Independent models for estimation of daily global solar radiation: A review and a case study

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Independent models for estimation of daily global solar radiation: A review and a case study ⁎

Muhammed A. Hassan , A. Khalil, S. Kaseb, M.A. Kassem Mechanical Power Department, Faculty of Engineering, Cairo University, Giza, Egypt

A R T I C L E I N F O

A BS T RAC T

Keywords: Solar energy Global radiation Independent model Regression analysis Empirical models Day number

Independent models are models that estimate daily horizontal global solar radiation without need for prior records of any solar or meteorological parameters such as sunshine duration, cloud cover or temperature. These models are based on the day number, which means that they are very simple, enabling the user to estimate global radiation using only a pocket calculator. Though, they are relatively newer and much fewer as compared to other dependent models such as sunshine-based models. In this study, a first comprehensive review of all such models is provided. Besides, daily radiation data integrated from highly accurate minutely irradiance data measured at Cairo (Egypt) is used to recalibrate and compare independent models statistically. All models are trained and cross-validated using a detailed novel approach. The results show that, despite their simplicity, independent models have very good estimations of solar irradiation with coefficients of determination no less than 82%.

1. Introduction Estimating incident solar radiation on earth's surface using empirical models is an old approach. This is mainly due to the lack of solar radiation records at most of locations around the world [1]. Solar radiations records are much scarcer than those of other common meteorological parameters such as temperature and relative humidity, due to many technical and financial issues such as the expensive costs, need for highly skilled labor, and requirements of periodical maintenance, cleaning and calibration of the solar sensors [2]. Models of solar radiation can be categorized based on different criteria. For instance, model output (global, beam or diffuse radiation), model input(s) (meteorological data, climatological data or other radiation components), time scale (daily, monthly average daily, hourly, monthly average hourly or even minutely basis), spatial coverage (site-dependent or global model), methodology (stochastic or time-series modeling), approach (physical, semi-physical or empirical), inclination of surface (horizontal, tilted or tracking surfaces), and type of sky (all sky or clear sky conditions) [2]. Other classifications may also be considered, including the algorithm used (statistical analysis or machine-learning algorithms), time coverage (all-year or seasonal), interaction of time-scales (modeling solar components from others in the same time scale, or from lower timeresolution records). In the next paragraphs, we introduce a concise review of empirical models of daily global horizontal solar radiation.

The oldest method to estimate daily horizontal global solar irradiation was introduced by Angstrom [3] in 1924, in which the ratio between actual global horizontal irradiation (H) and clear day global irradiation (Hc) was correlated linearly to the sunshine fraction (SF). The sunshine fraction is defined as actual to theoretical sunshine duration ratio. The actual sunshine duration (S) can be either measured or calculated from direct normal irradiance data, while the theoretical sunshine duration (So) is calculated as So = (2/15) cos−1(tan φ tan δ), where φ and δ are the latitude and declination angles, accordingly. However, the definition and estimation of clear day global irradiation are also challenging. Thus, the original correlation was modified later by Prescott [4], by replacing the clear day global irradiation term with horizontal extraterrestrial irradiation (Ho), which can be determined mathematically, to form the well-known AngstromPrescott equation:

H = Ho(a + bSF ),

(1)

where a and b are the well-known Angstrom coefficients. The model was refitted later for different locations and it was frequently reported as a site-dependent model since the Angstrom coefficients vary significantly according to the location and type of climate [5]. Other modified forms of the model have been suggested for sake of better estimations, for instance: quadratic [6], cubic [7], logarithmic [8], linear-logarithmic [9], exponential [10], linear-exponential [11], ex-

⁎

Correspondence to: Mechanical Power Department, Faculty of Engineering, Cairo University, P.O. Box:12613, Giza, Egypt. E-mail addresses: [email protected], [email protected] (M.A. Hassan), [email protected], [email protected] (A. Khalil), [email protected] (S. Kaseb), [email protected] (M.A. Kassem). http://dx.doi.org/10.1016/j.rser.2017.07.002 Received 21 December 2016; Received in revised form 29 April 2017; Accepted 1 July 2017 1364-0321/ © 2017 Elsevier Ltd. All rights reserved.

Please cite this article as: Hassan, M.A., Renewable and Sustainable Energy Reviews (2017), http://dx.doi.org/10.1016/j.rser.2017.07.002

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R2 Coefficient of determination MBE Mean bias error, MJ/m2.day RMSE Root mean square error, MJ/m2.day MPE Mean percentage error, % NDRMSE Non-dimensional root mean square error, % r Correlation coefficient

Nomenclature

H Ho δ N I Kt

Global horizontal solar irradiation, MJ/m2.day Extraterrestrial horizontal solar irradiation, MJ/m2.day Declination angle, degrees Day number Global horizontal irradiance, W/m2 Daily global clearness index

solar radiation have been developed to replace sunshine duration and cloud cover by air temperature. The first temperature-based model was suggested by Hargreaves and Samani [22], by estimating global radiation using only maximum daily temperature difference (δT) and latitude:

ponent [11] and simple power forms [12]. Other modified models included geographical parameters such as latitude angle or was developed in seasonal forms [13], but they are less common. These models are usually called: sunshine-based models. It worth noting that the equation is frequently reported in daily and monthly average daily time scales, where using the equation for modeling global radiation in a different time scale may lead to under- or over-estimation of the actual global radiation values. The inequality of the two time scales was demonstrated analytically by Muneer in [1]. Another category of empirical global radiation models, called: cloud-based models, was developed by correlating daily global clearness index (Kt = H/Ho) to another meteorological parameter: cloud cover (C). Black [14] firstly introduced the method by fitting a quadratic correlation between the two variables:

H = Ho(a + bC + cC 2 ).

