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Renewable Energy 66 (2014) 56e61

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Renewable Energy

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Technical note

Modeling the hourly solar diffuse fraction in Taiwan

Chia-Wei Kuo a, Wen-Chey Chang a, Keh-Chin Chang b,*

a Energy Research Center, National Cheng Kung University, Tainan, TaiwanbDepartment of Aeronautics & Astronautics, National Cheng Kung University, Tainan, Taiwan

a r t i c l e i n f o

Article history:Received 25 March 2013Accepted 28 November 2013Available online

Keywords:Diffuse fractionMultiple regressionModel assessmentMathematical modeling

* Corresponding author. No.1, University Rd., Taina2757575x63679; fax: þ886 6 2389940.

E-mail address: kcchang@mail.ncku.edu.tw (K.-C.

0960-1481/$ e see front matter � 2013 Published byhttp://dx.doi.org/10.1016/j.renene.2013.11.072

a b s t r a c t

Using the data for global and diffuse radiation in Tainan, Taiwan, for the years of 2011 and 2012,respectively, four correlation models with five predictors: the hourly clearness index (kt), solaraltitude, apparent solar time, daily clearness index and a measure of persistence of globalradiation level, are constructed to relate the hourly diffuse fraction on a horizontal surface (d) to theclearness index. Two models use a single logistic equation for all kt values, Eqs. (6) and (7), and theother two models use a set of piece-wise linear equations for four kt intervals, Eqs. (8) and (9). Theproposed models are compared respectively with the fourteen models available in the literature, interms of the four statistical indicators: the mean bias error, the root-mean-square error, the t-sta-tistic and the Bayesian Information Criterion, using the out-of-sample dataset for Tainan, Taiwan. It isconcluded from the analysis that the proposed piece-wise linear models perform well inpredicting the diffuse fraction, while the performances of the proposed logistic models are morecase-dependent. Among those fourteen models considered in this study, the models developed byErbs et al., Chandrasekaran and Kumar, and Boland et al. have competitive performances asthe proposed piece-wise linear models do, when applying to the prediction of diffuse fraction inTainan, Taiwan.

� 2013 Published by Elsevier Ltd.

1. Introduction

Global radiation (Iglobal) consists of two parts: diffuse radiation(Idiffuse) and beam radiation (Ibeam). Information about hourlydiffuse fraction (d), defined in Eq. (1), is prerequisite to evaluate theperformances of concentrating solar thermal systems. Since themeasurements of diffuse or beam radiation are not frequentlypossible on a site of interest, it is necessary to find amodel of diffusefraction correlating the diffuse radiation to the global radiationwhich is usually available in the reports from the meteorologicalstations.

d ¼ IdiffuseIglobal

(1)

An early model developed by Liu and Jordan [1] estimates d interms of the hourly sky clearness index (kt), which is an indicator tomeasure how clear the skies are, given by

n 701, Taiwan. Tel.: þ886 6

Chang).

Elsevier Ltd.

kt ¼ IglobalHo

(2)

where Ho is the hourly extraterrestrial radiation and can be theo-retically determined by specifying the site latitude, the day of eachyear and the hour of each day [2]. The idea of such early model israther simple: when kt is large (clear sky), diffuse fraction is smalldue to less obstruction by small droplets and particulates sus-pended in the atmosphere; when kt is small (cloudy sky), diffusefraction is large because of the greater scattering frequency.Following the same idea, quite a few other models in terms of thesingle predictor kt [3e12] were developed later, each of whichsuccessfully fit the data of diffuse fractionwithin a specified region,to varying degrees.

Studies that include multiple predictors in modeling to achievea better data fit were also conducted. For example, Reindl et al. [13]suggested that the solar altitude (a), ambient temperature (Ta) andrelative humidity (RH) are the other three effective ones, by per-forming a statistical analysis of twenty-eight possible predictors,using the data from cities worldwide.

Recently, Boland and his coworkers [14,15] demonstrated thatthe performance of the time-dependent model was better than theother and corroborated the significance of time (apparent solar

C.-W. Kuo et al. / Renewable Energy 66 (2014) 56e61 57

time) as an extra model predictor. On the basis of these in-vestigations, they recently proposed the BolandeRidleyeLauretmodel (abbreviated as the BRL model hereafter) [16] using a total offive predictors: kt, a, the apparent solar time (t), the daily clearnessindex (KT) and ameasure of persistence of global radiation level (j).The last two predictors are defined as follows.

