwise. Moreover, making simple product of all indicators could lead to pretty wide range of values of GPI (in this case 1:8  10  7 –2:7  1012 ), which makes comparisons of models more difficult.

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Table 5 Ranking of different sunshine radiation models according to GPI. Rank

Model

GPI

Type

Rank

Model

GPI

Type

Rank

Model

GPI

Type

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

65 2 60 13 41 86 3 66 6 49 45 62 19 15 20 8 22 21 23 25 63 61 26 5 101 46 55 90 1 32 11 33 71 9

0.327 0.311 0.310 0.306 0.304 0.299 0.279 0.276 0.269 0.267 0.267 0.264 0.261 0.261 0.260 0.258 0.254 0.252 0.238 0.231 0.223 0.217 0.214 0.189 0.188 0.182 0.170 0.159 0.156 0.153 0.152 0.147 0.146 0.131

L P L LOG/L P L L L P L L L P L P P E LOG P P L L P P P P P P L L A/L P L A/L

35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68.

91 40 47 79 77 78 64 92 29 28 27 59 24 7 74 94 95 70 12 93 97 88 99 100 14 30 76 73 31 96 10 53 68 69

0.129 0.125 0.108 0.097 0.096 0.093 0.085 0.062 0.061 0.060 0.058 0.056 0.054 0.046 0.018 0.000  0.002  0.008  0.013  0.014  0.038  0.043  0.043  0.052  0.053  0.059  0.059  0.063  0.064  0.064  0.109  0.137  0.151  0.181

P A/L P P L P L P P P L L P A/P L P P L AL P LOG/P L L A/L/H A/P L L L P P L L/H L L

69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101.

38 39 37 48 36 34 51 35 52 67 42 56 43 16 57 87 75 18 4 80 82 72 84 50 58 44 54 89 81 83 85 98 17

 0.236  0.269  0.271  0.272  0.276  0.282  0.292  0.296  0.328  0.329  0.387  0.387  0.393  0.400  0.405  0.435  0.506  0.539  0.765  0.797  0.829  0.847  0.888  0.921  1.058  1.826  1.918  2.462  2.620  2.620  2.752  3.747  9.058

E O P L P L L LOG P L P L P L L L A/L/H L P/H A/L/H A/L/H L A/P/H L L P P P A/L/H A/L/H A/P/H P P

L: linear; P: polynomial; A: angular; LOG: logarithmic; E: exponential; H: altitude; O: power.

hybrid LOG/L model is also highly ranked (fourth place). For the sake of completeness, the descriptive statistics of different types of models is given in Table 6.

Table 6 Descriptive statistics for different types of models. Type

Count

Best GPI

Worst GPI

Average GPI

A/L A/L/H A/P A/P/H E P/H L L/H LOG LOG/L LOG/P O P

4 6 2 2 2 1 40 1 2 1 1 1 38

0.152  0.052 0.046  0.888 0.254  0.765 0.327  0.137 0.252 0.306  0.038  0.269 0.311

 0.013  2.62  0.053  2.752  0.236  0.765  1.058  0.137  0.296 0.306  0.038  0.269  9.058

0.099  1.237  0.004  1.820 0.009  0.765  0.078  0.137  0.022 0.306  0.038  0.269  0.441

The ranking of investigated sunshine radiation models according to GPI is given in Table 5. All models are classified in 13 groups according to their types. There are 5 pure types of models: linear (L), polynomial (P), logarithmic (LOG), exponential (E), and power (O), and 8 hybrid types of models that represent a combination of specific terms or include extra angular or altitude terms. For example LOG/P model (model 97) is a combination of polynomial and logarithmic terms. If model types are compared, then based on Table 5, it could be concluded that better GPI demonstrate linear and polynomial models, while models that include altitude are amongst the worst. Amongst the top 10 ranked models, there are six linear models and three polynomial models. In addition,

7. Conclusion In the prediction, study, and design of solar energy systems, data on solar radiation and its components at a particular location are very essential. However, for many sites, particularly in developing countries, solar radiation measurements are not easily available due to the cost and maintenance and calibration requirements of the measuring equipment. For the designers and manufactures of solar equipment but also for other professionals, such as architects and agriculturists, it is very important to have methods to estimate the solar radiation based on readily available meteorological data. The number of such methods that have been developed is relatively high, which makes it difficult to choose the most appropriate correlation for a particular purpose and site [26]. The main objective of this study was to provide evaluation of different solar radiation models, which could be helpful in the selection of the most appropriate and accurate model based on the available sunshine data. In this study, a detailed statistical analysis and comparison of available solar radiation models is conducted. A total of 101 different solar radiations model were tested on long term meteorological data corresponding to 924 sites throughout the world. Ten statistical indicators were used to assess models’ performance. The results show wide range of calculated statistical indicators, from very poor to satisfactory. This indicates existence

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of models that cannot be recommended although they might have better performance with the initial data used for model development, but also existence of models that are acceptable for global use. In addition, we introduced specific global performance indicator (GPI), by means of which all analyzed models are depicted with a single parameter. In this way models are easily ranked to assist in the selection of most appropriate and accurate one. Based on the value of GPI, the top three models are models 65, 2 and 60.

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