Smart model for accurate estimation of solar radiation

Abstract

Prediction of solar radiation has drawn increasing attention in the recent years. This is because of the lack of solar radiation measurement stations. In the present work, 14 solar radiation models have been used to assess monthly global solar radiation on a horizontal surface as function of three parameters: extraterrestrial solar irradiance (G₀), duration sunshine (S) and daylight hours (S₀). Since it has been observed that each model is adequate for some months of the year, one model cannot be used for the prediction of the whole year. Therefore, a smart hybrid system is proposed which selects, based on the intelligent rules, the most suitable prediction model of the 14 models listed in this study. For the test and evaluation of the proposed models, Tamanrasset city, which is located in the south of Algeria, is selected for this study. The meteorological data sets of five years (2000–2004) have been collected from the Algerian National Office of Meteorology (NOM), and two spatial databases. The results indicate that the new hybrid model is capable of predicting the monthly global solar radiation, which offers an excellent measuring accuracy of R² values ranging from 93% to 97% in this location.

Keywords

global solar radiation / statistical indicator / hybrid model / spatial database / correlation coefficients

Introduction

According to the renewable energy program that is announced in 2011, the government of Algeria is aimed to develop solar energy to produce about 37% of domestic electricity from both photovoltaics and concentrating solar power (CSP) . This is due to the huge potential of the solar energy and the longer sunshine duration in Algeria. Unfortunately, the solar radiation data are not available due to the high costs involved in buying and maintaining solar measuring equipment. Therefore, the accurate prediction of solar radiation intensity and knowledge of the available solar resource in the country is an important issue for the design, optimization and performance evaluation of solar energy systems for any particular area. Indeed, it is necessary to develop models to estimate the intensity of solar radiation based on other climate data available. A number of techniques for prediction which have been developed so far are widely used by researchers to get accurate information.

Many researchers have developed monthly mean global solar radiation models using several approaches that include classical empirical regression, artificial neural networks (ANN), and autoregressive moving–average (ARMA) time–series regression techniques. The use of empirical correlation for predicting monthly mean daily global solar radiation (GSR) on horizontal surface dated as far back as 1924 was carried out by Angstrom [¹].This equation is the simplest and most widely used and is related to the monthly average daily radiation to clear day radiation in a given location and average fraction of possible sunshine hours. Later in 1940, Prescott [²] modified a formula of Angstrom equation and focused on the extraterrestrial radiation on a horizontal surface, rather than on clear day radiation. After that, many have been developed through these two models, Angstrom–Prescott, to estimate monthly mean daily radiation utilizing available meteorological, geographical and climatological parameters such as sunshine hours, air temperature, latitude, precipitation, relative humidity, and cloudiness. However, the most commonly used parameter for estimating global solar radiation is sunshine duration. Glover and McCulloch [³] suggested an empirical linear regression based on the latitude of location, the monthly average daily sunshine duration, and the monthly average maximum possible daylight hours. Chegaar and Chibani [⁴] proposed two models for estimating monthly mean daily (GSR) on a horizontal surface of four Algerian locations. Salmi et al. [⁵] also proposed three models based on the sunshine duration for estimating monthly daily (GSR) in horizontal surface, and applied them to four locations in Algeria. Koussa [⁶] compared a statistical 10 existing models for estimating monthly mean daily GSR in three main Algerian cities. Mecibah et al. [⁷] also compared 11 empirical models available in the literature for computing the monthly mean daily GSR on a horizontal surface. The models were tested and verified using the data recorded in six Algerian cities.

Khatib et al. [⁸] gave an overview of the solar energy modeling techniques, and classified them into linear, nonlinear and artificial intelligence models. They presented the sunshine ratio, ambient temperature and relative humidity as the most correlated coefficients for modeling solar energy. The concept of ANNs attracted much more interest in the last decade to estimate GSR in distinct places around the world. Mellit [⁹] applied an ANN for predicting solar radiation in terms of sunshine and ambient temperature in Algeria. Yacef [¹⁰] developed new combined models utilizing a Bayesian neural network (BNN). Maximum and minimum air temperatures were used by authors to estimate GSR. They have proven that the new calibrated models are able to estimate GSR with an excellent accuracy in Algeria.