KT ¼P24

i¼1 IglobalP24i¼1 Ho

(3)

j ¼

8>><>>:

kt�1þktþ12 for sunrise < t < sunset

ktþ1 for t ¼ sunrise

kt�1 for t ¼ sunset

(4)

Despite the abundance of models, such as those summarized inTables 1 and 2, there is still a need for developing correlationmodels for the Taiwan area. According to the comparative studies ofmodel performance made by Jacovides et al. [11], Torres et al. [17]and Dervishi and Mahdavi [18], most of the existing correlationmodels are not applicable to all geographical regions, withoutmodification. Therefore, the developments of applicable but accu-rate correlation models for a specific region, which account for thegeographical and climatic conditions, are still preferable.

This study correlates four models with two different mathe-matical formats: logistic and piecewise linear equations, using thetwo available yearly sets of data in Tainan, Taiwan. The logistic

Table 1Summary of the models from the literature.

Constraints

Orgill and Hollands, 1977 [3] kt < 0:350:35 � kt � 0:75kt > 0:75

Erbs et al., 1982 [4] kt � 0:220:22 < kt � 0:80kt > 0:80

Bugler, 1977 [5] 0 < kt � 0:40:4 < kt � 1:0

Hawlader, 1984 [6] kt � 0:2250:225 < kt < 0:775kt � 0

Miguel et al., 2001 [7] kt � 0:210:21 < kt � 0:76kt > 0:76

Karatasou et al., 2003 [8] 0 < kt � 0:78kt > 0:78

Chandrasekaran and Kumar, 1994 [9] kt � 0:240:24 < kt � 0:80kt > 0:80

Soares et al., 2004 [10] kt � 0:170:17 < kt � 0:75kt > 0:75

Jacovides et al., 2006 [11] kt � 0:10:1 < kt � 0:8kt > 0:8

Zhou et al., 2004 [12] kt < 0:200:20 � kt < 0:75kt > 0:75

Reindl et al., 1990 [13] 0 � kt � 0:30:3 < kt < 0:78kt � 0:7

Reindl et al., 1990 [13] 0 � kt � 0:30:3 < kt < 0:78kt � 0:7

Boland et al., 2001 [14] None

Boland et al., 2008 [15] None

Ridley et al. (BRL model), 2010 [16] None

models, which are based on the recent study of Ridley et al. [16], usea single logistic equation for all values of kt. In contrast, the otherrelatively simple models use a set of piece-wise linear equations fordifferent intervals of kt that use the same group of predictors as thelogistic models. The models listed in Table 1 are also tested usingthe available database in Taiwan, for the purpose of comparisonwith the proposed models developed in this study.

2. Experimental set-up and database

Because of the lack of diffuse radiation data in daily reports fromall meteorological stations of the Taiwan Central Weather Bureau,this modeling work uses the in situ data for global and diffuse ra-diation, measured at the Kuei-Jen campus of the National ChengKung University, Tainan, Taiwan (23�N 120�E), from 1 January 2011to 31 December 2012.

Two sets of devices from Eppley Laboratory, Inc., each of whichincluded a pyranometer (Model PSP) without and with a shadowband stand (Model SBS), were used to measure global and diffuseradiation, as shown in Fig. 1(a) and (b), respectively. The samplingrate was of 1 Hz. The method of data checking followed partially theideas of Reindl et al. [13], to identify themissing data and datawhichviolated physical limits. The missing data for individual secondsmostly occurred while transmitting were filled using a linear inter-polation of the neighboring data in the time sequence. After all of themissing data had been filled, the data for individual seconds wereconverted into an hourly value by integration with respect to time.

