9
New Technique for Estimating the Monthly Average Daily Global Solar Radiation Using Bees Algorithm and Empirical Equations Hajar Bagheri Tolabi, a Shahrin Bin Md Ayob, b M.H. Moradi, c and Mehrnoosh Shakarmi d a Faculty of Engineering, Islamic Azad University of Khorramabad, Iran; [email protected] (for correspondence) b Faculty of Electrical Engineering, Universiti Teknologi Malaysia, Johor, Malaysia c Faculty of Engineering, Bu Ali Sina University, Hamadan, Iran d Faculty of Engineering, Lorestan University, Khorramabad, Iran Published online 00 Month 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/ep.11858 Measurement of solar radiance demands expensive devi- ces to be used. Alternatively, estimator models are used instead. In this article, a new method based on the empirical equations is introduced to estimate the monthly average daily global solar radiation (GSR) on a horizontal surface. The proposed method uses Bees algorithm as a heuristic and population-based search technique. The best coefficients of linear and nonlinear empirical models and GSR are calcu- lated for seven different climate regions of Iran using pro- posed algorithm written in MATLAB software. The results of the proposed method are compared with other techniques. The result shows that the proposed method is more accurate in estimating the monthly average daily GSR. V C 2013 Ameri- can Institute of Chemical Engineers Environ Prog, 00: 000– 000, 2013 Keywords: Bees algorithm, global solar radiation, empiri- cal coefficients, statistical regression techniques, empirical- based models, intelligent-based models INTRODUCTION Solar radiation (SR) is the most essential parameter for design and development of various solar energy systems [1]. SR data provide information on the amount of the sun’s energy strikes a surface at a region on earth, during a partic- ular time period. The measured data are also important as they are used to aid in system design that is related to mete- orology, agricultural sciences, and engineering, as well as in the health sector and natural sciences applications [2–4]. In developing countries, global solar radiation (GSR) measure- ments are usually made at few sites, due to small number of solar observation stations, as well as high installation cost and maintenance [5]. Hence, to alleviate the problems, math- ematical models are used to accurately estimate the GSR. Recently, numerous studies have been conducted to esti- mate SR based on available geographic and meteorological parameters such as minimum and maximum temperature, relative moisture, elevation, SR hours, rainfall, cloudiness, and wind speed [6]. They can be typically categorized under two types, namely, empirical-based models and intelligent- based models [7]. For empirical-based models, Angstrom [8] proposed the first empirical relation for GSR estimation using sunshine hours for a long time. Prescott [9] modified Ang- strom model and is known as Angstrom–Prescott model. On the other hand, Page [10] reported the coefficients of the Angstrom–Prescott model, which is believed to be univer- sally applied. Bahel et al. [11] developed a worldwide corre- lation based on radiation data and sunshine hours for 48 stations around the world, with different meteorological and geographical conditions. A new time-dependent model was proposed by Yeboah-Amankwah and Agyeman [12]. Nino- miya [13] considered the effect of rainy days on the GSR esti- mation. Burari et al. [14] developed a model for GSR estimation in Bauchi with special regression coefficients. Chandel et al. [15] proposed a model based on temperature. Other multiparameter models were presented by Trabea et al. [16], Ojosu and Komolafe [17], and Garg and Garg [18]. Because of the complex and nonlinear nature of proposed empirical models, it is customary to solve the problem using statistical regression techniques (SRTs). Zabara [19], Samuel [20], Newland [21], Yazdanpanah et al. [22], and Sivamadhavi and Samuel Selvaraj [23] tried to estimate GSR using SRTs based on the aforementioned models or new proposed mod- els for different places in the world. For intelligent-based models, Mellit et al. [24] proposed an artificial neural network (ANN) model for the prediction of SR data with the application of sizing stand-alone photovol- taic power system. Moghaddamnia et al. [25] used an adapt- ive neuro-fuzzy inference system for SR prediction. In 1998, Sen [26] evaluated daily solar irradiance from the hours of sunlight using fuzzy set theory. In this study, the empirical equations are replaced by a set of fuzzy-rule bases and applied for three sites with monthly averages of daily irradi- ances in the western part of Turkey [27]. Su et al. [28] used genetic algorithm (GA) to improve the neural networks per- formance in SR estimation. Because the back-propagation (BP) neural networks are apt to converge at local optimal point, they used GA to optimize BP neural networks’ weights and threshold values. They proved that this compound algo- rithm’s prediction precision yields better performance for SR estimation. Wang et al. [3] proposed a GA optimization of wavelet neural network (GAO-WNN) model for daily SR V C 2013 American Institute of Chemical Engineers Environmental Progress & Sustainable Energy (Vol.00, No.00) DOI 10.1002/ep Month 2013 1