Recently, some researchers have used techniques of hybridization in order to predict global solar radiation. Voyant et al. [¹¹] have developed a hybrid ARMA/ANN model to estimate the GSR in five main Mediterranean places. The results show that the RMSE in the hybrid model is 14.9% while that in the original ANN model is 18.4%. Wu et al. [¹²] have applied a novel hybrid SOM-OPELM model with three–time series strategies (Recursive, DirRec, and MISMO). The results of hybrid models outperform and are much more accurate than those of the time series models alone.

Güçlü et al. [¹³] have recently combined application of the harmonic and the classical linear regression model (HarLin) to forecast solar radiation in three Turkish locations. They have concluded that the HarLin model is more efficient in comparison to the adaptive-neuro fuzzy inference system (ANFIS) model based on the Sugeno fuzzy logic inference system and the Angstrom-Prescott model. Hassan et al. [¹⁴] proposed, validated and compared the 17 new ambient-temperature-based models with the three models indicated in the literature for the measurement of the monthly average of the GSR in ten locations around Egypt. The results demonstrate that the ambient-temperature-based models are more accurate than the sunshine-based models. Ayodele et al. [¹⁵] have utilized the probability distribution of clearness index in order to imply the monthly average of the GSR in Ibadan, Nigeria. The experimental results indicate that the logistic distribution provides the best fit for the clearness index whereas the method proposed is effective in the prognostication performance rather than the three other empirical models. To improve the prediction of the monthly mean of the GSR, a support vector machine (SVM) has been developed by Belaid and Mellit [¹⁶]. Different combinations of (T_min, T_max, T_mean, T_diff, H₀ and S₀) in the inputs are used to develop the SVM models. The results unveil that the SVM based models developed deliver the best prediction rather than the ANN based model and other published models in the literature. In Ref. [¹⁷], the accuracy of the ANFIS model has been validated and checked in comparison with the expanded-improved Bristow-Campbell Model (E-IBCM) and the improved Yang Hybrid Model (IYHM). Instead of the E-IBCM and the IYHM models, the study uncover that the ANFIS model provides suitable evaluation for the prediction of the GSR in Hunan province of China. The RMSE and the MAE of the ANFIS model range from 0.59–1.60 MJ/(m²∙d) to 0.42–1.21 MJ/(m²∙d), respectively. Ling et al. [¹⁸] have studied the ability of the multilayer perceptron neural network (MLP) NN for predicting the GSR, regarding China, by using the back-propagation BP learning algorithm that is based on the spatial interpolation. Compared with the two improved empirical models, the (MLP) NN model provides a better result.

Through previous research, many researchers have shown that a careful analysis of complex models, which seems at first sight more sophisticated, are not necessarily most accurate; but rather simpler models, which depend on a limited number of parameters, are the most suitable. This is an important fact that must be taken into account. In this study, 14 different models are classified into two categories. The first category is nominated from the literature, while the second category is based on analytical proposed models, such as the analysis of models containing of Fourier series, equation of Weibull, sin functions, developed power equation, rational equations which are dependent on dividing the cubic regression model himself, but by different correlation coefficients and Gaussian series. Finally, by using ten statistical indicators, it is concluded that each model is adequate for some months of the year. Therefore, among the 14 models, there is no “one-for-all model” that can be used to predict. This is the main reason that has led to the proposal of a smart hybrid model, which is composed, based on the intelligent rules, of the most suitable prediction model of various simple models mentioned in the present paper. However, the parameter most commonly used in this study is sunshine duration.

The main objective of this study is establishing 14 empirical models with new coefficients of correlation based on the sunshine duration data for Tamanras set. By using ten statistical indicators, the models are evaluated in order to select the best model for each month. This is to create a new hybrid model, for estimating the global solar radiation data in the absence of the measured data. A database of monthly mean of sunshine duration and total solar radiation data, which has been recorded for 5years (2000–2004) are used. A set of 4-year known data are used for training the smart hybrid models, while a set of data of one year is used for testing and validating the suitability and capability of each hybrid model. The data of the modeled GSR in addition to two solar databases available for Tamanrasset, Algeria: NASA-SSE and HelioClim-1 (HC1) are validating their uncertainties in comparison with the measured data of the ground station in Tamanrasset. The goal of this contribution is to evaluate the accuracy of the monthly mean of GSR in order to choose the best data for CSP and CPV application in Algeria.