Diffuse fraction (d)

1:0� 0:249kt1:557� 1:84kt0:177

1:0� 0:09kt0:9511� 0:1604kt þ 4:388k2t � 16:638k3t þ 2:336k4t0:165

0:94ð1:29� 1:19ktÞ=ð1:00� 0:334ktÞ

:7750:9151:135� 0:9422kt � 0:3878k2t0:215

0:995� 0:081kt0:724þ 2:738kt � 8:32k2t þ 4:967k3t0:18

0:9995� 0:05kt � 2:4156k2t þ 1:4926k3t0:20

1:0086� 0:178kt0:9686þ 0:1325kt þ 1:4183k2t � 10:1862k3t þ 8:3733k4t0:197

1:00:90þ 1:1kt � 4:5k2t þ 0:01k3t þ 3:14k4t0:17

0:9870:94þ 0:937kt � 5:01k2t þ 3:32k3t0:177

0:9871:292� 1:447kt0:209

81:000� 0:232kt þ 0:0239 sinðaÞ � 0:000682Ta þ 0:0195RH1:329� 1:716kt þ 0:267 sinðaÞ � 0:00357Ta þ 0:106RH0:426kt � 0:256 sinðaÞ þ 0:00349Ta þ 0:0734RH

81:020� 0:254kt þ 0:0123 sinðaÞ1:400� 1:749kt þ 0:177 sinðaÞ0:486kt � 0:182 sinðaÞ

�0:039þ 1:039=½1þ expð�8:769þ 7:325kt þ 0:377tÞ�1=½1þ expð8:60kt � 5:00Þ�1=½1þ expð�5:38þ 6:63kt � 0:007aþ 0:006t þ 1:75KT þ 1:31jÞ�

Table 2The function types and the weather stations used for development of the models in Table 1.

Function type Weather stations where data are collected

Orgill and Hollands [3] Piecewise linear Toronto (43�N)Erbs et al. [4] Piecewise 4th-order polynomial Fort Hood (31�N), Livermore (38�N), Raleigh (36�N), Maynard (42�N), and Albuquerque (35�N)Bugler [5] Piecewise fractional Melbourne (38�S)Hawlader [6] Piecewise 2nd-order polynomial Singapore (1�N)Miguel et al. [7] Piecewise 3rd-order polynomial Athens (38�N), Lisbon (39�N), Coimbra (40�N), Evora (39�N), Faro (37�N), Porto (41�N),

Carpentras (44�N), Pau (43�N), Perpignan (43�N), Madrid (40�N), and Seville (37�N)Karatasou et al. [8] Piecewise 3rd-order polynomial Athens (38�N)Chandrasekaran and Kumar [9] Piecewise 4th-order polynomial Madras (13�N)Soares et al. [10] Piecewise 4th-order polynomial Sao Paolo (23�S)Jacovides et al. [11] Piecewise 3rd-order polynomial Athalassa (35�N)Zhou et al. [12] Piecewise linear 78 weather stations in ChinaReindl et al. [13] Piecewise linear Albany (43�N), Cape Canaveral (28�N), Copenhagen (56�N), Hamburg (54�N),

Valentia (52�N), and Oslo (60�N)Boland et al. [14] Logistic Geelong (38�S)Boland et al. [15] Logistic Adelaide (35�S), Darwin (12�S), Geelong (38�S), Maputo (26�S), Bracknell (51�N),

Lisbon (39�N), Macau (22�N), and Uccle (51�N)Ridley et al. (BRL model) [16] Logistic Adelaide (35�S), Darwin (12�S), Maputo (26�S), Bracknell (51�N), Lisbon (39�N),

Macau (22�N), and Uccle (51�N)

C.-W. Kuo et al. / Renewable Energy 66 (2014) 56e6158

Any hourly data that violated a physical limit, including those withnegative values for radiation or with a value for global radiation thatexceeded the extraterrestrial radiation, were then eliminated fromthe dataset. The final datasets that had passed the quality controlchecks mentioned produced 3621 and 3382 hourly data points, forthe yearly databases of 2011 and 2012, respectively.