New technique for estimating the monthly average daily global solar radiation using bees algorithm and empirical equations

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Page 1: New technique for estimating the monthly average daily global solar radiation using bees algorithm and empirical equations

New Technique for Estimating the Monthly Average

Daily Global Solar Radiation Using Bees Algorithm

and Empirical EquationsHajar Bagheri Tolabi,a Shahrin Bin Md Ayob,b M.H. Moradi,c and Mehrnoosh Shakarmida Faculty of Engineering, Islamic Azad University of Khorramabad, Iran; [email protected] (for correspondence)b Faculty of Electrical Engineering, Universiti Teknologi Malaysia, Johor, Malaysiac Faculty of Engineering, Bu Ali Sina University, Hamadan, Irand Faculty of Engineering, Lorestan University, Khorramabad, Iran

Published online 00 Month 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/ep.11858

Measurement of solar radiance demands expensive devi-ces to be used. Alternatively, estimator models are usedinstead. In this article, a new method based on the empiricalequations is introduced to estimate the monthly averagedaily global solar radiation (GSR) on a horizontal surface.The proposed method uses Bees algorithm as a heuristic andpopulation-based search technique. The best coefficients oflinear and nonlinear empirical models and GSR are calcu-lated for seven different climate regions of Iran using pro-posed algorithm written in MATLAB software. The results ofthe proposed method are compared with other techniques.The result shows that the proposed method is more accuratein estimating the monthly average daily GSR. VC 2013 Ameri-can Institute of Chemical Engineers Environ Prog, 00: 000–000, 2013

Keywords: Bees algorithm, global solar radiation, empiri-cal coefficients, statistical regression techniques, empirical-based models, intelligent-based models

INTRODUCTION

Solar radiation (SR) is the most essential parameter fordesign and development of various solar energy systems [1].SR data provide information on the amount of the sun’senergy strikes a surface at a region on earth, during a partic-ular time period. The measured data are also important asthey are used to aid in system design that is related to mete-orology, agricultural sciences, and engineering, as well as inthe health sector and natural sciences applications [2–4]. Indeveloping countries, global solar radiation (GSR) measure-ments are usually made at few sites, due to small number ofsolar observation stations, as well as high installation costand maintenance [5]. Hence, to alleviate the problems, math-ematical models are used to accurately estimate the GSR.

Recently, numerous studies have been conducted to esti-mate SR based on available geographic and meteorologicalparameters such as minimum and maximum temperature,relative moisture, elevation, SR hours, rainfall, cloudiness,and wind speed [6]. They can be typically categorized undertwo types, namely, empirical-based models and intelligent-

based models [7]. For empirical-based models, Angstrom [8]proposed the first empirical relation for GSR estimation usingsunshine hours for a long time. Prescott [9] modified Ang-strom model and is known as Angstrom–Prescott model. Onthe other hand, Page [10] reported the coefficients of theAngstrom–Prescott model, which is believed to be univer-sally applied. Bahel et al. [11] developed a worldwide corre-lation based on radiation data and sunshine hours for 48stations around the world, with different meteorological andgeographical conditions. A new time-dependent model wasproposed by Yeboah-Amankwah and Agyeman [12]. Nino-miya [13] considered the effect of rainy days on the GSR esti-mation. Burari et al. [14] developed a model for GSRestimation in Bauchi with special regression coefficients.Chandel et al. [15] proposed a model based on temperature.Other multiparameter models were presented by Trabea etal. [16], Ojosu and Komolafe [17], and Garg and Garg [18].Because of the complex and nonlinear nature of proposedempirical models, it is customary to solve the problem usingstatistical regression techniques (SRTs). Zabara [19], Samuel[20], Newland [21], Yazdanpanah et al. [22], and Sivamadhaviand Samuel Selvaraj [23] tried to estimate GSR using SRTsbased on the aforementioned models or new proposed mod-els for different places in the world.