Description of data used

In this study, Tamanrasset under tropical Algerian climate is the ground site to simulate 14 empirical models for forecasting the monthly mean global solar radiation on a horizontal surface. The Algerian National Office of Meteorology (NOM) and two satellite-derived data that have been recorded over a period of five years (2000–2004) have been used to determine the coefficients constants of the 14 linear regression models and the smart hybrid model. As follows, the explanation of the site description and the definition of the data employed in the analysis are explained.

Site location

Indeed, the ambitious renewable energy program of Algeria has been expected to increase the concentrating solar thermal power plants (CSP) and photovoltaic systems (CPV) that will be systematically implanted until 2030. Many locations in Algeria are considered to hold solar thermal power plants in the near future. Tamanras set is the largest city in Algeria. It has an area of about 557906 km² (23.42% of the total area of Algeria) with a latitude of+22.783°, a longitude of+5.517°, and an altitude of 1377 m above the mean sea level. It is characterized by higher radiation than other Algerian ground sites. The climate in this ground site is desert arid with a minimum, maximum, and annual average of GSR ranging from 17.4 MJ/m² in winter, 29.2 MJ/m² in summer and 23.5 MJ/m², respectively. Therefore, it offers a promising opportunity to install solar thermal power projects that has been announced in the Algerian renewable energy program.

The geographical and climatic characteristics of the selected location are reported in Table 1.

Databases employed

The measured experimental data of GSR (MJ/(m²∙d)) and the sunshine duration (S) used in this work was obtained from three meteorological data, the NOM and two databases spatial NASA-SSE and HelioClim-1. Figure 1 shows the monthly evolution of GSR and extraterrestrial radiation throughout five years (2000–2004), while the daily and monthly mean of duration sunshine are given in Fig. 2.

Methodology

The main originality of this methodology is to use 14 solar radiation models and a new smart hybrid system to assess the monthly GSR using three main parameter inputs: extraterrestrial solar irradiance (G₀), duration sunshine (S), and daylight hours (S₀). These parameters are the most commonly used in the literature and provide the best results compared with other parameters. The procedure is described as follows:

Strategy of computation

To estimate the monthly mean daily global solar radiation G (MJ/m²) and to compare the models below, the data of sunshine duration and global solar radiation are taken from NOM and the two spatial databases, between 2000 and 2004. The monthly mean daily extraterrestrial radiation G₀, the maximum possible sunshine hours S₀, and the declination angle δ_s, for using the average days of the month, are calculated using the standard relations in Ref. [¹⁹].

The monthly mean daily extraterrestrial radiation on a horizontal surface G₀ are calculated for days giving the mean of each month. G₀ is calculated using

(1)G_{\mathrm{0}}={\displaystyle \frac{\mathrm{24}\times \mathrm{3600}\cdot G_{\mathrm{sc}}}{\pi }}\cdot G_t\cdot [\cos \phi \cdot \cos {\delta }_s\cdot \sin {\omega }_s+{\displaystyle \frac{\pi \cdot {\omega }_s}{\mathrm{180}}}\cdot \sin \phi \cdot \sin {\delta }_s],

where G_sc is the solar constant as 0.082 MJ/(m²∙min^–1) (1367 W/m²), G_t is the relative correction of the earth–sun distance, δ_s is the solar declination, ω_s is the mean sunrise hour angle for the given month, and φ is the latitude of the site.

G_t can be calculated by

(2)G_t=\mathrm{1}+\mathrm{0.034}\cos {\displaystyle \frac{\mathrm{360}\cdot nj}{\mathrm{365}}},

where nj is the number of day in the year starting from the 1st of January.