3. Model development

It is noted [16] that the BRL model using a single logistic equa-tion for the entire range of kt, instead of a set of piecewise linearequations, could evaluate the diffuse fraction satisfactorily. Thus,the approach of the BRLmodel is firstly used for themodelingwork.As expressed in Eq. (5), the other predictors used in the BRL modelinclude solar altitude (a, in rad), apparent solar time (t, in h), dailyclearness index (KT, defined in Eq. (3)) and a measure of thepersistence of the global radiation level (j, defined in Eq. (4)).

d ¼ 1=½1þ expðg1 þ g2kt þ g3aþ g4t þ g5KT þ g6jÞ� (5)

where gi are coefficients to be determined. Using the hourly dektdatabases collected in 2011 and 2012, respectively, together withfour other corresponding predictors, and the coefficients of the BRLmodel (see Table 1) as the initial guessed values, the optimal formsof the modified BRL models for the Taiwan area are given as

Fig. 1. (a) The pyranometer and (b) the pyranometer þ shadow band

Model 1 (with fitting dataset of 2011)

d ¼ 1=½1þ expð � 4:5274þ 5:6956kt � 0:0814a� 0:0464t

þ 2:4162KT þ 1:0125jÞ�(6)

Model 2 (with fitting dataset of 2012)

d ¼ 1=½1þ expð � 4:5312þ 5:7627kt � 0:0882a� 0:0391t

þ 1:9998KT þ 1:1521jÞ�(7)

As seen from the models compiled in Table 1, most of theexisting models split the data into sub-regions defined for differentkt values, before any regression is attempted. Two relatively simplepiece-wise linear models that use the same five predictors as theprevious modified BRLmodels are suggested as Eqs. (8) and (9). Themodels use four sub-regions, instead of the two or three sub-regions commonly used in the existing piece-wise models(Table 1), in order to produce a better fit for the presentdatabase. Note that the variable form of “sin (a)” instead of “a” usedin Eqs. (6) and (7), is used in Eqs. (8) and (9). This follows themethod used in the model of Reindl et al. [13].

Model 3 (with fitting dataset of 2011)

stand used to measure global and diffuse radiation, respectively.

d ¼ 0:9885; 0 � kt < 0:2d ¼ 1:0981� 0:3769kt þ 0:0233 sinðaÞ þ 0:0027t � 0:1451KT � 0:1727j; 0:2 � kt < 0:3d ¼ 1:4185� 1:1897kt þ 0:0100 sinðaÞ þ 0:0071t � 0:3891KT � 0:2181j; 0:3 � kt < 0:75d ¼ 0:1922; kt � 0:75

(8)

C.-W. Kuo et al. / Renewable Energy 66 (2014) 56e61 59

Model 4 (with fitting dataset of 2012)

d ¼ 0:9896; 0 � kt < 0:2d ¼ 1:0874� 0:3936kt þ 0:0359 sinðaÞ þ 0:0035t � 0:1899KT � 0:1253j; 0:2 � kt < 0:3d ¼ 1:4188� 1:2191kt þ 0:0150 sinðaÞ þ 0:0063t � 0:3403KT � 0:2125j; 0:3 � kt < 0:75d ¼ 0:2775; kt � 0:75

(9)

4. Assessment of model performance

Fig. 2(a)e(d) shows graphical comparisons between the pre-dicted diffuse fraction and the measured data versus the skyclearness index. The comparisons reveal that the predictions madewith Eqs. (6)e(9) match the data well. Three commonly used sta-tistical indicators: the mean bias error (MBE), the root-mean-

Fig. 2. (a) and (b) Comparison of the 2011-year dataset with those predicted using Models

square error (RMSE) and the t-statistic defined in Eqs. (10)e(12),respectively, are firstly used to evaluate the performances of theproposed models against those summarized in Table 1, except forthe model of Reindl et al. which uses four predictors (kt, a, Ta, RH),due to lack of the ambient temperature (Ta) and relative humidity(RH) data in the presentmeasurements. In other words, a total of 14models from Table 1 are compared here.

1 and 3; while (c) and (d) the 2012-year dataset against those using Models 2 and 4.

Table 3Statistical performances of the models considered, using the out-of-sample data in

C.-W. Kuo et al. / Renewable Energy 66 (2014) 56e6160

MBE ¼ 1 Xnðdest � dmeaÞ (10)

Table 4Statistical performances of the models considered, using the out-of-sample data in2011, Tainan, Taiwan.