For intelligent-based models, Mellit et al. [24] proposed anartificial neural network (ANN) model for the prediction ofSR data with the application of sizing stand-alone photovol-taic power system. Moghaddamnia et al. [25] used an adapt-ive neuro-fuzzy inference system for SR prediction. In 1998,Sen [26] evaluated daily solar irradiance from the hours ofsunlight using fuzzy set theory. In this study, the empiricalequations are replaced by a set of fuzzy-rule bases andapplied for three sites with monthly averages of daily irradi-ances in the western part of Turkey [27]. Su et al. [28] usedgenetic algorithm (GA) to improve the neural networks per-formance in SR estimation. Because the back-propagation(BP) neural networks are apt to converge at local optimalpoint, they used GA to optimize BP neural networks’ weightsand threshold values. They proved that this compound algo-rithm’s prediction precision yields better performance for SRestimation. Wang et al. [3] proposed a GA optimization ofwavelet neural network (GAO-WNN) model for daily SRVC 2013 American Institute of Chemical Engineers

Environmental Progress & Sustainable Energy (Vol.00, No.00) DOI 10.1002/ep Month 2013 1

Page 2: New technique for estimating the monthly average daily global solar radiation using bees algorithm and empirical equations

estimation. It uses a combination of global optimality of GAand favorable local properties of WNN to train the model. Itis believed to produce an accurate and reliable method topredicate daily SR. In addition, it provides an effective wayfor the initialization of parameters and reaching the globaloptimum quickly. Mohandes [29] used particle swarm optimi-zation (PSO) for modeling GSR. In his work, PSO is used totrain an ANN using data from available measurement stationsto estimate monthly mean daily GSR in the Kingdom ofSaudi Arabia.

In this study, the Bees algorithm (BA) is used for the firsttime to estimate monthly average daily GSR on horizontalsurface for seven different climate cities of Iran based on thedifferent linear and nonlinear empirical equations. Theremainder of this article is organized in the following man-ner: linear and nonlinear empirical equations are discussedin the “Empirical Equations for SR Estimation” section. Con-cept of BA and its overall progress is reviewed in the “BeesAlgorithm and Its Overall Progress” section. Proposed meth-odology to find the optimal empirical coefficients and theestimation of SR based on the BA are investigated in the“Proposed Methodology” section. The obtained results ofapplying the new method on the sample cities, discussionsand comparisons between the generated outcomes, othertechniques, and actual data are presented in the “Results andDiscussion” section. Finally, a summary of the results is givenin the “Conclusion” section.

EMPIRICAL EQUATIONS FOR SR ESTIMATION

Over the past decades, many researchers have designed dif-ferent empirical equations for estimating SR using meteorologi-cal parameters. Some of these empirical equations are basedon the cloudiness, and others are based on sunshine hours,temperature, and so forth. In an overall view, they can bedivided into two categories: linear and nonlinear equations.

Linear Empirical EquationsIn the literature, the original models expressed the rela-

tionship between SR and the sunshine duration as a straightline. This pioneering relationship was first presented by Ang-strom in 1924 [8]. Prescott [9] in 1940 modified Angstrommodel and is known as Angstrom–Prescott model, which canbe expressed as a linear regression expression:

H

H05a1b

S

S0

� �(1)

where H is the GSR, H0 is the extraterrestrial SR, S is theactual sunshine hours, S0 is the maximum possible sunshineduration, and a and b are the empirical coefficients.

Afterward, some models have been proposed with addi-tional meteorological parameters. Swartman and Ogunlade[30] and Abdallah [31] proposed linear models, which arepresented in Eqs. 2 and 3, respectively:

H5a1bS

S0

� �1cRH (2)

H

H05a1b

S

S0

� �1cRH1dT (3)

where RH is the mean relative humidity, T is the daily meanair temperature, and a–d are the empirical coefficients.