The declination angle (δ_s) is defined by

(3){\delta }_s={\mathrm{23.45}}^{°}\sin {\displaystyle \frac{\mathrm{360}\times (\mathrm{284}+nj)}{\mathrm{365}}}.

The sunset hour angle by

(4){\omega }_s={\cos }^{-\mathrm{1}}(-\tan \phi \cdot \tan {\delta }_s).

The maximum possible sunshine duration (S₀) can be obtained as

(5)S_{\mathrm{0}}={\displaystyle \frac{\mathrm{2}\cdot {\omega }_s}{\mathrm{15}}}.

A description of the mathematical expressions of the regression models used to predict the monthly mean daily GSR on a horizontal surface can be divided into two categories:

Category one: Category one contains eight empirical models: linear, quadratic and cubic, logarithmic, exponential and exponent, linear latitude and known constants related. These empirical models are the most used ones to estimate the monthly mean global solar radiation on horizontal surface in the literature and have the form of the Angstrom equation (see Table 2). This type of regression equations depends of the monthly mean GSR to extraterrestrial radiation at the location and the monthly mean fraction of possible sunshine hours, where a, b, c and d are the regression coefficients used in these models. These models (#1, #2, #3, #4, #5, #6, #7 and #8) have already been tested in Refs. [¹–⁴,²⁰–²⁴].

Category two: Category two includes six proposed models which are based on Fourier series (#9), equation of Weibull (#10), sin functions (#11), power series (#12), rational equations (#13), and Gaussian model (#14), which have the same input correlated parameters(H₀, S and S₀) as those of the first category with different regression coefficients (a₀, a₁,…,a₇, b₁,…,b₇, c₁,…,c₇, ω). The regression models proposed in this work are listed in Table 3.

A hybrid model is proposed and adopted, which combines 14 empirical models in one model, to estimate the monthly mean GSR from the measured sunshine duration. The flowchart for the developed model is suggested below.

where G_HYBRID is the global solar radiation of a new hybrid system in (MJ/(m²∙d)) at month t and the model specified m. Error_m=1,…,14% is the main criterion designed to assess the performance of a hybrid model for computing the monthly mean GSR. It is a calculated part of three fundamental criteria, MBE|_t=1,…12, RMSE|_t=1,…,12 and R²|_t=1,…,12 of each month to validate the analysis, so that the Error% in the first indicator is the difference between the calculated and measured data, while the Error(%) in the second indicator is the root of the square of the first error and in the last indicator, the Error(%) is the same as that of the first criterion divided by the difference between the measured and mean measured data. A model preferred to compute monthly mean GSR provides good performance if the MBE|_t=1,…12 and RMSE|_t=1,…,12 have as low values in each month, while R²|_t=1,…,12 has the highest value.

Accuracy indicators

Ten statistical indicators are used in order to compare the closeness of the measured data of GSR to the modeled values, RPE, MPE, MAPE, MBE, MABE, RMSE, U₉₅, TT, R², and GPI. As defined in Refs. [²⁴–²⁶], both of MBE, MABE and RMSE are expressed in (MJ/(m²∙d)), while RPE, MPE, MAPE,U_95, TT, R^2.and GPI in (%).

(6)\mathrm{RPE}=\left({\displaystyle \frac{H_c^i-H_m^i}{H_m^i}}\right)\times \mathrm{100},

(7)\mathrm{MPE}={\displaystyle \frac{\mathrm{1}}{k}}{\displaystyle \sum _{i=\mathrm{1}}^k\left({\displaystyle \frac{H_c^i-H_m^i}{H_m^i}}\right)}\times \mathrm{100},

(8)\mathrm{MAPE}={\displaystyle \frac{\mathrm{1}}{k}}{\displaystyle \sum _{i=\mathrm{1}}^k\left|{\displaystyle \frac{H_c^i-H_m^i}{H_m^i}}\right|}\times \mathrm{100},

(9)\mathrm{MBE}={\displaystyle \frac{\mathrm{1}}{k}}{\displaystyle \sum _{i=\mathrm{1}}^k(H_c^i-H_m^i)},

