A review on classification and comparison of different models in solar radiation estimation

REVIEW PAPER

A review on classification and comparison of differentmodels in solar radiation estimationHajar Bagheri Tolabi1,*,†, M.H. Moradi2 and Shahrin Bin Md Ayob3

1Faculty of Engineering, Department of Electrical Engineering, Islamic Azad University, Khorramabad Branch, Iran2Faculty of Engineering, Department of Electrical Engineering, Bu Ali Sina University, Iran3Faculty of Electrical Engineering, Universiti Teknologi Malaysia, Malaysia

SUMMARY

This paper introduces a new classification scheme for the solar radiation estimation techniques based on three categories:empirical models (based on statistical regression techniques), simulated models (based on training), and optimized models(based on optimization algorithms). For the optimized model category, a novelty bees algorithm estimation based on alinear empirical model is developed. Eight different methods from three classes have been tested on three samplegeographic positions of Iran in order to compare the efficiency, complexity, sensed parameters, and required prior trainingof each category with others by implementing in the Matlab software. Among all tested models, the best properties areobtained for optimized empirical models by optimization algorithms. The main advantages of this model type are that iteliminates the training stage and therefore reduces the complexity rather than simulated models yet offers high accuracyestimation. Copyright © 2014 John Wiley & Sons, Ltd.

KEY WORDS

solar radiation; empirical; simulated and optimized models; statistical regression techniques; optimization algorithms

Correspondence

*Hajar Bagheri Tolabi, Faculty of Engineering, Department of Electrical Engineering, Islamic Azad University, Khorramabad Branch, Iran.†E-mail: [email protected]

Received 12 April 2013; Revised 2 September 2013; Accepted 23 December 2013

1. INTRODUCTION

The energy from the sun propagates in electromagneticwaveform. However, half of the energy is reflected andscattered to the atmosphere while the other half is directedand diffused to hit the earth’s surface. The radiated solarenergy (hereafter referred as solar radiation) governs ourearth’s climate/weather system, keeping the hydrologiccycle in motion and vital as the basis of life in the earth.Referring to these, the knowledge on the solar radiationis extremely important. The knowledge can be widelyapplied in various applications such as meteorological,agriculture science, biological, engineering, architecture,etc. to enhance the human quality of life.

The global solar radiation (GSR) data for an area can bedirectly obtained using sophisticated devices. However,the high cost of GSR measurement tools installation hashalted the developed countries to make it available at everymeteorological office in their territory. It is also impracticalto have the GSR installed in remote or rural regions, whichchiefly have capability of solar installation [1–5]. Thus,GSR estimation techniques are proposed as the lessexpensive alternative [6]. It calculates the GSR estimation

based on the available geographic and meteorologicalparameters such as minimum and maximum temperature,sunshine hours, relative moisture, elevation, rainfall,cloudiness, and wind speed as the input parameter [7].The techniques are mostly accurate but they may have littlediscrepancy compared to the real-time database collectedusing GSR devices.

Angstrom–Prescott estimation is one of the establishedsolar radiation estimation models. In 2011, Wong andChow [8] reviewed seven models that deployed theAngstrom–Prescott equation to predict the average dailyglobal radiation with hours of sunshine. Bagheri Tolabiet al. propose an imperialist competitive GSR estimationbased on Angstrom model [9]. Several reviews of the solarradiation models have also been conducted and can befound in Ulgen and Hepbasli (2004), Ahmad and Tiwari(2011), and Katiyar and Pandey (2013) [10–12].Meanwhile, Harrouni et al. (2002, 2003, 2005), Maafiet al. (2003), and Badescu (2008) reviewed on the classifi-cation for fractal solar irradiances [7,13–16].

This paper first proposes a critical review on severalprominent GSR estimation models. The review classifiesthe models under three new categories, namely empirical

INTERNATIONAL JOURNAL OF ENERGY RESEARCHInt. J. Energy Res. 2014; 38:689–701

Published online 12 February 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/er.3161

Copyright © 2014 John Wiley & Sons, Ltd. 689

models (based on statistical regression techniques (SRTs)),simulated models (based on training), and optimizedmodels (based on optimization algorithms).A total of eightGSR models are tested on sample geographic positions ofIran in order to compare the efficiency, complexity, sensedparameters, and required prior training. For the optimizedmodel category, a novelty bees algorithm (BA) estimationbased on a linear empirical model and particle swarmoptimization (PSO) based on a nonlinear empirical modelare deployed. Both models have been optimized based onthe minimization an objective function.