Nonlinear Empirical EquationsAtmospheric turbidity and transmission, cloud thickness,

planetary boundary layer turbulence, and temporal and spa-tial variations cause embedding of nonlinear elements in the

SR phenomena. Hence, most often the linear models aremodified with adding extra terms to the linear equation.€Ogelman et al. [32] and Akinoglu et al. [33] suggested addinga nonlinear term to the Angstrom model; therefore, the fol-lowing quadratic equation was obtained:

H

H05a1b

S

S0

� �1c

S

S0

� �2

: (4)

Bahel et al. [11] proposed higher-order polynomial non-linear model for estimating GSR as follows:

H5H0½a1bS

S0

� �1c

S

S0

� �2

�1dS

S0

� �3

(5)

Almorox and Hontoria (1967) suggested the following expo-nential relationship between SR and sunshine hours [34]:

H5a1bexpS

S0

� �: (6)

Bakirci [35] developed the following model for GSRestimation:

H5a1bS

S0

� �1c exp

S

S0

� �: (7)

A logarithmic equation for SR estimation was proposedby Ampratwum and Dorvlo [36] as follows 1999:

H

H05a1b log

S

S0

� �: (8)

Behrang et al. [37] introduced five nonlinear equations forSR estimation, and the results showed that the new nonlinearmodels have better performance than the existing models.

BEES ALGORITHM AND ITS OVERALL PROGRESS

A new heuristic optimization technique mimicking thebee behavior has been developed in 2005 by Pham et al.[38]. To exploit larger number of food sources in nature, thecolony of bees can extend itself simultaneously over longdistances and in multiple directions. In fact, flower patcheswith more nectar that acquire less effort should observemore bees, whereas patches with less nectar should receivefewer bees. The BA equilibrates between the global and thelocal search. The BA randomly explores the solution spacelooking for areas of potential optimality. The neighborhoodsearch is based on a random distribution of bees in a prede-fined neighborhood range. For each selected site, bees arerandomly distributed to find a better solution. Only the bestbee is chosen to advertise its source after which the centerof the neighborhood field is shifted to the position of thebest bee. Then BA exploits the optimal areas by conductinga local search, until either a satisfactory solution is found ora defined number of iterations has been reached. The pro-cess begins by scout bees dispatched randomly from onepatch to another in search of suitable flower patches.The algorithm includes neighborhood and global search[38,39].

The algorithm needs a number of parameters to be set:number of scout bees (n), number of sites selected out of nviewed sites (m), number of best sites out of m chosen sites(e), number of bees employed for the best e sites (nep),number of bees used for the other (m–e) selected sites (nsp),and initial size of patches (ngh) that includes site, its neigh-borhood, and stop criterion [38].

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The process of finding the optimum solution using BA isshown in Figure 1 as a flow chart and described by the fol-lowing steps [38–40]:

Step 1. Initialize population by random solutions.Step 2. Evaluate the fitness of population.Step 3. While (stopping condition not met)/creating new

population.Step 4. Choose sites for neighborhood search.Step 5. Employ bees for chosen sites and evaluate fitness.Step 6. Select the best bee from each patch.Step 7. Allocate remaining bees to search randomly and

evaluate their fitness.Step 8. End while.The algorithm starts with the number of scout bees (n)

being placed randomly in the search space in Step 1. The fit-nesses of the sites visited by the scout bees are evaluated inStep 2. In Step 4, the bees that have the highest finesses arechosen as “selected bees” and sites visited by them are cho-sen for neighborhood search. Then, in Steps 5 and 6, thealgorithm conducts searches in the neighborhood of theselected sites, assigning more bees to search near to the beste sites. The bees can be chosen directly according to the fit-ness associated with the sites they are visiting. Alternatively,the fitness values are used to determine the probability ofthe bees being selected. Searches in the neighborhood of thebest e sites, which represent more promising solutions, aremade more detailed by recruiting more bees to follow themthan the other selected bees. Together with scouting, this dif-ferential recruitment is a key operation of the BA. However,in Step 6, for each patch only the bee with the highest fitnesswill be selected to form the next bee population. In nature,there is no such restriction. This restriction is introducedhere to reduce the number of points to be explored. In Step7, the remaining bees in the population are randomlyassigned around the search space scouting for new potentialsolutions. These steps are repeated until a stopping criterionis met. At the end of each iteration, the colony will have two

parts to its new population, representatives from eachselected patch and other scout bees assigned to conduct ran-dom searches [38].