H = Ho(a δT ),

(3)

where δT = Tmax - Tmin, and Tmax and Tmin are the maximum and minimum daily temperatures, respectively, in °C. The empirical coefficient (a) was initially set to 0.17 for arid and semi-arid regions, while modified later by Hargreaves [23] to 0.16 and 0.17 for interior and coastal regions, accordingly. Samani [24] suggested a modified form of the original equation by replacing the empirical coefficient with a second order polynomial function of δT. Chen et al. [25] and Hunt et al. [26] added other empirical coefficients for better estimations of the model. Another important model in the temperature-based category is the well-known Bristow and Campbell model [27], in which the global irradiation is evaluated based on another temperature difference term (ΔT) and three empirical coefficients (a-c):

(2)

Here, C is the mean total cloud cover (octal). Paltridge and Proctor [15] proposed a more complex set of equations that estimates the instantaneous global irradiance and total daily global irradiation using solar zenith angle, day length and cloud factor (a parameter that varies from zero during clear sky days to 1 during overcast days). It can be determined using the cloud cover and the number of cloudy days in each month. The authors also provided an equation to convert cloud cover records to cloud factor. Badescu [16] modified the original Black's equation by refitting it into different order polynomials. Other modified or refitted models are available in literature [17–19]. However, this category of models is less common and usually less accurate than the former one [13]. Despite their strong relationship with global radiation, sunshine duration and cloud cover records are less available if compared to more common meteorological parameters such as temperature and relative humidity [20,21]. In addition, their measurements encounter more inherent sources of error. To deal with this situation, simpler models of

H = Ho(a(1 − exp(− b ∆T c ))).

(4)

Here, ΔT is defined as ∆Tj = Tmax j −0.5(Tmin j + Tmin j+1), where j is a day index (j for current day and j + 1 for the following day). The regression coefficients are mainly dependent on the environment type (e.g. humid, arid,.. etc.), with typical values of: a = 0.700, b = 0.004– 0.001 and c = 2.400 [20,27]. The equation was modified later by Goodin et al. [28], by inserting another Ho term in the exponential as a scaling factor. Simpler models that replace sunshine fraction in the Angstrom-Prescott equation with temperature terms: H = Ho(a + b(Temperature Term )) were also suggested. The temperature term could be the daily average temperature (T) [29] or the temperature ratio (Tratio, minimum to maximum daily temperature ratio) [30]. An important multivariate linear equation of this type was suggested by

Fig. 1. Graphical representation of the relation between global horizontal radiation and: (a) day number, (b) declination angle, and (c) horizontal extraterrestrial radiation.

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of measurements) at 68 locations in Turkey using a trigonometric function of the day number. Li et al. [45] developed a relatively complex sine-cosine form of the correlation between global irradiation and day number. They validated the performance of the model by comparing its results with measurements at nine major stations in China. They also compared its performance to that of the models proposed by Kaplanis and Kaplani [43], Bulut and Buyukalaca [44], and the sine-wave model by Al-Salaymeh [42]. The new model was found more accurate at the considered locations. Zang et al. [46] suggested a close, but simpler, sine-cosine model for estimation of global radiation over 35 stations in six climatic zones in China to be used for generation of typical meteorological years. Their results showed that the new sine-cosine model performs better than the cosine model by Kaplanis and Kaplani [43] and the trigonometric model by Bulut and Buyukalaca [44]. Finally, Quej et al. [47] refitted the previous models by Bulut and Buyukalaca [44], Kaplanis and Kaplani [43] and Li et al. [45] for Yucatan Peninsula in Mexico. They also refitted the sine wave model by Al-Salaymeh [42] but failed to refit his Gaussian model. Consequently, they modified it by making use of summation of two Gaussian function, which resulted in very good predictions. Based on the available measured data, they found that their new modified Gaussian model performs better than all other models. In addition to the previously mentioned studies that were dedicated to developing independent global radiation models, some other studies have been carried out to calibrate former models for other locations and compare their performance statistically. Khorasanizadeh and Mohammadi [48] used long-term global solar radiation records to test six day number-based models (trigonometric model by Bulut and Buyukalaca [44], cosine model by Kaplanis and Kaplani [43], sinecosine model by Li et al. [45], and the Gaussian, 4th order polynomial and sine-wave models by Al-Salaymeh [42]) at four cities in Iran (Bandarabass, Isfahan, Kerman and Tabass). The sine-cosine wave model was found better in three cities, while the 4th order polynomial model showed best results at the fourth city. They also assessed the performance of independent models by comparing their estimations with the predictions of some existing monthly averaged models and found that the newly refitted independent models are superior. Khorasanizadeh et al. [49] carried out a study to compare the accuracy of day number-based models with other more common categories, namely: sunshine- and temperature-based models. The city of Birjand (Iran) was considered as a case study. Models by Kaplanis and Kaplani [43], Li et al. [45] and Zang et al. [46] have been refitted for the new location, validated using long-term measured radiation data, and compared to the other dependent models. The results revealed the superiority of day number-based models as compared to temperaturebased models. Sunshine-based models, however, were slightly more accurate. The authors concluded that day number-based models are qualified as proper alternatives to sunshine-based models due to their simplicity. In a recent study, Hassan et al. [50] used a dataset of more than 20 years of daily global solar radiation records to refit and validate different day number-based models for ten cities in Egypt. The validated models are those of Bulut and Buyukalaca [44], Kaplanis and Kaplani [43], Li et al. [45] and the four models proposed by AlSalaymeh [42]. They found that the sine-cosine wave model by Li et al. [45] and the 4th order polynomial model by Al-Salaymeh [42] have the best performance measures. Machine learning algorithms have been adopted recently in many studies in order to refine solar radiation estimations. Despite being widely used to develop sunshine-, temperature- and meteorologicalbased models [51–56], very few studies are available for independent models. An adaptive neuro-fuzzy inference system (ANFIS) model was developed by Mohammadi et al. [57] using long-term measured data at Tabass, Iran. Their results assessed the capability of that algorithm in modeling global solar irradiation. In addition, the authors revealed the superiority of the developed ANFIS model as compared to six

