A total of 6 years data [Appendix], monthly average value of temperature, humidity, and sunshine duration are obtained from the solar radiation handbook and NREL (National renewable energy laboratory) are used for training and the remaining 1 year is used for testing. Recent datasets of 2015–17 have also been prepared with the help of IMD Bhubaneswar center for further studies and experimentation.

As we know that solar energy is unpredictable and uncertain, there is an urgent need to mitigate the uncertain nature of solar radiation. Conventional methods fail to predict the solar irradiation properly because of the uncertain behavior of Sun. So soft computing happens to be an innovative approach with an ability of a human mind. The application of various soft computing tools such as multi-layer perceptron (MLP), ANFIS, RBF, RNN, NARX, GNN, FL, FG, NFG, NG, and SVM are suitably employed to predict and estimate the solar irradiance.

An adaptive neuro-fuzzy inference system has been used to predict the daily global solar radiation of the eastern zone of India. The data on daily solar radiation, sunshine duration, humidity, and temperature for the period of 5 years are collected from the renewable energy source laboratory, NASA, and the solar radiation handbook. A total of 2,190 day (2000–2005) datasets are used in the A N F I S model. Out of 2,190 days, 1825 days are considered as training and the rest 365 days are considered for testing. Later, the latest data have been applied for further research work.

The typical structure of A N F I S has been divided into three parts: (I) a rule based, (II) a database, and (III) reasoning mechanism. ANFIS as shown in Figure 1 is a hybrid method which combines algorithms such as back propagation, least-square algorithm, and gradient-descent for optimizing the system output.

The ANN network is depicted in Figures 1A,B. Possessing lots of nodes joined through the directional linking. In order to minimize the error, couple of basic learning rule–based method has been used in the network such as back-propagation technique. A fuzzy model having rules is as follows: R u l e   I : I f   x 1 = A 1 & y 1 = A 2 ⇒ M 1 = p 1 x + q 1 y + r 1 , R u l e   I I : I f   x 1 = B 1 & y 1 = B 2 ⇒ N 1 = p J x + q J y + r 2 , (1)

where x 1 , y 1 symbolizes input values.

M 1 & N 1 symbolize outputs.

A 1 & A 2 stand for the fuzzy sets (Figures 1A,B)

The model (as shown in Figures 2, 3) uses six dissimilar membership type function such as (Gauss mf, triang mf, two side Gaussian mf, Bell mf, Difsig mf, and Trap mf) along with (i.e., linear &constant) membership function. The dataflow has been explained from Eqs 3–8).

Layer 1. Every node i in this layer is an adaptive node with a node function. o 1 , i = { μ A i ( x )   f o r   i = 1,2 μ B i − 2 ( y )   f o r   i = 3,4 } . (2)

Y or X = input node I.

A_i or B_i-2 = linguistic value.

O_{1, i} = membership grade.

Membership grade satisfies the quantifier A. μ A i ( x ) = { 0 x ≤ a i x − a i b i − a i a i ≤ x ≤ b i c i − x c i − b i b i ≤ x ≤ c i 0 x ≥ c i } , (3)

Layer no 2: Every node in the layer is fixed, where the output symbolizes product of all input signals. o 2 , i = w i = μ A i ( x ) μ B i ( y )   i = 1,2. (4)

The node output indicates the firing strength. Any other T-norm which performs fuzzy is used as a node function. Layer no three is fixed and is labeled as N. Furthermore, the ith node decides about the ith rule’s firing capacity for the sum of all the rule’s firing capacity. o 3 , i = w i ¯ = w i w 1 + w 2 i = 1,2 − w here   the   outputs   idicate   normalized   firing   strength . (5)

Layer 4. Every node i in this layer is an adaptive node with function. o 4 , i = w i ¯ f i = w i ¯ ( p i x + q i y + r i )   i = 1,2 , (6) where w i ¯ is a normalized firing strength from layer three and = parameter sets of the node.

Layer no 5. Here, in this case; a single node is a fixed node and is given as Σ. This calculates the overall output as sum of all the incoming signals. o u t p u t   o 5 = ∑ i w i ¯ f i = ∑ i w i f i ∑ i w i . (7)