PROPOSED METHODOLOGY

In this method, the BA is used to find the best coefficientsof empirical models (both linear and nonlinear equations)and to estimate the GSR based on the measured data. Thedetailed description of this method is summarized in the fol-lowing steps.

Step 1: Grouping of Measured DataAll measured data provided by meteorological offices are

divided into two different parts: installation and validationdata series. The period of installation data should be differ-ent from the validation data series [e.g., in this study, forEsfahan city, 111 months are considered for installation datafrom 1985 until 2001, and 45 valid months (valid months hasbeen described in the Appendix) are considered from 2002until 2005 for validation data series].

Step 2: Calculation of the Required Values UsingMeasured Data

The values of HH0

(the fraction of possible monthly averagedaily GSR) and S

S0(the fraction of possible monthly average

daily sunshine duration) are calculated using measured data,in both installation and validation periods.

Step 3: Estimation of Empirical Coefficients and GSRCalculated values for installation period from Step 2 are

used in BA program to find the candidates of the best coeffi-cients for the empirical equations to minimize the fitnessfunction, which is defined as follows:

Figure 1. The process of finding the optimum solution using BA.

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Page 4: New technique for estimating the monthly average daily global solar radiation using bees algorithm and empirical equations

F5Xmi51

Yi2Xið Þ2 (9)

where Yi5HH0

� �ci and Xi5

HH0

� �ei are the calculated and esti-

mated fraction of possible monthly average daily GSR,respectively, for the ith observation, H is the GSR, H0 theextraterrestrial SR, and m illustrates the cumulative observa-tions. [Calculation of the extraterrestrial SR (H0) is discussedin Ref. [41].

The BA process continues until the stopping criterion(herein number of iterations) is satisfied.

Step 4: Validation of ResultsAfter running each program, the obtained results of BA

(candidates of the best empirical coefficients) are validated

using calculated values in the validation period. If theobtained GSR values using BA are in good agreement withthe calculated GSR values in the validation period (require-ment of minimum 80% agreement is considered in thisstudy), thus the obtained empirical coefficients are the bestotherwise this process is repeated from Step 3.

Figure 2 shows the general pattern for estimating SR usingthe combination of BA and empirical equations.

The accuracy of obtained empirical coefficients has beeninvestigated by using two statistical indicators, absolute frac-tion of variance (R2) and root mean square error (RMSE). R2

and RMSE are described by Eqs. 10 and 11, respectively, asfollows:

R2512

Xm

i51Xi2Yið Þ2Xm

i51Xið Þ2

; (10)

RMSE5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXmi51

Xi2Yið Þ2,m

vuuuut : (11)

(Xi;Yi and m have been defined in Eq. 9).

RESULTS AND DISCUSSION

In this section, BA is implemented to estimate themonthly average daily GSR on horizontal surface for sevendifferent climate cities of Iran by programming in the MAT-LAB software [3]. The geographical positions of the sample

Figure 2. General pattern of SR estimation using combina-tion of BA and empirical equations.

Figure 3. Geographical positions of seven sample cities ofIran. [Color figure can be viewed in the online issue, whichis available at wileyonlinelibrary.com.]

Table 1. Information of seven sample cities.