Li et al. [31], in which the daily maximum and minimum temperatures were used:

H = Ho(a + bTmax + cTmin )

(5)

A hybrid method is frequently adopted, in which different meteorological variables are used as model's inputs. These models are usually called meteorological parameters-based models, and the commonly used meteorological variables are sunshine duration, cloud cover, temperature, daily average pressure (P), water vapor pressure (Pv), wind speed (W), relative humidity (RH) and precipitable water vapor [13]. Perhaps the most important models in this category are those proposed by Abdalla [32], who has correlated the global clearness index with different meteorological variables using a set of multivariate linear equations:

H = Ho (a + bSF + cδT ),

(6)

H = Ho (a + bSF + cδT + d RH ),

(7)

H = Ho (a + bSF + cδT + d RH + e(P / Pv )),

(8)

where a–e are the regression coefficients. There are a numerous number of such models in the literature with different types of inputs and different degrees of complexity [13,33]. In most cases, these models provided better estimations of global radiation. However, complex models are not usually the best choice for model users, since some of these variables may not be recorded at the considered location. Besides, adding more inputs does not always improve the model estimations [13], and whenever it does, the improvements are not always significant [34]. Thus, these models must be subjected to a careful cross-validation procedure to estimate the optimum number and type of inputs. In order to simplify global radiation models, especially for locations where no solar or meteorological variables are recorded, independent models have been recently developed. These models make use of the semi-linear relation between global irradiation and extraterrestrial horizontal irradiation (or declination angle). Alternatively, they can be developed based on the sinusoidal behavior of global radiation throughout the year, as a function of the day number (Fig. 1). Since these models have no meteorological variables that vary (with random components) from day to day, they have rough estimations of global radiation. Nevertheless, they are frequently reported with good or very good statistical measures. There are many reviews on other dependent models [20,21,35–39]. However, there are no comprehensive reviews on independent models. An extensive review of such models is provided below. Despite their simplicity, independent models are relatively newer than other types of global irradiation models. In 1999, Ertekin and Yaldiz [40] proposed a simple-linear model correlating global radiation to declination angle. The model was tested using global radiation records of six-years at Antalya, Turkey. In the same year, Togrul and Onut [41] suggested another close model by replacing the declination angle with its sine term. Additionally, they proposed another simplelinear equation between global irradiation and daily horizontal extraterrestrial radiation. The two models were tested at Elazig (Turkey) using solar records of one-year duration. Al-Salaymeh [42] started another method by correlating global irradiation directly to the day number. This is a simpler method since it excludes the prior calculations of declination angle or extraterrestrial radiation. The author proposed four different forms of the correlation between the two variables, namely: sinusoidal, 4th order polynomial, Lorentzian and Gaussian forms. The four correlations were tested using measured data at Amman, Jordan. It was found that the first three forms provide excellent estimations of global radiation at the considered location. Kaplanis and Kaplani [43] proposed a cosine-wave equation between global irradiation and day number, which was fitted using solar data of six years at six locations in Greece. Bulut and Buyukalaca [44] simulated long-term measured radiation data (from 10 up to 19 years 3

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A close model was suggested by Togrul and Onut [41]:

recalibrated empirical models. Gani et al. [58] adopted the neural network auto-regressive model with exogenous inputs (NN-ARX) for predicting irradiation values using only day number as a sole input. Seven Iranian cities have been considered as case studies. They also developed ANFIS models for the different locations for comparison. Their results proved the accuracy and reliability of predicting daily global solar radiation using only day number. The superiority of NNARX model over ANFIS was indicated. Hassan et al. [59] investigated the potential of four machine-learning algorithms, namely: multi-layer perceptron neural networks, support vector machines, ANFIS and decision trees, in modeling daily global irradiation. The authors developed different models of global irradiation for Cairo (Egypt) based on those machine-learning algorithms and other empirical equations and categorized the developed models in four groups: sunshine-, temperature-, meteorological parameters- and day number-based models. Among their results, the authors found that temperature and day number-based models are reliable in modeling solar radiation at locations where no sunshine records are available. In contrast to other groups of models, the proposed decision tree algorithms were found more accurate in modeling global radiation independently. Based on this careful review of available literature on empirical modeling of daily horizontal global radiation, independent models seem undervalued considering their simplicity and very good estimations. The next sections provide a comparative study of all previous empirical independent models using solar radiation data measured at Cairo (Egypt) as a case study. All models are refitted for the new location and cross-validated using a novel detailed technique of data portioning in order to determine the true performance of each model and choose the most suitable model for quantifying global radiation at the considered location.

(11)

H = a + bsinδ.

Model: M3 They also developed a new method by correlating global radiation to corresponding extraterrestrial radiation using the following simplelinear equation [41]:

H = a + bHo.