(10)\mathrm{MABE}={\displaystyle \frac{\mathrm{1}}{k}}{\displaystyle \sum _{i=\mathrm{1}}^k\left|H_c^i-H_m^i\right|},

(11)\mathrm{RMSE}={\left[{\displaystyle \frac{\mathrm{1}}{k}}{\displaystyle \sum _{i=\mathrm{1}}^k{(H_c^i-H_m^i)}^{\mathrm{2}}}\right]}^{\mathrm{1}/\mathrm{2}},

(12)U_{\mathrm{95}}=\mathrm{1.96}{(\mathrm{2}\cdot {\mathrm{RMSE}}^{\mathrm{2}}-{\mathrm{MBE}}^{\mathrm{2}})}^{\mathrm{1}/\mathrm{2}},

(13)\mathrm{TT}={\left[{\displaystyle \frac{(k-\mathrm{1}){\mathrm{MBE}}^{\mathrm{2}}}{{\mathrm{RMSE}}^{\mathrm{2}}-{\mathrm{MBE}}^{\mathrm{2}}}}\right]}^{\mathrm{1}/\mathrm{2}},

(14)R^{\mathrm{2}}={\displaystyle \frac{{\displaystyle {\sum }_{i=\mathrm{1}}^k\left(H_c^i-{\bar{H}}_c\right)\cdot \left(H_m^i-{\bar{H}}_m\right)}}{\sqrt{{\displaystyle {\sum }_{i=\mathrm{1}}^k{\left(H_c^i-{\bar{H}}_c\right)}^{\mathrm{2}}\cdot {\displaystyle {\sum }_{i=\mathrm{1}}^k{\left(H_m^i-{\bar{H}}_m\right)}^{\mathrm{2}}}}}}},

(15)\mathrm{GPI}=\mathrm{MBE}\cdot \mathrm{RMSE}\cdot U_{\mathrm{95}}\cdot \mathrm{TT}\cdot (\mathrm{1}-R^{\mathrm{2}}),

where H_m^i, {\bar{H}}_m and H_c^i are the ith measured clearness index (G_m/G₀), the average of measured clearness index, the predicted of clearness index (G_d/G₀) and the average of predicted of clearness index values of solar radiation respectively, and k is the total number of data points.

Results and discussions

In this paper, the measurements of the NOM and two spatial databases NASA-SSE and HelioClim-1 are taken in order to demonstrate the effectiveness of the six proposed models and the smart hybrid model, and to compare the accuracy of satellite-derived data with ground–based measurements. In this context, the monthly mean daily of GSR are analyzed using 14 regression models, for the Tamanrasset city in Algeria, where the predicted values with the measured values are compared using the only main parameter input, i.e., the sunshine duration. It is the most commonly used parameter in the literature and provides the best results compared with other parameters.

For this purpose, a regression analysis is conducted in order to establish 14 models for the Tamanrasset city in Algeria with measured test data set for the years 2000 to2004 (60 months). For validation, ten statistical indicators are used in this study, i.e., RPE, MPE, MAPE, MBE, MABE, RMSE, U₉₅, TT, R² and GPI. The regression coefficients (a, b, c and d) obtained from the eight models in the first category are presented in Table 4, while the regression coefficients ({a₀,…,a₇}, {b₁,…,b₇}, {c₁,…,c₇} and ω) generated by the six proposed models in the second category are tabulated in Table 5. The accurate indicators of the selected models of the first and the second category are presented in Tables 4 and 6, respectively. The simulated results from each model during the selected months are visualized in Figs. 3 and 4.

A comparison of the results in Tables 6 and 7 indicates that the majority of the proposed models are more accurate than those models used in the literature. On this ground, for each station databases, the best model among all of the nominated models is recognized and introduced in Table 7 in bold italics. Table 7 ranks of the 14 models utilizing the four error measures, MAPE, RMSE, R² and U₉₅, because of their similar results.