Model MBE (%) RMSE (%) tstat. tstat. � tcrit.a BIC (�104)

Model 2, Eq. (7) �0.23 10.45 1.35 Yes �1.63Model 4, Eq. (9) �0.36 10.40 2.08 No �1.63Orgill and Hollands [3] �0.50 11.20 2.71 No �1.58Erbs et al. [4] 0.34 11.42 1.77 Yes �1.56Bugler [5] 12.28 21.29 42.50 No �1.12Hawlader [6] �6.18 12.93 32.77 No �1.48Miguel et al. [7] �0.38 11.07 2.09 No �1.59Karatasou et al. [8] �5.84 13.01 30.22 No �1.47Chandrasekaran and

Kumar [9]�0.21 11.10 1.14 Yes �1.59

Soares et al. [10] �6.84 13.64 34.88 No �1.44Jacovides et al. [11] �3.92 11.86 21.09 No �1.54Zhou et al. [12] �3.57 11.86 18.97 No �1.54Reindl et al. [13] �0.17 12.33 0.85 Yes �1.51Boland et al.- Time-dependent [14] �6.26 23.70 16.47 No �1.04- Time-independent [15] 0.20 11.24 1.06 Yes �1.58Ridley et al.

(BRL model) [16]�2.33 11.06 12.99 No �1.59

a tcrit. ¼ 1.96 at a level of confidence of 95% and n � 1 degrees of freedom,n ¼ 3621.

2012, Tainan, Taiwan.

Model MBE (%) RMSE (%) tstat. tstat. � tcrit.a BIC (�104)

Model 1, Eq. (6) �1.00 10.49 5.56 No �1.52Model 3, Eq. (8) �0.36 10.40 2.01 No �1.52Orgill and Hollands [3] �0.94 11.16 4.89 No �1.48Erbs et al. [4] �0.07 11.39 0.35 Yes �1.46Bugler [5] 11.57 20.63 39.40 No �1.06Hawlader [6] �6.66 13.07 34.43 No �1.37Miguel et al. [7] �0.79 11.06 4.17 No �1.48Karatasou et al. [8] �6.35 13.16 32.00 No �1.37Chandrasekaran and

Kumar [9]�0.63 11.06 3.33 No �1.48

Soares et al. [10] �7.24 13.85 35.70 No �1.33Jacovides et al. [11] �4.36 11.96 22.79 No �1.43Zhou et al. [12] �3.98 11.98 20.48 No �1.43Reindl et al. [13] �0.61 12.25 2.89 No �1.41Boland et al.- Time-dependent [14] �7.11 23.63 18.34 No �0.97- Time-independent [15] �0.25 11.18 1.30 Yes �1.48Ridley et al.

(BRL model) [16]�2.72 11.20 14.56 No �1.48

a tcrit. ¼ 1.96 at a level of confidence of 95% and n � 1 degrees of freedom,n ¼ 3382.

ni¼1

RMSE ¼"1n

Xni¼1

ðdest � dmeaÞ2#1=2

(11)

tstat: ¼� ðn� 1ÞMBE2

RMSE2 �MBE2

�1=2(12)

The tests for MBE and RMSE respectively provide informationabout the long-term and short-term performance of any. In general,the smaller the absolute value of the MBE and the RMSE are, thebetter the model performs. However, there possibly appear con-flicting results such as simultaneously having a large value of RMSEand a small value of MBE, or vice versa. Thus, the t-statistic isintroduced to give an additional indicator to measure the statisticalsignificance of a model at a specified confidence level [19]. To fulfillthis purpose, the critical t value (tcrit.) at a particular level of sig-nificance and n � 1 degrees of freedommust be determined, whichcan be obtained from standard statistical tables. The greater thedegree to which tstat. is smaller than tcrit., the better is the model’sperformance.

Next, to study the trade-off between the goodness of fit and thecomplexity of the models, the Bayesian Information Criterion (BIC),which takes the fitting errors and the number of parameters (k) intoaccount, is calculated using the following equation, as with thestudies of Ridley et al. [16] and Torres et al. [17]. When comparingthe performance of the model, a lower value for the BIC indicates abetter model.