(12)

Daily horizontal extraterrestrial radiation (Ho) is the integration of instantaneous horizontal extraterrestrial irradiance over the daylight period [60]:

Ho =

⎞ ⎛ πω ⎞ 24*3600*Ic ⎛ *⎜cos φ cos δ sin ωs + ⎜ s ⎟sin φ sin δ ⎟ . ⎠ ⎝ π 180 ⎠ ⎝

(13)

The solar constant (Ic) is equal to 1367 W/m [2], while ωs is the sunset hour angles, respectively: 2

ωs = cos−1(−tan φ tan δ ).

(14)

Model: M4 Al-Salaymeh [42] proposed four different equations correlating global radiation and day number. The first one is a sine-wave equation:

⎞ ⎛ 2π H = a + b sin ⎜ N + d ⎟ , ⎠ ⎝ c

(15)

where a, b, c, and d are the amplitude, the phase shift, the wavelength, and the intercept. Model: M5 The second equation is a 4th order polynomial equation:

H = a + bN + cN 2 + d N 3 + eN 4,

(16)

where a–e are the regression parameters. Model: M6 The third equation has a Lorentzian structure:

2. Materials and methods 2.1. Independent models

H=

In this section, we present all empirical independent models that have been suggested in literature chronologically. Model: M1 The first model found was a simple linear model, proposed by Ertekin and Yaldiz [40], between global radiation (H) and declination angle (δ):

N−b 2 c

)

. (17)

2⎞ ⎛ 1⎛N − b⎞ ⎟ H = a exp ⎜⎜ − ⎜ ⎟ ⎟, 2⎝ c ⎠ ⎠ ⎝

where a and b are the regression parameters. The declination angle is calculated as a function of the day number (N) as follows:

⎞⎞ ⎛ 360 ⎛ δ = 23.45sin⎜ ⎜N +284⎟⎟ . ⎠⎠ ⎝ 365 ⎝

1+(

Here, a, b and c are the amplitude, the center, and the width. Model: M7 The fourth function has a Gaussian form:

(9)

H = a + bδ,

a

(18)

with parameters a, b and c standing for the amplitude, the center, and the width. Model: M8 Kaplanis and Kaplani [43] suggested a different cosine-wave equation based on day number, with three regression coefficients (a-c):

(10)

Model: M2 Table 1 Original regression coefficients and reported accuracy of independent models. Model

a

b

c

d

e

f

g

Accuracy

City

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12

13.580 4.089 −1.320 5250.650 8220.000 7905.650 2706.825 15.140 21.410 16.440 15.976 10.269

0.333 6.459 0.675 2782.090 183.400 183.210 −15.554 10.326 2.570 −7.889 1.934 7.633

– – – 375.430 119.800 107.620 0.836 9.405 – 0.909 0.975 109.048

– – – 4.783 – – −0.004 – – 8.360 −3.374 41.191

– – – – – – 0.000 – – 0.519 0.883 5.355

– – – – – – – – – 1.946 – 226.989

– – – – – – – – – 7.915 – 43.932

r = 0.984 r = 0.975 r = 0.974 r = 0.964 r = 0.961 r = 0.960 r = 0.964 r = 0.996 r = 0.840 r = 0.972 r = 0.826 R2 = 0.863

Antalya Elazig Elazig Amman Amman Amman Amman Athens Istanbul Minqin Sanya Calakmul

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⎞ ⎛ 2π H = a + b cos ⎜ N + c⎟ . ⎠ ⎝ 364

heliometer is used for measuring direct normal irradiance. The three sensors are mounted on a Kipp & Zonen-SOLYS2 sun tracker with shading ball assembly. Detailed specifications of the two sensors are provided in Table 2. The sunshine duration is computed from recorded pyrheliometer's data instead of being measured directly since it is defined by the World Meteorological Organization (WMO) as the time during which the beam normal irradiance exceeds the threshold of 120 W/m2 [60]. Alongside with the radiation sensors, the station is equipped a CS215 temperature and relative humidity sensor, an NRG #40 C anemometer, an NRG #200 P wind direction vane, and a Setra278 barometric pressure sensor. A National Instruments data acquisition system is used to monitor, record and integrate all measured variables. The data is recorded in minutely time-resolution. This high resolution enables the user to detect any suspicious data, where atmospheric parameters exceed their physical limits. It addition, the user can monitor small variations of variables and take instant actions, e.g. cleaning the sensors. Besides, when the user needs to estimate hourly or daily total radiation, the high resolution of collected data results in more accurate integrated values. In the solar station described above, solar radiation data has been recorded from October 2012 to December 2016, with one major interruption (April 2013 – December 2013).

(19)

Model: M9 Bulut and Buyukalaca [44] proposed a trigonometric form of the underlying function between incident radiation and day number:

⎞ 1.5 ⎛ π H = a + (b − a ) sin ⎜ (N +5)⎟ . ⎠ ⎝ 365

(20)

Model: M10 Li et al. [45] developed a sine-cosine form of the relation between global radiation and day number, with seven regression parameters (ag):

⎞ ⎛ 2πf ⎞ ⎛ 2πc H = a + b sin ⎜ N + d ⎟ + e cos ⎜ N + g⎟ . ⎠ ⎝ 365 ⎠ ⎝ 365

(21)

Model: M11 Zang, H. et al. [46] suggested a close, but simpler model (fewer regression parameters):

⎛ 2πe ⎞ ⎛ 2πc ⎞ H = a + b sin ⎜ N ⎟ + d cos ⎜ N ⎟. ⎝ 365 ⎠ ⎝ 365 ⎠

(22)