According to the statistical performances of the 14 aforementioned models, the Fourier series model (#9) represents the best model for assessment of the monthly mean global solar radiation for station data, as it yields the lowest error parameter values (MPE= –0.3111%, MAPE= 4.5037%, MBE= 0, MABE= 0.0306 MJ/m², RMSE= 0.0376 MJ/m², TT= 0.0072%, RPE_min = –8.7868%, RPE_max = 10.8271%, U₉₅ = 8.0037% and GPI= –3.0582E–08) and a coefficient of determination R² = 0.5981%.

Moreover, the model (#7) is assessed as the worst model for forecasting the monthly mean GSR because it has the highest values of MPE, MAPE, MBE, MABE, and RMSE, which are 4.5915, 8.5264, 0.0335, 0.0588 and 0.0725, respectively.

For the NASA–SSE data, the Fourier series (#9) is also assessed as the best model for estimating the monthly mean global solar radiation, with MPE= –0.2431, MAPE= 3.9024, MBE= 0.0002, MABE= 0.0252, RMSE= 0.0338, TT= 0.0381, RPE_min = –8.8038, RPE_max = 7.3424, U₉₅ = 6.6317, GPI= –5.4049E–07 and R² = 0.6225. While the worst result is for (#8) model with the highest values of TT= 2.5333, U₉₅ = 16.5794 and GPI= 0.1114.

In the case of HelioClim–1 data, the Gaussian series model (#14) has the best performance for evaluating the monthly mean of GSR, which presents the lowest values of MPE, MAPE, MBE, MABE, RMSE and TT such as –0.3141, 4.1810, 0, 0.0274, 0.0366 and 0.0004, respectively. The maximum and minimum values of RPE are 11.3943% and –8.8906%, respectively. The worst prediction model is (#13) with R² = 0.2961.

As shown in Table 6, it can be seen that the new statistical indicator GPI is similar to that of the most widely applied indicator MBE, where the lowest values of GPI are derived by the same best models mentioned in the three cases of meteorological data (#9, #9 and #14), as can be seen in Table 6.

According to numerical simulation, in order to evaluate the monthly performance of the previous 14 models, a statistical analysis is performed by calculating three main statistical indicators, MBE, RMSE and R² for each month. It has been observed that each model is adequate for some months of the year. Therefore, one model cannot be used for prediction. That is just the reason why a smart hybrid system is proposed which selects, based on the intelligent rules, the most suitable prediction model from the 14 models listed in this present paper. It must be noted that this study is intentionally conducted to find the more accurate model that could be chosen to create the smart hybrid system. For the elaboration of the intelligent rules, a set of 48 months of the years (2000–2003) is used to estimate the performance of the smart hybrid system. Subsequently, to test this new proposed smart hybrid system based on these rules, the data for the last year are employed. The results for the best model of each month have been highlighted in colored boxes, as can be seen in Table 8. It is one of the best models recognized by studying the statistical indicators presented in Tables 4 and 6.

Table 8 shows the accuracy of the models for computing the monthly GSR. MBE, RMSE and R² are used as the main accuracy indicators. As is stated before, a model preferred to compute the monthly mean GSR provides good performance if the MBE|_t=1,…,12 and RMSE|_t=1,…,12 have low values in each month, while R^2| _t=1,…,12 has the highest value. The models that have been arranged through intelligent rules in Table 8 can be introduced as the most precise models.

In fact, 14 regression models have been investigated, for building the smart hybrid model, using intelligent rules. Therefore, the best model for each month is selected by calculating the three main indicators (MBE, RMSE, and R²). In this work, for new smart hybrid models are adopted for designed and implemented SHBM1, SHBM2, SHBM3 and SHBM4, each model from the first three models is designed for each set of databases: Station, NASA-SSE and HelioClim-1, respectively. The last model is adopted to estimate the GSR to be used for each set of meteorological databases. Detailed results of comparative study to evaluate the performance of the four established SHBM models applied on the three meteorological databases are depicted in Table 9. Figure 5 shows the comparison of the estimated monthly mean GSR values for all smart hybrid models with measured data of the three meteorological databases, during the training and testing phases.