BIC ¼ n,ln

"1n

Xni¼1

ðdest � dmeaÞ2#þ k,lnðnÞ (13)

To evaluate the performances of the four proposed models, Eqs.(6)e(9), the out-of-sample data are used. In other words, Models 1and 3, which were constructed by using the training dataset of2011, use the dataset of 2012 as the out-of-sample data for theirmodel assessments, and vice versa for Models 2 and 4.Tables 3and 4 compare the statistical performances and the BICvalues of the proposed models of two different formats, i.e. Models1 and 3 as well as Models 2 and 4, against the 14 existing modelsconsidered in Table 1, using the collected dataset of 2011 and 2012,respectively, in Fig. 2.

As seen from Tables 3and 4, the proposed piece-wise linearmodels (Models 3 and 4) performwell in terms of the value of MBE,RMSE, and BIC among all the models considered, while their tstat.values exceed the tcrit. value slightly. In contrast, the modified BRLmodels (Models 1 and 2) show more case-dependent perfor-mances, particularly, for the evaluation of tstat., as compared to theproposed piece-wise linear models. Model 2 performs slightlybetter than Model 4 by using the dataset of 2011 for the out-of-sample data, whereas Model 1 performs worse than Model 3,particularly with respect to the tstat., by using the dataset of 2012 forthe out-of-sample data. Another better performing modelsconcluded from the comparison results in Tables 3and 4 includetwo piece-wise higher-order polynomial models with a single ktpredictor, respectively developed by Erbs et al. [4] as well asChandrasekaran and Kumar [9], and one time-independent logisticmodel with a single kt predictor, developed by Boland et al. [15].

It is agreed that, to avoid the use of the extreme case of weatherconditions, the best choice of the dataset for modeling work is theone based on a typical meteorological year, which is constructed ofeach representative month in a year over a long period (usually 10

or 15 years) weather records. However, due to lack of availablelong-period database in Tainan, Taiwan, it is hard tomake a definiteassessment of the model performance at the present time.

As noted from Table 2, the model of Boland et al. was con-structed with data from cities worldwide and the model of Erbset al. was developed with data from cities in USA. In contrast, themodel of Chandrasekaran and Kumar used the data from a singlecity (Madras, India) which is the same situation as for developmentof the present models (Tainan, Taiwan). Thus, the models of Erbset al. and Boland et al. possess more potential to be applied to otherplaces in the world in comparison with the presently proposedmodels.

5. Conclusions

Four correlation models with five predictors: kt, a, t, KT and j, assuggested in the BRL model of Ridley et al. [16], are constructed

C.-W. Kuo et al. / Renewable Energy 66 (2014) 56e61 61

respectively by using two different yearly datasets to relate thehourly diffuse fraction on a horizontal surface to the clearness in-dex for the Taiwan area. Two models, Eqs. (6) and (7), which followthe mathematical format of the BRL model, use a single logisticequation for all values of kt, and the other two models, Eqs. (8) and(9), use a set of piece-wise linear equations for four intervals of kt.The proposed models are compared respectively with the fourteenmodels available in the literature using the four widely used sta-tistical indicators: the MBE, the RMSE, the t-statistic and the BIC,using the out-of-sample dekt databases for Tainan, Taiwan. Fromthe analyses, it is found that the proposed piece-wise linear modelsperform well in diffuse-fraction predictions in Tainan, Taiwan,while the proposed logistic (or modified BRL) models show morecase-dependent performance. Another better performing modelsamong the others considered in Table 1, include two piece-wisehigher-order polynomial models respectively developed by Erbset al. [4] as well as Chandrasekaran and Kumar [9], and one time-independent logistic model developed by Boland et al. [15], whenapplying to the estimation of diffuse fraction in the Taiwan area.

Acknowledgments

This work was financially supported by the Bureau of Energy,the Ministry of Economic Affairs, Taiwan under Grant No. 102-D0304.

Nomenclatured hourly horizontal diffuse fractiondest estimated value of ddmea measured data of dHo extraterrestrial radiation (kJ/h m2)Ibeam horizontal beam radiation (kJ/h m2)Idiffuse horizontal diffuse radiation (kJ/h m2)Iglobal horizontal global radiation (kJ/h m2)KT daily sky clearness indexk number of coefficients to be estimated in the BIC testkt hourly sky clearness indexn number of data pointsRH relative humidityTa ambient temperature (�C)t apparent solar time (h)

tcrit. critical value of tstat.tstat. value of the t-statistica solar altitude (degree)j measure of persistence of global radiation level

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