Model: M12 Finally, Quej, V. et al. [47] modified the Gaussian model of AlSalaymeh for Yucatan Peninsula in Mexico by making use of sum of two Gaussian functions: 2⎞ 2⎞ ⎛ ⎛ 1⎛ N − c⎞ ⎟ 1 ⎛N − f ⎞ ⎟ H = a + b exp ⎜⎜ − ⎜ ⎟ ⎟ + e exp ⎜⎜ − ⎜ ⎟ ⎟. 2⎝ d ⎠ ⎠ 2⎝ g ⎠ ⎠ ⎝ ⎝

2.3. Data filtering and smoothing In the majority of studies dedicated to modeling daily global solar radiation, the dataset is usually available in daily time-resolution from meteorological administrations. In this study, however, the solar irradiance (in W/m2) is measured in minutely time-resolution in order to be integrated to daily time-resolution (in MJ/m2.day). Therefore, a detailed procedure is described below for filtering and smoothing measured data. While ground measurements of solar radiation are the best method for the knowledge of incident radiation, errors can be expected as in any type of measurements. Solar radiation measurements, in particular, are more likely to have errors than measurements of other meteorological parameters. Usually, questionable values appear even when sensor cleaning, maintenance, and calibration procedures are performed periodically [1,61]. In order to make sure that all measured data points are reliable, a hybrid procedure is adopted based on the restrictions suggested by Jacovides et al. [62] Reindl et al. [63], and Scharmer and Greif [64]. Firstly, data points of night or blocked radiation periods are excluded: I (global irradiance) ≥ 0. Periods of low sun elevations are also excluded due to the cosine response: α (solar altitude angle) ≥ 5°, and I ≥ 5 W/m2. A 10% allowance is considered for shading ring correction: Id /I (diffuse fraction, kd: diffuse to global irradiance ratio) ≤ 1.1. Global irradiance values have to be within the range of the expected clear-sky extreme values by considering the influence of the atmospheric layer: I/Io,h (global clearness index, kt: global to extraterrestrial horizontal irradiance ratio) ≤ 1.2. Here, the value of the clearness index may exceed 1 for minutely data due to

(23)

The original regression coefficients, as well as the reported accuracy of all previously discussed models, are listed in Table 1. In case if regression coefficients are reported for different locations, only one location is selected to be listed in the table. The correlation coefficient (r) is defined as: n

r=

∑i =1 (Hc − Hc )(Hm − Hm ) n

n

[∑i =1 (Hc − Hc )2][∑i =1 (Hm − Hm )2 ]

(24) th

where n is the number of observations. Hc and Hm are the i calculated and measured values of global irradiation, respectively. While Hc and Hm are the means of calculated and measured values, all in MJ/m2.day. 2.2. Measured data The solar radiation dataset used in this study has been collected at Faculty of Engineering, Cairo University (El-Sheikh Zayed annex, latitude: 30.04°N, longitude: 31.01°E). A solar-meteorological station is operated since 2012 to measure radiation and atmospheric parameters. As for solar radiation sensors, the station is equipped with two Kipp & Zonen-CMP21 pyranometers for measuring horizontal global and diffuse irradiance components, while a Kipp & Zonen-CHP1 pyrTable 2 Specifications of CMP21 pyranometer and CHP1 pyrheliometer.

ISO classification Response time (95%) Zero offsets due to ambient temperature change (5 K/h) Non-stability (change/year in percentage of full scale) Non-linearity Temperature dependence of sensitivity Expected daily uncertainty (in clean state) Full opening view angle Operating temperature Spectral range (50% points)

CMP21 pyranometer

CHP1 pyrheliometer

Secondary standard <5s ± 2 W/m2 ± 0.5% ± 0.2% (at 500 W/m2 within 100–1000 W/m2) ± 1% (−20 °C to + 50 °C) ± 2% 180° −40 °C to + 80 °C 310–2800 nm

First class 5s ± 1 W/m2 ± 0.5% ± 0.2% (0–1000 W/m2) ± 0.5% (−20 °C to + 50 °C) ± 1% 5° ± 0.2° −40 °C to + 80 °C 200–4000 nm

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sample are saved, and the regression coefficient considered for this sample are those of the fold that has the closest RMSE to the average RMSE of all folds. At this point, the model has been cross-validated using a single random sample. The performance of the model is highly dependent on the order of observations in the data matrix, especially when the data points are serially correlated [69]. In order to get a more honest performance indicators of model, the whole procedure de-