The remainder of this paper is organized in the follow-ing manner: the motivation of the solar radiation researchcan be found in Section 2. In Section 3, a critical reviewon the models is conducted. Performance investigationfor eight GSR models based on sample of cities in Iran iscarried out in Section 4. From the comparison, it wasshown that the optimized models are superior comparedto the other two model types.

2. SOLAR RADIATION (SR) SPECIES

Solar radiation from the sun propagates through space andenters the atmosphere at the space–atmosphere interface,where the ionization layer of the atmosphere ends.Afterwards, the atmosphere, clouds, and particles in theatmosphere absorb a certain amount of solar radiation orphotons. A certain amount is, however, reflected back intothe space, and a certain amount is absorbed by the earth’ssurface [17,18].

Solar radiation can be divided into direct, diffuse, andreflected radiation as shown in Figure 1.

Direct radiation is also called beam radiation. It is usedto describe solar radiation traveling on a straight line fromthe sun down to the surface of the earth.

Diffuse radiation is described as the sunlight that has beenscattered by molecules and particles in the atmosphere butthat still made it down to the surface of the earth.

Reflected radiation is mainly reflected from the terrain andis therefore more important in mountainous areas [18–20].

Direct radiation is the most important component ofglobal radiation because it contributes the most to theenergy balance and also the other components depend onit, either directly or indirectly [21].

3. REVIEW AND CLASSIFICATIONOF SR ESTIMATION MODELS

In this section, SR estimation models will be reviewed byclassifying them under three categories: empirical models(based on SRTs), simulated models (based on training),and optimized models (based on optimization algorithms).The general patterns of SR estimation and researchesavailable for each category will be reviewed. In addition,the advantages and drawbacks of each category arehighlighted and briefly discussed.

3.1. Empirical models (based on SRTs)

Over the past few decades, a number of researchers haveproposed different empirical models for SR estimatingusing meteorological parameters, i.e. the cloudiness,sunshine hours, temperature, etc. [22–25]. These modelscan be linear or nonlinear models. Typically, an SRT isused to obtain the coefficient of the empirical models.Figure 2 shows the overall SR estimating procedure usingempirical models based on SRTs in this category. Thesimplicity and non-training stage requirement are the mainadvantages of empirical models. However, they sufferfrom low output response’s accuracy and selection of theappropriate empirical equation with respect to specialclimate of the region. Techniques that fall under thiscategory can be divided into two main sub-categories:linear and nonlinear empirical models as discussed in thefollowing sections.

3.1.1. Linear empirical modelsIn the solar energy literature, the original models

expressed the relationship between solar radiation and thesunshine duration as a straight line. Such pioneeringrelationship was firstly derived by Angstrom in 1924 [24].

Prescott [25] modifies the Angstrom model and isknown as the Angstrom–Prescott model, which can beexpressed as a linear regression expression:

Rs ¼ Ra aþ bn

N

� �� (1)

where Rs is the GSR, Ra is the extraterrestrial solarradiation, n is the actual sunshine hours, N is the maximumpossible sunshine duration, and a and b are the empiricalcoefficients [26].

To increase the accuracy of the SR estimation, severalmodels have been proposed with additional meteorologicalparameters. Swartman and Ogunlade [27] and Abdalla [28]proposed such linear empirical models which are presentedby equations (2) and (3), respectively:

Rs ¼ aþ bn

N

� �þ cRH (2)

Rs ¼ Ra aþ bn

N

� �þ cRH þ dT

� �(3)

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

Further reviews on the linear empirical models can befound in [11].