City name Longitude �E Latitude �N Altitude (m)Installationdata period

Validationdata period

Esfahan 51.67 32.62 1550.4 1985–2001 2002–2005Hamadan 48.53 34.87 1741.5 1985–2001 2002–2005Kerman 56.97 30.25 1753.8 1984–2001 2002–2005Mashhad 59.63 36.27 999.2 1980–2000 2001–2003Orumieh 45.05 37.67 1328.0 1985–2001 2002–2004Khoorbiabanak 55.08 33.78 845.0 1988–2001 2002–2005Tabriz 46.28 38.08 1361.0 1987–2001 2002–2005

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regions are shown on the Iran map in Figure 3. These sam-ple cities are Esfahan, Hamadan, Kerman, Mashhad, Tabriz,Khoorbiabanak, and Orumieh. All required data such as min-imum and maximum temperature, SR hours, relative mois-ture, and elevation have been provided by themeteorological office of Iran. The information of the sevencities considered in this study is given in Table 1. In Table 1,the information of longitude, latitude, altitude, and all data-collected ranges have been given for all seven cities. Col-lected data are divided in two installation and validationparts. Details of both data collection ranges are presented inTable 1. As can be seen in Table 1, the data-collected rangesare so great to achieve meaningful results. To check theaccuracy of the measured data, the method presented byMoradi [42] is used. H

H0and S

S0values have been calculated

separately for both installation and validation data series forall seven cities. The H

H0and H

H0calculated values on 16 sample

months for Hamadan and Tabriz cities are given in Tables 2and 3, respectively. These sample months are 1, 11, 21, 31,41, 51, 61, 71, 81, 91, 101, and 111 months based on installa-tion data and 121, 131, 14, and 151 months for validationdata series.

Among the equations presented in “Empirical Equationsfor SR Estimation” section, two linear and two nonlinearempirical models have been selected to evaluate the per-formance of the proposed technique to find the best empiri-cal coefficients and GSR estimation on sample cities.

Linear selected empirical equations include equations pro-posed by Prescott (Eq. 1) [9] and Abdallah (Eq. 3) [31], here-after defined as Model 1 and Model 2, respectively.Nonlinear selected empirical equations are equations pro-posed by Akinoglu and Ecevit (Eq. 4) [33] and Ampratwumand Dorvlo (Eq. 8) [36], hereafter defined as Model 3 andModel 4, respectively.

Table 2. Sample calculated values of HH0

and SS0

for both datatypes on Hamadan city.

Data series Months HH0

SS0

Installation 1 0.510 0.447Installation 11 0.485 0.485Installation 21 0.513 0.512Installation 31 0.424 0.421Installation 41 0.423 0.398Installation 51 0.500 0.423Installation 61 0.681 0.779Installation 71 0.554 0.878Installation 81 0.529 0.891Installation 91 0.844 0.850Installation 101 0.759 0.633Installation 111 0.796 0.774Validation 121 0.620 0.619Validation 131 0.597 0.598Validation 141 0.591 0.678Validation 151 0.693 0.700

Table 3. Sample calculated values of HH0

and SS0

for both datatypes on Tabriz city.

Data series Months HH0

SS0

Installation 1 0.419 0.523Installation 11 0.493 0.212Installation 21 0.655 0.735Installation 31 0.873 0.820Installation 41 0.700 0.791Installation 51 0.670 0.813Installation 61 0.715 0.895Installation 71 0.403 0.319Installation 81 0.512 0.600Installation 91 0.697 0.732Installation 101 0.478 0.600Installation 111 0.581 0.591Validation 121 0.400 0.400Validation 131 0.493 0.641Validation 141 0.448 0.823Validation 151 0.500 0.615

Table 4. Parameters used for the Bees algorithm.

Parameter Value

n: number of scout bees 70m: number of sites selected out of n visited sites 8e: number of best sites out of m selected sites 2nep: number of bees recruited for best e sites 26nsp: number of bees recruited for other

(m–e) selected sites 6ngh: neighborhood size 5Number of iterations 300

Table 5. Obtained empirical coefficients using BA.