sunlight reflecting off clouds and into the sensor. However, as the time resolution decreases (e.g. hourly or daily means), the number of data points with values near or greater than 1 significantly decreases [65]. The normal irradiance must be less than its corresponding extraterrestrial component: In ≤ Io. Measured data must lie within the expected kd-kt envelop: Id/Io,h ≤ 0.8, Id/I ≥ 0.9 for kt < 0.2, and Id/I ≤ 0.8 for kt > 0.6. Another set of restrictions is considered for integrated or averaged daily values. Upper and lower limits of daily mean clearness index (Kt) are considered as 1 and 0.015, respectively. Daily total sunshine duration is also checked such as to be less than or equal to its corresponding astronomical value (SF < 1). Missing data is caused by equipment failure, cleaning and maintenance routines, equipment being offline, or due to discarding recorded values when applying the quality control procedure. Different methods have been used for estimating missing values such as Average Nearest Observation (ANO) [66]. The ANO is a very simple method since it replaces missing values with the mean of nearest previous and following observations. However, it generates different estimations depending on the marching direction (i.e. forward or backward). Therefore, it has a poor performance when applied to time series data that has weak autocorrelation and/or strong daily seasonality [67]. The ANO method, however, is the basis of the advanced Twodirectional Exponential Smoothing method (TES) used here, which estimates the missing observations based on the autocorrelations of the time series to account for the fact that the missing values occur at nonrandom times. Firstly, all missing observations are generated using the ANO method in forward and backward directions. Then, the missing values are estimated using Holt's linear trend method: Hi = 0.5 (Hi,forward + Hi,backward) [68]. 2.4. Data portioning The common approach in modeling solar radiation using empirical models is to split measured data without randomization into two subdatasets for training and validating the models. For instance, if a dataset of four years is available, three years of measurements may be chosen for training stage while the fourth year is kept for model validation. This approach, however, does not account for the annual variations of solar radiation, especially when the dataset is too large, i.e. covering a decade or more. Another issue is that choosing the optimum model, in case of comparing different models, is subjective. The performance of each model highly depends on which data points have been chosen for validation, and which have been chosen for training. Another simple technique of portioning measured data and validating the models is suggested here based on the repeated K-fold crossvalidation method. Instead of splitting the data chronologically, measured data are randomized using the software (MATLAB R2016a®), and the only interference is choosing the number of folds (K) and the number of samples (U) in the cross-validation process. As shown in Fig. 2, solar radiation data is collected and the quality control procedure, described above, is applied to raw data. Next, TES method is used to handle missing data. Measured data is now used to construct an n ×P + 1 matrix, where P is the number of inputs (the extra column in the matrix is for the target output). Afterward, the rows (observations) of the matrix are randomized for the first sample, and the K-folds cross-validation method is applied. Using 10 folds (K = 10), the randomized data matrix is split into 10-equally sized sub-matrices. The first nine sub-matrices are used to construct the training data matrix, while the remaining one is kept for validation. Each model is fitted using the training matrix and then validated using the validation matrix [69]. The resulted regression coefficients and validation statistical indicators are saved. After that, another sub-matrix is preserved for validation, while the other nine are used for training. This process is repeated until all data points are used for training and validating the model. The average statistical indicators of all folds of this random

Fig. 2. A flowchart of the procedure used for training and validation of independent models.

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Table 3 Recalibrated regression parameters of independent models. Model

R2 = 1−

a

b

c

d

e

f

g

− Hm )2

(29)

20.507 20.486 −2.173 19.969 11.843 28.906 28.015 20.546 10.869 20.297 20.512 21.240

0.357 21.044 0.711 −8.704 0.027 172.492 173.574 −8.422 28.241 −0.566 41.718 −9.269

– – – 1.003 0.002 143.213 123.267 6.454 – 2.487 −0.005 5.906

– – – 13.782 0.000 – – – – −5.641 −2.542 −23.907

– – – – 0.000 – – – – −8.421 2.237 −3.331

– – – – – – – – – 0.966 – 51.486

– – – – – – – – – 6.553 – −14.839

It worth noting that a conflict usually appears when comparing different models based on different statistical indicators, where a model may have the best performance based on one statistical indicator (e.g. MBE), while another model is the best based on another statistical indicator (e.g. RMSE) [2]. In such cases, the model with lowest RMSE is preferred, since the RMSE is usually better in showing the true accuracy of the model. Table 3 shows the recalibrated regression parameters of all considered models for the new location. Table 4, on the other hand, provides the statistical measures of training and validation stages. It is clear that all refitted models, except for models M11 and M12, have very good estimations of global solar radiation with all R2 values greater than 82%, and all NDRMSE values less than 8.5%. The results also suggest that the recalibrated 4th order polynomial model is the best choice for modeling daily global radiation without need to meteorological data. It has validation RMSE and R2 of 7.761% and 86.231%, respectively, which is very appealing regarding its simplicity. It also performs better in training stage. Its competitors are the recalibrated sine-cosine model by Li et al. [45] and the two declination angle-based models. Models M11 and M12, on the other side, failed to generalize to the new data even when single training/validation step is performed. The original models (without recalibration) also showed poor performance in the new location, suggesting that they have poor generalization characteristics as compared to elder models, which have been refitted before for different locations. Fig. 3 shows scatter plots of measured daily global irradiation against estimated values for all trained models. Data points are distinguished according to their random assignment (training or validation). The shown plots are for the fold (k) with closest RMSE to the average RMSE of all folds, in the sample (u) with closest RMSE to the average RMSE of all samples. The failure of models M11 and M12 in generalization to new data is shown visually. Fig. 4, on the other hand, depicts the distribution of models’ residuals against estimated radiation values. All models have almost random distributions, except for models M11 and M12 whose residuals have apparent patterns. A time-series plot of measured and estimated values of global radiation is shown in Fig. 5 for all successfully validated models. Since all models depend mainly on the day number, all curves have close trends throughout the year with some significant deviations at the beginning and the end of the year. Model M3 has a different behavior due to the sine term of sunrise angle in the extraterrestrial radiation equation. The inset in the same figure is for model M5, where only 20 random days in 2014 are shown. For these days, the uncertainty bounds of measurements, as well as the 95% confidence interval of the

scribed is repeated using different random order of observations. In this study, the process is repeated 1000 times for each model (U = 1000). For each sample, the regression coefficients and statistical indicators are saved. The reported statistical indicators are the average of corresponding statistical indicators of all samples, while the reported regression coefficients are those of the sample that has the closest RMSE to the average RMSE of all samples.