3.1.2. Nonlinear empirical modelsAtmospheric turbidity and transmission, cloud thickness,

planetary boundary layer turbulence, and temporal andspatial variations cause embedding of nonlinear elements inthe solar radiation phenomena. Hence, most often, linearmodels are modified by adding an extra terms into the linearmodels [17]. Ogelman et al. (1984) and Akinoglu and Ecevit

A new classification for solar radiation estimation techniquesH. B. Tolabi, M. H. Moradi and S. Bin Md Ayob

690 Int. J. Energy Res. 2014; 38:689–701 © 2014 John Wiley & Sons, Ltd.DOI: 10.1002/er

(1990) suggest an addition of nonlinear terms into Angstrommodel. Thus, the following quadratic equation [29,30] wasobtained:

Rs ¼ Ra aþ bn

N

� �þ c

n

N

� �2� �

(4)

Bahel et al. (1987) proposed a higher-order polynomialnonlinear model for estimating GSR as follow [31]:

Rs ¼ Ra aþ bn

N

� �þ c

n

N

� �2� �

þ dn

N

� �3(5)

Almorox and Hontoria (1967) suggested the followingrelationship between solar radiation and sunshine hours [32].

Rs ¼ aþ b expn

N

� �(6)

Bakirik (2009) developed the following model for GSRestimation [33]:

Rs ¼ aþ bn

N

� �þ c exp

n

N

� �(7)

Behrang et al. (2011) introduced five nonlinear equationsfor SR estimation and the results showed that the newnonlinear models have better performance than existingmodels [34]. Further review study on nonlinear empiricalmodels can be found in [11].

3.2. Simulated models (based on training)

For the GSR models that fall under this category, estimationof the solar radiation is done by training and testing. For

Figure 1. Solar radiation types: (a) direct, (b) diffuse, and (c) reflected.

Figure 2. SR estimating procedure using empirical models.

A new classification for solar radiation estimation techniques H. B. Tolabi, M. H. Moradi and S. Bin Md Ayob

691Int. J. Energy Res. 2014; 38:689–701 © 2014 John Wiley & Sons, Ltd.DOI: 10.1002/er

several techniques found in this class, training stage can beaccomplished by a trial and error method. Figure 3 showsthe SR estimating procedure using simulated models. Theyare more efficient as compared to the empirical models butyield greater complexity rather than the latter modelcategory. In general, models under this type of categorycan be divided into local linear regression (LLR) andartificial intelligence (AI)-based models.

3.2.1. LLR modelsThe LLR models are the simplest simulated models

based on non-parametric regression method. This methodis based on locally fitting a line rather than a constant.Local linear estimation removes a bias term from thekernel estimator that makes it to have better behavior.The advantage of the LLR technique is that it yieldsreasonably reliable statistical modeling that can beperformed locally using a small set of sample data. At thesame time, LLR can produce very accurate predictions inregions of high data density in the input space. The onlyproblem with LLR is the size determination of Pmax (thenumber of near neighbors to be included for the local linearmodeling). The method of choosing Pmax for linearregression is called influence statistics.

Given a neighborhood of Pmax points, the followinglinear matrix equation should be solved:

Xm ¼ y (8)

where X is a Pmax × d matrix of the Pmax input points in ddimensions, xi (1≤ i≤Pmax) are the nearest neighborpoints, y is a column vector of length Pmax of thecorresponding outputs, and m is a column vector ofparameters that must be determined to provide the optimalmapping from X to y [35].

The LLR models do not require training in the sameway as that of artificial neural network (ANN) or fuzzylogic (FL). Its optimal number of nearest neighbors for

LLR (principally depending on the noise level) usually isdetermined using a simple training via trial and error method.The LLR can be implemented using a minimum number ofdirect evaluations using a kd-tree (short for k-dimensionaltree) to organize the input training data [36].

Remesman et al. (2008) used an LLR model with 16nearest neighbors for the prediction of solar radiation withthe gamma test (GT). A kd-tree is used to organize theinput training data [35].

Moghaddamnia et al. (2009) used LLR to estimate dailyglobal irradiation values with the GT, based on differentmeteorological input data [37].

3.2.2. AI modelsAI techniques are becoming very popular as an alternative

approach to conventional techniques or as components ofintegrated systems [38]. They have been employed to solvecomplicated practical problems in various areas [39]. In thispaper, three well-known applications of ANN, FL, andadaptive neuro fuzzy inference system (ANFIS) are investi-gated in SR modeling.

3.2.2.1. ANN model. An ANN is a collection ofelectrical neurons (Figure 4) interconnected to each otherin various topologies. The most common application ofan ANN involves identification and modeling of thesystem using nonlinear and complex functions. Duringthe learning process in ANNs, the Weights (Wi) aredetermined. The ANN undergoes an adaptation cycle,during which the weights are updated until the networkreaches a state of equilibrium. The efforts for the introduc-tion of ANN model in estimating solar irradiance, usingrecorded weather parameters as the input and otherenvironmental variables, started in the late 1990.