As seen from Table 8, the models proposed have been used many times in creating the smart hybrid models. The Fourier series model (#9) has been used 13 times to create the SHBM1 and 14 times to create the SHBM2 for the case of Station and NASA-SSE measured data, respectively. In HelioClim1 data, the Gaussian model (#14) was used 9 times in order to create the SHBM3. On the other hand, it is found that some models used in the literature are as good as the models proposed, and have the potential to provide accurate predictions similar to the models proposed. For instance, cubic (#3) and linear known constant (#8) models, have been used many times in creating the smart hybrid model (case of Tamanrasset station), models (#7)and(#8) in the case of NASA–SSE database, and models (#3) and(#7) in the case of HelioClim-1 database. But on the other hand, it is noted that there are models in the same category which have not been used in creating the smart hybrid model, for instance, models (#1) and (#2) in the case of Tamanrasset station, models (#1), (#2), (#3), (#4), and (#5) in the case of NASA-SSE database, and models (#1) and (#6) in the case of HelioClim-1 database.

Figure 6 illustrates the scatter plots of measured data versus the estimated monthly mean GSR values via the SHBM1, SHBM2, SHBM3, and SHBM4 models proposed, respectively, for the testing phase of data. All test SHBM models yield high R² values (>87%). Clearly, the SHBM2 model provides the best long–term prediction with a R² value higher than 95%.

As seen in Fig. 6, SHBM1, SHBM2, and SHBM3 performs much better compared to the fourth SHBM4 model in estimating the values of the monthly GSR, but SHBM4 has higher R² values (87.56%–92.68%).

Based on Table 8, the flowing intelligent rules have been dressed, for predicting of GSR in each month of the year:

Rule 1: if t=1 then G_HYBRID=G_Model.09

Rule 2: if t=2 then G_HYBRID=G_Model.08

Rule 3: if t=3 then G_HYBRID=G_Model.09

Rule 4: if t=4 then G_HYBRID=G_Model.10

Rule 5: if t=5 then G_HYBRID=G_Model.08

Rule 6: if t=6 then G_HYBRID=G_Model.12

Rule 7: if t=7 then G_HYBRID=G_Model.09

Rule 8: if t=8 then G_HYBRID=G_Model.09

Rule 9: if t=9 then G_HYBRID=G_Model.11

Rule 10: if t=10 then G_HYBRID=G_Model.03

Rule 11: if t=11 then G_HYBRID=G_Model.09

Rule 12: if t=12 then G_HYBRID=G_Model.10

Another statistical analysis is performed by calculating RPE for each model, where the model that has values of RPE from –10% to+ 10% can be considered as the most acceptable one. Table 9 confirms that the use of the RPE has given the best results for the new smart hybrid models as demonstrated in Fig. 7.

It can be seen that in Fig. 7, the 4 SHBM models can obtain the maximum and the minimum values of RPE from –6.449% to+ 7.555% and –8.616% to+ 19.68%, for training and testing phases of all meteorological measured data, respectively. Figure 7 reveals that there are higher numbers of months falling in the lower ranges of RPE, which indicates the high potential of all smart hybrid models in estimating the monthly mean global solar radiation.

Conclusions

For many years, predicting future values of solar radiation at isolated sites poses a challenge to researchers and scientists. This paper proposed a set of SHBM models for prediction of the monthly mean global solar radiation at Tamanrasset, Algeria. The accurate results reported in the paper have been accomplished in several steps. First, 14 empirical models are established to forecast the monthly mean GSR. Next, by using 10 statistical indicators, the empirical models are tested to select the best model for each month to build a base of rules of an intelligent system. Finally, for the testing data, the most suitable algorithm for building a new smart hybrid system is identified and applied to evaluate global solar radiation.

The results of numerical simulation of the monthly mean GSR data of all the empirical models for a long-term period from 2000 to 2004 yield coefficients of determination (R²) higher than 65%. The comparison of the new strategy proposed shows that the SHBM models are the best with a R² higher than 84%, and with the lowest statistical error parameters. The results also confirm that the new models proposed provide a very good estimation for the monthly and daily average GSR data, especially the two models (#9) and (#14). These computational results confirm that the new smart hybrid model configuration with the only input parameter which is the sunshine duration produces an accurate prediction and can be used for other locations in Algeria.

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