3. Results and discussion The previously discussed independent models are recalibrated here for the new location and cross-validated according to the procedure provided in the previous section. The performance of these models is evaluated based on different statistical measures, namely: Mean Bias Error (MBE), Mean Percentage Error (MPE), Root Mean Square Error (RMSE), Non-Dimensional Root Mean Square Error (NDRMSE) and Coefficient of Determination (R 2 ), defined as: n

MPE =

n ∑i =1 (Hm

Model parameters

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12

MBE =

∑i =1 (Hc − Hm )2

∑i =1 (Hc − Hm ) n

1 n

RMSE =

, MJ/m2.day,

(25)

⎛ Hc − Hm ⎞ ⎟*100%, Hm ⎠ i =1 ⎝ n

∑⎜ 1 n

NDRMSE =

(26)

n

∑ (Hc − Hm )2 ,MJ/m2.day,

(27)

i =1

1 n

⎛ Hc − Hm ⎞2 ⎟ *100%, Hm ⎠ i =1 ⎝ n

∑⎜

(28)

Table 4 Training and validation statistical measures of re-fitted models. Best values are shown in bold font. Model

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12

Training

Validation

MBE

RMSE

R2

MPE %

MBE

RMSE

NDRMSE %

R2

0.000 0.000 0.000 0.000 0.000 0.008 −0.028 0.000 0.000 0.000 0.053 0.000

2.347 2.345 2.374 2.488 2.318 2.519 2.435 2.345 2.403 2.319 5.189 5.668

0.862 0.862 0.859 0.832 0.866 0.828 0.852 0.862 0.855 0.866 0.266 0.201

0.002 0.002 0.000 −0.006 0.001 0.039 −0.094 0.001 0.000 −0.002 0.125 0.005

0.000 0.000 0.000 −0.002 0.000 0.011 −0.028 0.000 0.000 −0.001 0.036 0.000

2.330 2.328 2.355 2.473 2.305 2.507 2.419 2.336 2.387 2.312 5.176 5.651

7.845 7.837 7.927 8.328 7.761 8.442 8.143 7.863 8.037 7.784 17.422 19.024

0.860 0.860 0.856 0.828 0.862 0.825 0.849 0.859 0.853 0.861 0.255 0.188

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Fig. 3. Scatter plots of estimated and measured radiation values.

Fig. 4. Scatter plots of models’ residuals.

and the non-linear meteorological parameters-based model by Ododo et al. [72]. All dependent models have been trained and validated using the same procedure of independent models. Table 5 provides a list of those dependent models, alongside with their corresponding refitted parameters and statistical measures. The provided results demonstrate the importance of sunshine duration in estimating the global solar irradiation. The two models with the sunshine fraction is an input (models: D1 and D5) have superior performance as compared to other

model, are displayed. Good agreement between the two intervals can be noticed. For further assessment of the prediction performance, the independent models are compared to five different dependent models, namely: the 3rd order polynomial sunshine-based model by Zabara [7], the linear temperature-based model by Chen et al. [25], the non-linear meteorological parameters-based model by Allen [70], the multivariate linear meteorological parameters-based model by El-Sebaii et al. [71],

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Fig. 5. Time-series plot of models’ estimations in comparison with measured radiation values. Table 5 Statistical results of refitted dependent models. Best values are shown in bold font. Model

D1 D2 D3 D4 D5

Ref.

[7] [25] [70] [71] [72]

Equation

H = Ho(a + b SF + c SF2 + d SF3) H = Ho(a + b δT0.5) H = Ho(a(P/Po)0.5 δT0.5) H = Ho(a + bT + c RH) H = Ho(a + b SF + cTmax + d RH + e Tmax SF)

Refitted parameters

Training

Validation

a

b

c

d

e

MBE

RMSE

R2

MBE

MPE %

RMSE

NDRMSE %

R2

0.437 0.345 0.172 0.727 0.579

−0.125 0.080 – 0.001 0.228

0.858 – – −0.002 −0.005

−0.451 – – – −0.001

– – – – 0.005

−0.036 0.008 −0.259 0.041 0.006

1.362 2.162 2.462 2.311 1.324

0.954 0.883 0.848 0.866 0.956

−0.120 0.022 −0.871 0.142 0.021

−0.036 0.006 −0.259 0.042 0.006

1.364 2.158 2.427 2.298 1.322

4.593 7.265 8.172 7.737 4.452

0.952 0.879 0.845 0.863 0.955

Fig. 6. A graphical comparison between models D2, D5 and M5: (a) statistical comparison plot using validation dataset, (b) one-year time-series plot.

the best independent model (model M5) is shown in Fig. 6. Fig. 6a shows a statistical comparison using validation dataset (validation data points are different for each model due to the randomization procedure), while Fig. 6b shows a one-year time-series plot of the three models. While model D5 is capable of capturing the variation of global radiation during overcast days, the independent model provides rough estimations of global radiation behavior throughout the year. Finally, Fig. 7 demonstrates the importance of cross-validating the models using random samples. Histograms of validation RMSE are displayed for three models: M4, M5, and M10. The histograms show the distribution of all RMSEs of the 1000 samples, while the vertical lines indicate the average of these RMSEs (as given in Table 4). Here,

dependent and dependent models in both training and validation stages. On the other side, when the sunshine fraction is excluded (models: D1-D3), the performance of the model drops significantly to the accuracy level of independent models. This can be also noticed when comparing the statistical measures of models D1 and D5, where adding temperature and relative humidity terms slightly improved the predictions. This suggests that complicating the models by adding some meteorological parameters such as temperature, mean sea-level pressure or relative humidity is not worthy, and the independent models are preferred in locations where sunshine duration is not recorded. A graphical comparison between model D2 (temperaturebased model), model D5 (meteorological parameters-based model) and 9