Mohandes et al. (1998) used latitude, longitude, altitude,and sunshine duration as inputs of multi-layer perceptron andradial basis function neural networks in order to predict GSRfor some stations over Saudi Arabia [40–42].

Al-Alawi and Al Hinai (1998) used location parameters,month, average of pressure, temperature, vapor pressure,relative humidity, wind speed, and sunshine duration asinputs into ANN models to predict GSR [43].

Tymvios and his colleagues (2005) utilized four inputparameters for SR modeling using neural network. In this

Figure 3. SR estimating procedure using simulated models. Figure 4. The basic neuron.



work, several neural network models have been investi-gated, and ANN models have been compared by Angstrommodel outputs [44]. Bosch et al. (2008) proposed neuralnetworks to estimate the parameters of radiation in moun-tainous areas of Spain, and the results are promising. Theresults revealed that ANN is suitable as an SR estimatorfor areas with complex topography [45].

On the other hand, Rehman and Mohandes (2008) useair temperature, number of day, and relative humidity asinputs to neural networks in order to estimate daily GSRfor Abha city in Saudi Arabia. Results showed a meanabsolute percentage error of 4.49% [46].

Sozen et al. (2005) used neural network to estimatesolar radiation in Turkey. They considered six differentclimate structures in the research [47].

Azadeh et al. (2009) present an integrated artificialneural approach for predicting solar global radiation byclimatological variables. The proposed approach is partic-ularly useful for locations where there is no available mea-surement equipment. The method is used for SR estimationin the six different regions of Iran, and the outputs havebeen compared by Angstrom model. The results haveproved the superiority of ANN than other conventionaltechniques [48].

Jiang (2009) has compared ANN predicted values ofaverage daily GSR with other empirical models in variousregions of China, and as a result, the high precision ofANN is emphasized [49].

Behrang et al. (2011) used different ANN techniques topredict daily (GSR) on a horizontal surface, based onmeteorological variables, for Dezful city of Iran [50].

3.2.2.2. FL model. FL was introduced in 1965 byZadeh [51,52]. FL is a form of probabilistic logic that isapproximate rather than fixed and exact. As compared totraditional binary sets, the FL variables may have a truthvalue that ranges in degree between 0 and 1. Figure 5shows the flow diagram for fuzzy inference system. Thissystem implements the FL control in three stages:fuzzification, decision making, and defuzzification [53].

The main feature of fuzzy models is their capability todescribe the knowledge in a descriptive human-likemanner in the form of simple linguistic rules. In thistechnique, the empirical models or any other type ofregression equations can be replaced by a set of fuzzy-rulebases. The fuzzy algorithm developed in these researchesdoes not provide an equation but adjusts itself to each typeof linear or nonlinear form through fuzzy subsets oflinguistic SR variables [54].

Sen [55] evaluates daily solar irradiance from the hoursof sunlight, hence does not correspond to the type of modelanalyzed in the present study. In this study, the empiricalequations are replaced by a set of fuzzy-rule bases andapplied for three sites with monthly averages of dailyirradiances in the western part of Turkey [54].

Meanwhile, Santamouris et al. (1999) propose a fuzzy-ruled model of global solar irradiance. This model calculatesthe GSR on a horizontal surface based on measured data

such as the air temperature, the relative humidity, and thesunshine duration. This method is tested and compared usingvarious sets of SR measurements. The comparison showedthat the proposed intelligent techniques offer good estima-tion of GSR during the warm period of the year. Duringthe cold period, the atmospheric deterministic model wasfound to offer better estimation [56].

Gomez and Casanovas (2003) proposed a model ofsolar irradiance on arbitrarily oriented inclined surfacesbased on FL procedures. The performance of the proposedmodel is superior to similar models, though it requiresminor adjustments. The model considers overlappingclusters and allows an improved description of the skysituations close to the transition zone between contiguouscategories [57].