City nameEmpirical

model a, b, c, d

Esfahan Model 1 0.39906, 0.32148Model 2 0.07851, 0.42319, 0.21954,

0.12943Model 3 0.53276, 20.00124, 0.18459Model 4 0.69416, 0.20893

Hamadan Model 1 0.36710, 0.30821Model 2 0.23096, 0.14398, 0.309432,

0.28409Model 3 0.50125, 20.06171, 0.33186Model 4 0.73761, 0.26813

Kerman Model 1 0.32507, 0.50238Model 2 0.17320, 0.19320, 0.21062,

0.24945Model 3 0.22454, 0.80448, 20.19675Model 4 0.82525, 0.37466

Mashhad Model 1 0.32846, 0.30162Model 2 0.31830, 0.38306, 0.29408,

0.31093Model 3 0.95321, 21.03523, 1.0009Model 4 0.69562, 0.09413

Orumieh Model 1 0.36413, 0.34172Model 2 0.18545, 20.16936, 0.19206,

0.17043Model 3 0.67349, 20.48877, 0.33151Model 4 0.46213, 20.06955

Tabriz Model 1 0.33372, 0.42148Model 2 0.38065, 0.34265, 0.39629,

0.30284Model 3 0.13279, 0.74376, 20.31547Model 4 0.55728, 0.21934

Khoorbiabanak Model 1 0.33293, 0.39008Model 2 0.0.4126, 0.37305, 0.43538,

20.41972Model 3 0.34525, 0.34723, 0.05423Model 4 0.72027, 0.28468

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By applying the BA program written in the MATLAB soft-ware based on the procedures described in the “Bees Algo-rithm and Its Overall Progress” and “Proposed Methodology”sections, it was observed that the performance of BA is satis-factory using the parameters values shown in Table 4. This isapplicable for all cities. The obtained coefficients of a, b, c,and d for four tested empirical models on the seven samplecities using BA are given in Table 5. Similar data are used toensure a fair comparison between the performance of BA,SRTs, and ANN on GSR modeling. The empirical coefficients

for four empirical models are separately calculated for sevensample cities using SRTs (least absolute deviations method)[9,31,33,36], and an ANN model trained using the Levenberg–Marquardt algorithm with sigmoid and linear transfer func-tions in the hidden and output layers, respectively, wasselected by using neural network toolbox of MATLAB soft-ware [43].

RMSE and R2 values for obtained GSR values using differ-ent models based on BA, SRT, and ANN are given in Table6. The results show the superiority of combination of BA

Figure 4. Comparison between actual and BA, SRT (based on Angstrom model), and ANN values of monthly average dailyGSR for seven sample cities. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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and linear Angstrom model (BA and Model 1) rather thanother models for SR estimation with R2 greater than 0.96 andRMSE smaller than 0.03 on seven sample cities. Among allsample cities, the best result is obtained for Esfahan with R2

5 0.9931 and RMSE 5 0.0009 for combination of BA andAngstrom model (BA and Model 1), and the worst result isobtained for Orumieh with R2 5 0.8387 and RMSE 5 0.2610for SRT and Abdallah model (SRTs and Model 2).

By calculating the average of R2 and RMSE values for alltested models on seven sample cities from Table 6, the fol-lowing conclusions have been obtained:� Combination of BA and all linear and nonlinear tested

empirical models has generated acceptable results withR2

average > 0.95 and RMSEaverage < 0.16 for all samplecities.� Among BA, SRT, and ANN models, the best result is

obtained by BA and Angstrom model (BA and Model 1)with R2

average 5 0.982 and RMSEaverage 5 0.010, and theworst result is obtained by SRT and Abdallah model (SRTsand Model 2) with R2

average 5 0.884, RMSEaverage 5 0.074.� Combination of linear empirical models (Models 1 and 2)

and BA had R2average values greater than the combination

of nonlinear empirical models (Models 3 and 4) and BA,whereas such a conclusion does not apply for RMSEaverage

values.� The performance of ANN with R2

average 5 0.948 andRMSEaverage 5 0.012 is better than SRTs and very close toBA results, whereas BA does not need a complex trainingstage, and thus, it can be said that using the BA is supe-rior to the ANN.Figure 4 shows the comparisons between GSR estimations

obtained by BA and Angstrom model (BA and Model 1), SRTand Angstrom model (SRT and Model 1), and ANN withactual data for all seven sample cities.