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Fig. 7. Histograms of validation RMSE: models M4, M5 and M10. [4] Prescott JA. Evaporation from a water surface in relation to solar radiation. Trans R Soc South Aust 1940;64: p. 114–25. [5] Myers DR. Solar radiation: practical modeling for renewable energy applications, 1st ed.. New York: CRC Press; 2013. [6] Ogelman H, Ecevit A, Tasdemiroglu E. A new method for estimating solar radiation from bright sunshine data. Sol Energy 1984;33:619–25. [7] Zabara K. Estimation of the global solar radiation in Greece. Sol Wind Technol 1986;3:267–72. [8] Ampratwum B, Dorvlo AS. Estimation of solar radiation from the number of sunshine hours. Appl Energy 1999;63:161–7. [9] Newland FJ. A study of solar radiation models for the coastal region of South China. Sol Energy 1988;31:227–35. [10] Almorox J, Benito M, Hontoria C. Estimation of monthly Angstrom–Prescott equation coefficients from measured daily data in Toledo, Spain. Renew Energy 2005;30:931–6. [11] Bakirci K. Correlations for estimation of daily global solar radiation with hours of bright sunshine in Turkey. Energy 2009;34:485–501. [12] Sen Z. Solar energy fundamentals and modeling techniques: atmosphere environment, climate change and renewable energy, 1st ed.. London: Springer; 2008. [13] Besharat F, Dehghan AA, Faghih AR. Empirical models for estimating global solar radiation: a review and case study. Renew Sustain Energy Rev 2013;21:798–821. [14] Black JN. The distribution of solar radiation over the earth's surface. Arch Met Geophys Bioklim 1956;7:165–89. [15] Paltridge GW, Proctor D. Monthly mean solar radiation statistics for Australia. Sol Energy 1976;18:235–43. [16] Badescu V. Correlations to estimate monthly mean daily solar global irradiation: application to Romania. Energy 1999;24:883–93. [17] Reddy SJ. An empirical method for estimating sunshine from total cloud amount. Sol Energy 1974;15:281–5. [18] Supit I, Van Kappel RR. A simple method to estimate global radiation. Sol Energy 1998;63:147–60. [19] Ehnberg JSG, Bollen MHJ. Simulation of global solar radiation based on cloud observations. Sol Energy 2005;78:157–62. [20] Almorox J, Hontoria C, Benito M. Models for obtaining daily global solar radiation with measured air temperature data in Madrid (Spain). Appl Energy 2011;88:1703–9. [21] Benghanem M. Solar radiation estimated from measured air temperature: a review and proposed new model. Int J Renew Energy Technol 2013;4:191–211. [22] Hargreaves GH, Samani ZA. Estimating potential evapotranspiration. J Irrig Drain Eng 1982;108:225–30. [23] Hargreaves GH. Simplified coefficients for estimating monthly solar radiation in North America and Europe. Departmental Paper, Dept of Biol and Irrig Engrg, Utah State University, Logan; 1994. [24] Samani Z. Estimating solar radiation and evapotranspiration using minimum climatological data (Hargreaves-Samani equation). J Irrig Drain Eng 2000;126:265–7. [25] Chen R, Kang E, Lu S, Yang J, Ji X, Zhang Z, Zhang J. New methods to estimate global radiation based on meteorological data in China. Energy Convers Manag 2006;47:2991–8. [26] Hunt LA, Kuchar L, Swanton CJ. Estimation of solar radiation for use in crop modelling. Agric Meteorol 1998;91:293–300. [27] Bristow KL, Campbell GS. On the relationship between incoming solar radiation and daily maximum and minimum temperature. Agric Meteorol 1984;31:159–66. [28] Goodin DG, Hutchinson L, Vanderlip R, Knapp M. Estimating solar irradiance for

the performances of models M5 and M10 are appealing since they show fewer variations in their accuracies with change of input data. The two distributions are close to a normal one. Also, their RMSEs are much lower than that of model M4, which suffers from multimodal nature (its accuracy in estimating global irradiation is highly affected by the training and validation entries). The most important issue here is that there is an overlap in the distributions of the three models, which means that choosing the best model from a set of different models based on chronological portioning of data is somehow subjective. Model M5, which has the best generalization abilities among all models considered here, may result in a greater RMSE than other models if a single sample is used. Adopting the procedure described here leads to a more honest judgment of the true performance of validated models. 4. Conclusions In this paper, a comprehensive review of all studies concerned with modeling horizontal global radiation using independent models is provided. All empirical independent models have been recalibrated and cross-validated using a detailed novel method. The results assess the accuracy of independent models considering their simplicity, with coefficient of determination ranging from 82.5% up to 86.2%. When compared to other dependent models, the independent models were less accurate than sunshine-based models due to the strong relation between sunshine duration and global irradiation. However, they were at the same accuracy level of temperature- and meteorological parameters-based models (with sunshine fraction is not an input). This suggests that they are more suitable for modeling global radiation at locations where no sunshine records are obtainable. The importance of adopting the provided procedure of randomizing and portioning the data and cross-validating the models is demonstrated, to be used in studies dealing with modeling solar radiation empirically. References [1] Muneer T. Solar radiation and daylight models, 2nd ed.. New York: Elsevier; 2004. [2] Gueymard CA, Myers DR. Validation and ranking methodologies for solar radiation models. In: Badescu V, editor. Modeling solar radiation at the earth's surface1st ed.. New York: Springer; 2008. p. 479–509. [3] Angstrom A. Solar and terrestrial radiation. Report to the international commission for solar research on actinometric investigations of solar and atmospheric radiation. Q J R Meteorol Soc 1924;50: p. 121–5.

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