The application of Takagi–Sugeno (TS) fuzzy modelfor SR modeling has been proposed by Iqdout and Zeroual(2007). The TS fuzzy model is trained using data of dailysolar radiation recorded on a horizontal surface in Dakhlain Morocco. The predicting results indicated that the TSfuzzy model gives a good accuracy of approximately96% and a root mean square error lower than 6%. Inaddition, the performances of the identified TS fuzzymodel have been compared to a linear model using thesecond order statistics techniques. The results showed theeffectiveness of the model [58].

3.2.2.3. ANFIS model. ANFIS model is a kind ofneural network that is based on TS fuzzy inference system.Since it integrates both neural networks and FL principles,it has the potential to capture the benefits of both in a singleframework. Its inference system corresponds to a set offuzzy IF–THEN rules that have learning capability toapproximate nonlinear functions [59,60].

Moghaddamnia et al. (2009) used an ANFIS model for SRestimation. They concluded that the ANFIS model does nothave the ability for precisely SR estimation. This leads to theconclusion that the LLR and neural network auto-regressivewith exogenous inputs models are the most suitable modelsfor this study area [37].

Figure 5. Flow diagram of fuzzy inference system.



Sabziparvar and Bayat (2011) used an ANFIS model inprediction of GSR. The results showed that ANN andANFIS intelligent models are the powerful tools in predic-tion of GSR for the selected stations, but prediction byANN was found to be more accurate than ANFIS [34].

Rahoma et al. (2011) used ANFIS for SR estimation.They can prove that the combination of linguistic rules ofFL with the training algorithm used in neural networkscontributes in very qualitative prediction results, whichapproach the ‘best‘ neural predictor’s results [61].

3.3. Optimizedmodels (basedonoptimizationalgorithms)

The new category proposed in this study is the optimizedmodels. The optimization algorithms (OAs) designate acomputational method that optimizes a problem by itera-tively trying to improve a candidate solution with regard toa given measure of quality. The OAs established a few orno assumptions on the problem being optimized and do thesearch in very large spaces of candidate solutions. However,random optimization does not guarantee an optimal solution[62].Many optimization algorithms implement some form ofstochastic optimization. They have been designed and cre-ated to ‘jump out‘ of local optimal to reach the global one.They are supposed not to ‘get stuck‘ in local optimal. In otherwords, because optimization algorithms perform a wide ran-dom search, the chance of being trapped in local optimal isdeeply decreased [63]. Numerous optimization algorithmsexist, and new variants are continually being proposed. Someof the most significant contributions to the field are: randomoptimization, genetic algorithm, Tabu search, PSO, artificialbee colony algorithm, BA, imperialist competitive algorithm(ICA), and galaxy-based search algorithm [64–72].

Optimization algorithms can be used for both empiricaland simulated models in SR estimation. Accordingly, theoptimized models (based on OAs) can be divided intotwo groups: optimized empirical models by OAs and opti-mized simulated models by OAs.

3.3.1. Optimized empirical models by OAs (Optimizedmodels 1)

In this sub-category, installation data series are used tofind the best coefficients of the empirical models and SRintensity through the optimization algorithms (instead ofSRTs in first class).The credit of the obtained results willbe evaluated through a validation data series. Figure 6 (a)shows SR estimating procedure using this method. Thismethod is presented in detail in Section 4.2. The mainadvantages of this sub-category are the elimination oftraining stage and therefore reducing the complexity ratherthan simulated models, as well as more accurate computedcoefficients for empirical models and thereupon SRintensity estimation with more precise in compare withempirical models based on SRTs.

Behrang et al. (2011) used the combination of PSO andsome linear and nonlinear empirical models to estimatemonthly average daily GSR on horizontal surface for 17different regions of Iran [34]. The quantities of empiricalcoefficients for all models were determined using PSOtechnique for all cities. The models were validated usingvalidation data series. The results showed that the obtainedempirical coefficient for Angstrom model based on PSOhave more accuracy than conventional statistical tech-niques for all tested cities [34].

Bagheri Tolabi et al. (2013) proposed ICA based on theconventional Angstrom model for prediction of GSR infour different climate cities in Iran [9]. Since all Angstromcoefficients obtained by ICA in this study have absolutefraction of variance values (R2) more than statisticalconventional technique results, it is confirmed that newhybrid technique is more accurate than statistical conven-tional methods SR estimation for all tested locations [9].