CONCLUSION

This study proposed a new technique based on the com-bination of BA with linear and nonlinear empirical equationsto estimate the monthly average daily GSR on horizontal sur-face. It is implemented in MATLAB software. The proposedalgorithm is tested on seven different climate cities of Iranbased on two linear and two nonlinear empirical equationsto determine the best empirical coefficients and GSR estima-

tion by using installation and validation data series providedby the meteorological office of Iran. The results produced bythe proposed technique are compared with SRT and ANNtechnique using two introduced statistical indicators: absolutefraction of variance and RMSE. The comparison of resultsproved the superiority of combination of BA and linear Ang-strom model rather than other models for SR estimation withR2 greater than 0.96 and RMSE smaller than 0.03 in all sam-ple cities. Among all sample cities and compared methods,the best result is obtained for Esfahan in the center of Iranfor combination of BA and Angstrom model, and the weak-est result is obtained for Orumieh in the northwest of Iranfor SRTs based on Abdallah model. Although the perform-ance of ANN is better than tested SRTs and very close to BAresults, however, because of the need of complex trainingstage in ANN, it can be said that using the BA is superior tothe ANN.

APPENDIX

The limit check (i.e., higher limits of monthly average dailysunshine hours and monthly average daily limits of extrater-restrial SR of the location) is carried out on the monthlymean daily GSR and monthly mean daily sunshine durationto make sure that the data are homogeneous and the varia-tions of monthly mean daily GSR are caused only by climaticinfluences and not by other sources of errors (e.g., system-atic errors cased by instruments, calibration, and data trans-ferring) [37]. In this study, the number of valid data for eachregion is calculated from the following equation:

Nv5Nt2Nl;

where Nv is the number of valid data, Nt is the number of

total available data in the period, and Nl is the number of

data in the period out of limit.

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Table 6. Evaluation of accuracy of results through R2 and RMSE indicators.

City name Method R2 RMSE

Esfahan BA and Models 1–4 0.9931, 0.9807, 0.9604, 0.9520 0.0009, 0.0021, 0.0144, 0.0057SRTs and Models 1–4 0.9913, 0.8635, 0.8726, 0.9278 0.0135, 0.1046, 0.1012, 0.0481ANN 0.9377 0.0108

Hamadan BA and Models 1–4 0.9912, 0.9417, 0.9699, 0.9835 0.0011, 0.0364, 0.0047, 0.0019SRTs and Models 1–4 0.9147, 0.8817, 0.8915, 0.9167 0.0715, 0.0376, 0.0358, 0.0169ANN 0.9815 0.0012

Kerman BA and Models 1–4 0.9872, 0.9751, 0.9411, 0.9407 0.0083, 0.0136, 0.0063, 0.0071SRTs and Models 1–4 0.9864, 0.8934, 0.9318, 0.8916 0.0066, 0.0267, 0.0189, 0.0164ANN 0.9409 0.0282

Mashhad BA and Models 1–4 0.9786, 0.9639, 0.9618, 0.9484 0.0175, 0.0294, 0.0092, 0.0205SRTs and Models 1–4 0.8815, 0.8753, 0.8836, 0.8719 0.0518, 0.0710, 0.0303, 0.0615ANN 0.9639 0.0189

Orumieh BA and Models 1–4 0.9651, 0.9587, 0.9418, 0.9445 0.0256, 0.0261, 0.0167, 0.0211SRTs and Models 1–4 0.9312, 0.8387, 0.9170, 0.9235 0.0173, 0.2610, 0.0151, 0.0287ANN 0.9461 0.0179

Tabriz BA and Models 1–4 0.9745, 0.9736, 0.9643, 0.9572 0.0194, 0.0085, 0.0187, 0.0034SRTs and Models 1–4 0.8953, 0.9055, 0.9147, 0.8939 0.0734, 0.0167, 0.0223, 0.1001ANN 0.9716 0.0016

Khoorbiabanak BA and Models 1–4 0.9908, 0.9852, 0.9843, 0.9707 0.0014, 0.0017, 0.0056, 0.0189SRTs and Models 1–4 0.9463, 0.9303, 0.8945, 0.8974 0.0038, 0.0012, 0.0182, 0.0143ANN 0.9532 0.0058

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