3.3.2. Optimized simulated models by OAs(Optimized models 2):

In this sub-category, optimization algorithms are usedto improve the training stage to achieve an optimal simu-lated SR model. Figure 6 (b) shows the SR estimatingprocedure using this method. These models are more

Figure 6. SR estimating procedure using: (a) Optimized models 1, and (b) Optimized models 2.



efficient as compared to the empirical and simulatedmodels but yield greater complexity rather than the lattermodels.

Jianmin Su et al. (2009) used GA to improve the neuralnetwork performance in SR estimation [73]. Since theback-propagation (BP) neural networks are apt to convergeat local optimal point, they used genetic algorithm tooptimize BP neural networks’ weights and threshold values.They proved that this compound algorithm’s predictionprecision has better performance to estimate solar radiation.

Wang et al. (2011) proposed a genetic algorithm optimiza-tion of wavelet neural network (GAO-WNN) model for dailysolar radiation estimation [74]. They believed it is an accurateand reliable method to predicate daily solar radiation, becauseusing the combination of global optimality of GA andfavorable local properties ofWNN to train the model, not onlyprovides an effective way for the initialization of parametersbut also for finding the global optimum quickly.

Mohandes (2012) has used PSO formodelingGSR [75]. Inthis study, PSO has been used to train an ANN (PSO-ANN)using data from available measurement stations to estimatemonthly mean daily GSR in the Kingdom of Saudi Arabia.The comparison with BP-ANN and an empirical modelshowed the superiority of the PSO-ANN. Figure 7 showedthe comprehensive view of the proposed classification.

4. CHARACTERISTIC INVESTIGATION

In this section, eight different techniques from three categoriesare compared in terms of efficiency, complexity, sensedparameters, and required prior training. These eight techniquesinclude two methods of empirical models, three techniquesfrom simulated models and three methods from optimizedmodels. The following briefly describe the eight techniques.

Empirical models: Linear Angstrom–Prescott [25], andnonlinear Akinoglu and Ecevit [30] (these empiricalequations are selected because they are based onsunshine hours, and Iran is a sunny country with highsunshine hours [76]). These two models are hereafter

defined as tech1 and tech2, respectively. To estimatemonthly average daily GSR on three sample regions,the coefficients of empirical models of tech1 and tech2have been calculated using SRTs (least absolutedeviations method) by programming in Matlab2007software environment.

Simulated models: LLR [36], ANN [48], and ANFIS[37]. These three models are hereafter defined astech3, tech4, and tech5, respectively. For tech3, theLLR model uses a kd-tree to organize the input train-ing data in Matlab software. The optimal number ofnearest neighbors for LLR is determined by a trialand error method, and 16 nearest neighbors wereimplemented. Various ANN models were tested fortech4, and finally an ANN model trained using theLevenberg–Marquardt algorithm with sigmoid andlinear transfer functions in the hidden and outputlayers, respectively, is selected. A sub-clusteringmethod available in Matlab software is used forANFIS (tech5). In this method, the type of member-ship functions is determined by the model input dataand existing classifications.

Optimized models:Optimized models 1: BA with linear Angstrom–Prescott

empirical model and PSO with nonlinear Akinogluand Ecevit empirical model [34]. Hereafter, thesetwo models are defined as tech6 and tech7, respec-tively. It should be noted that, tech6 is a new modeland never been applied for GSR. Its design stepsand performance study will be discussed in detail un-der Section 4.1. For tech7, the quantities of empiricalcoefficients for Akinoglu and Ecevit model are deter-mined using PSO technique implemented in Matlabsoftware for three sample cities. The model is veri-fied using validation data series.

Optimized models 2: PSO-ANN model [75]. Hereafter,this model is defined as tech8. The PSO is used totrain ANN to estimate the monthly mean daily GSRvalues. The network design started with a networkof five inputs, two units of one hidden layer, andone output to learn all available input output pairs.

4.1. The design and performance analysis ofthe BA with Angstrom–Prescott model

As mentioned earlier, the BA with Angstrom-Prescottempirical model is a novel method for GSR estimationbased on the OAs. Thus, in this section, a brief descriptionon the application of the proposed model is carried out.

Step1: Grouping of measured data: All measured dataprovided by meteorological offices are divided intotwo different parts: installation and validation dataseries. The period of installation data should bedifferent from the validation data series.

Step2: Calculate the required values using measureddata: The values of Rs

Ra(the fraction of possibleFigure 7. Proposed classification for SR estimation techniques.



monthly average daily GSR) and nN (the fraction of

possible monthly average daily sunshine duration)are calculated using measured data, in both installationand validation periods.

Step3: Estimation of empirical coefficients of Angstrommodel and GSR: Calculated values for installationperiod from step 2 are used in BA program to findthe candidates of the best coefficients for the empiricalequations to minimize the fitness function that isdefined as equation (9).

F ¼ ∑m

i¼1Yi � Xið Þ2 (9)

where, Yi ¼ RsRa

� �ci and Xi ¼ Rs

Ra

� �ei are the calculated and

estimated fraction of possible monthly average daily GSR,respectively, for the ith observation, Rs is the GSR, Ra isthe extraterrestrial solar radiation, and m illustrates thecumulative observations (calculation of the extraterrestrialsolar radiation (Ra) is discussed in [26]).

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

Step4: Validation of results: After running the program,the obtained results of BA (candidates of the best

empirical coefficients of Angstrom model) arevalidated using calculated values in the validationperiod. If the GSR values based on the obtained Ang-strom coefficients using BA are in good agreementwith the calculated GSR values in the validation period(requirement of minimum 80% agreement is consid-ered in this study), the obtained empirical coefficientsare the best; otherwise, this process is repeated fromstep 3.

4.2. Case studies and data types

Three different climate cities of Iran, as depicted in Figure 8with the geographical and meteorological characteristicspresented in Table I and Table II, are chosen as data samplesused to evaluate eight different models.

Figure 8. Geographical positions of three samples.

Table I. Geographical information of tested cities.

City name Longitude °E Latitude °N Altitude (m)

Esfahan 51.67 32.62 1550.4Kerman 56.97 30.25 1753.8Orumieh 45.05 37.67 1328.0

Table II. Meteorological information of tested cities.

Minimumtemperature (°C)

Maximumtemperature (°C)

Mean relativemoisture (%)

Sunshinehours (h)

Solar radiation(J.cm�2.day�1)

City name min max ave min max ave min max ave min max ave min max ave

Esfahan �14.4 28.2 9.79 �1 43 24.7 8.5 98.8 34.15 0 13.9 9.4 273 3442 2068.6Kerman �16.2 26.6 7.4 �4 42 25.5 8.9 98.4 32.1 0 13.5 9.1 253 3430 2140.7Orumieh �15.8 23 5.2 �5.2 37 17.3 15.6 97 58.8 0 13.8 7.9 329 3478 1766.2



Data used in this study consisted of meteorological dailydata (minimum temperature, maximum temperature, mean rel-ative moisture, sunshine hours, and solar radiation) recorded atmeteorological stations of Esfahan, Kerman, and Orumieh

during the years 1992 to 2006. This period was chosen be-cause of the availability data in all stations where study wasconducted. All collected data is divided into two parts: trainingdata and validation data series. In order to check the accuracy

Figure 9. Comparison between estimated and actual monthly average daily GSR values for all tested cities.



of the measured solar radiation data, the method presented byMoradi (2009) [77] is used.

4.3. Results and discussion

The estimated monthly average daily GSR values for eighttechniques in comparison with actual data are presented inFigure 9. The proximity between actual and estimatedmonthly average daily GSR is calculated using absolutefraction of variance (R2) indicator for each technique bytesting on three sample cities. R2 is defined by the follow-ing equation:

R2 ¼ 1�∑n

i¼1Xi � Yið Þ2

∑n

i¼1Yið Þ2

(10)

where Xi and Yi have been defined in equation (9).Table III summarized the R2 values for three sample

cities. As can be seen in the table, maximum R2 value is99.31% achieved bytech6 in Esfahan city and minimumis 86.61% achieved by tech2in Orumieh city. Highest tolowest R2 values for eight tested techniques on each cityare organized as follows:

Esfahan: tech6, tech7, tech4, tech3, tech8, tech5, tech1,tech2.

Kerman: tech6, tech7, tech4, tech3, tech8, tech5, tech1,tech2.

Orumieh: tech8, tech6, tech7, tech4, tech3, tech5, tech1,tech2.

It is seen that the simulated and optimized models exhibithigherR2 values than the empirical models for all three samplecities. In the following sections, the efficiency, complexity,

sensed parameters, and required prior training characteristicsare investigated and compared for eight tested techniques.

4.3.1. EfficiencyThe average of R2 values on three sample cities is used to

measure the efficiency for each technique. Table IV tabulatesthe results. From the table, it is shown that tech6 has themaximum efficiency (98.18%) while tech2 yields the mini-mum efficiency of 88.70%.

4.3.2. ComplexityThe complexity assessment is defined as requirement for

special expertise engagement, initial training, and high pre-cision. Based on this definition, tech1 and tech2 from theempirical model category offer low complexity. A mediumcomplexity is offered by tech6 and tech7. The tech3, tech4,tech5, and tech8 exhibit high complexity. In contrary, theoptimized empirical models by OAs can be considered ashalf-complex; they do require expertise engagement butcomplex training stage is no longer required.

4.3.3. Sensed parametersAll eight techniques are compared in Table IV in terms

of required parameters to achieve SR estimation. FromTable IV, most of the data dependency is related to tech3,tech4, tech5, and tech8. This is because their modelingbasis is strongly related to the amount of correlation be-tween the parameters. In contrast, the limited dependencyon data for other techniques can be an advantage for them.

4.3.4. Prior trainingAll tested techniques under simulated models and a sub-

category of optimized models (PSO-ANN) require initialtraining to find a suitable nonlinear model for SR estimation.This note is worthy that in the simulated models, the initial

Table III. R2 values for three sample cities.

R2 values (%)

City (tech1) (tech2) (tech3) (tech4) (tech5) (tech6) (tech7) (tech8)

Esfahan 90.86 90.12 95.46 97.00 91.89 99.31 98.26 95.12Kerman 89.49 89.37 95.65 96.00 91.74 98.72 96.01 94.72Orumieh 87.05 86.61 92.22 92.31 88.12 96.51 94.70 96.57

Table IV. Main characteristics of SR estimation techniques.

Technique Category Sensed parameters Complexity Required prior training Efficiency

tech1 Empirical model (linear) Sunshine hours Low No 89.13 (Low)tech2 Empirical model (nonlinear) Sunshine hours Low No 88.70 (Low)tech3 Simulated model (LLR) Depends High Trial and error 94.44 (High)tech4 Simulated model (ANN) Depends High Yes 94.77(High)tech5 Simulated model (ANFIS) Depends High Yes 90.58(Medium)tech6 Optimized model 1 Sunshine hours Medium No 98.18 (High)tech7 Optimized model 1 Sunshine hours Medium No 96.32 (High)tech8 Optimized model 2 Depends High Yes 95.47(High)



training of tech3 (LLRmodel) is easier than other techniquesand usually accomplished by a trial and error method.

The comparison between the characteristics of eighttested techniques is summarized and presented as a selectionguide in Table IV.

5. CONCLUSION

This paper introduces a new classification scheme for solarradiation estimation models based on the three categories:empirical, simulated, and optimized models. Eight differentmodels from the three categories have been implemented inthe Matlab software environment. They are tested on threesample geographic positions of Iran to assess their efficiency,complexity, parameters, and required prior training. Amongthe models, the estimated values obtained by a novel BAbased on linear Angstrom–Prescott empirical model produceresult that is almost identical to the actual data. It yieldsabsolute fraction of variance (R2) value equal to 99.31%for Esfahan while nonlinear Akinoglu and Ecevit empiricalmodel based on least absolute deviations method showedthe most distinctive result from the actual data with R2 valueequal to 86.61% for Orumieh. For each model category, thefollowing models achieved the highest efficiency: linearempirical models (89.13%), ANN model (94.77%), andoptimized empirical model by OAs (98.18%) for empirical,simulated, and optimized models, respectively. It should benoted that the empirical models (linear Angstrom–Prescottand nonlinear Akinoglu and Ecevit models) exhibit thelowest complexity and require no prior training. However,this results in inaccurate GSR estimation. On the other hand,the LLR, ANN, ANFIS, and PSO-ANN produce high accu-racy estimation but demand special training. Above all, thenewly proposed BA based on linear Angstrom–Prescottmodel and PSO with Akinoglu and Ecevit models are thewell-balanced models; they offer medium complexity yethigh accuracy estimation.

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A review on classification and comparison of different models in solar radiation estimation