Based on the geographical coordinates and following meteorological parameters such as relative humidity and sunshine duration, the isolated places of Nigeria (Ojosu and Komolafe, 1987; Ododo et al., 1995) are studied for forecasting daily global solar radiation using ( R M S E ) and M A P E values. Further advantages of this model in (Olatomiwa et al., 2015a) the accuracy of the model are measured using the ANFIS-based soft computing technique for predicting solar radiation. The model uses the following meteorological parameters such as monthly mean maximum and minimum temperature and sunshine duration. Finally, the accuracy using ANFIS is compared with experimental results in terms of RMSE and coefficient of determination (R ²). Further research has been carried out using a hybrid machine learning technique for solar radiation prediction based on some meteorological data (Olatomiwa et al., 2015b; Olatomiwa et al., 2015c; Olatomiwa et al., 2015d). For this, a novel method named as SVM–FFA is developed by hybridizing the support vector machines (SVMs) with the firefly algorithm (FFA) to predict the monthly mean horizontal global solar radiation using three meteorological parameters such as sunshine duration (n¯), maximum temperature (T max), and minimum temperature (T min) as inputs. The prediction accuracy of the proposed SVM–FFA model is validated compared to those of artificial neural networks (ANNs) and genetic programming (GP) models. The root mean square error (RMSE), coefficient of determination (R²), correlation coefficient (r), and mean absolute percentage error (MAPE) are used as reliable indicators to assess the models’ performance. In this work, the authenticity of the soft computing method in forecasting based on the number of meteorological data of Nigeria is studied. The simulation work has been performed using the SVM where the inputs are monthly maximum temperature T max, monthly mean temperature T min, and monthly Sunshine (Bahel et al., 1987a; Asl et al., 2011). The sizing of the standalone photovoltaic system is designed with the help of a solar radiation pattern. Mohammadi et al. (2016a) have also used the A N F I S model for finding out the most suitable parameters for the forecasting of daily horizontal diffused solar radiation. Here, the author suggests a single input for case 1, both H&H _O combination for case 2 and H , H O & n combined value for the third case. A comparative study has been carried out between ANN&ANFIS to predict daily solar radiation G S R in different parts of Iran (Bahel et al., 1987b; Robaa, 2009; Abdo and EL-Shimy, 2011; Ramedani et al., 2014a; Mohammadi et al., 2016a). Mehmet et al. (Rahimikhoob, 2010; Koca et al., 2011; Demirhan, 2014; Demirhan and Kayhan Atilgan, 2015; Yıldırım et al., 2018) have drawn a comparison between statistical and neuro-fuzzy network models to forecast the weather of Istanbul. A long period of 9 years ranging from 2000 to 2008 has been considered taking parameters such as daily temperature average (dry–wet) and pressure of air and speed of wind. Different models such as ANFIS and autoregressive integrated moving average (ARIMA) have been incorporated in this particular research work. Further several training and testing datasets have been considered to find out the effectiveness of these models. The performance is determined after comparing several parameters with respect to the moving average error (MAE) and root mean square error (RMSE−R ²). Teke and Yıldırım (2014) estimates monthly global solar radiation for twelve cities of the eastern Mediterranean region based on meteorological data based on the following statistical test (MBE, RMSE, and MPE). The result shows that the Angstrom–Prescott model is most suitable for the calculation of GSR in the sites of Bonger, Pala, and Am-Timan mongo. Al-Mostafa et al. (2014) developed a sunshine based GSR model in Riyadh, Saudi Arabia as it is easily and reliably measured with wide availability of data. Almorox et al. (Quej et al., 2016) estimate empirical models for predicting daily GSR in Peninsula, Mexico. A total of 13 different models were developed based on following parameters such as temperature, rainfall, and air humidity. But by taking temperature as the input parameter, model performs the best result. On the basis of statistical indicators RMSE, MBE, MPE, and coefficient of determination, Prescott (1940) developed an empirical model to calculate monthly average daily global solar radiation on a horizontal surface from monthly average daily total insolation on an extra-terrestrial horizontal surface by using the following equation H/H₀ = a + b (S/S₀).

Yacef et al. (2012) prepare a comparative study between Bayesian neural network (BNN), classical neural network (CNN), and empirical models for estimating the daily global solar irradiation (DGSR) of AI-Madinah (Saudi Arabia) from 1998 to 2002. A comparative study has also been carried out between the Bayesian network with the classical neural network and the empirical model developed using the Angstrom–Prescott equation. Mellit (2005) and Mellit et al. (2007) applied an ANFIS model for estimating the sequence of monthly mean clearness index (k_t) and daily solar radiation data in isolated areas of Algerian location with some geographical coordinates (latitude, longitude, and altitude) and meteorological parameters such as temperature, humidity, and wind speed. The comparison has also been made between ANFIS and ANN by evaluating the RMSE and MAPE.

An ANFIS model (Mellit, 2004; Mellit et al., 2008) is presented for estimating the mean monthly clearness index (K_t) and total solar radiation data in isolated sites based on geographical coordinates. These data have been collected from 60 locations in Algeria. The magnitude of solar radiation is the most important parameter for sizing photovoltaic (PV) systems. The ANFIS model is trained using MLP based on fuzzy logic (FL) rules. The inputs of the ANFIS model are the latitude, longitude, and altitude, while the outputs are the 12-values of mean monthly clearness index K_t. The results show that the performance of the proposed approach in the prediction of mean monthly clearness index K_t is favorably compared to the measured values. The RMSE between measured and estimated values varies between 0.0215 and 0.0235 and the MAPE is less than 2.2%. Data from 60 locations in Algeria are taken into account, and the performance of the model is found out through the RMSE and mean relative error (MRE) (Angstrom, 1924; Garg and Garg, 1983; Takagi and Sugeno, 1985; Bahel et al., 1987c; Hawlader et al., 2001; Kalogirou, 2001; Iqdour and Zeroual, 2004; López et al., 2005; Tymvios et al., 2005; Bosch et al., 2008; Zounemat-Kermani and Teshnehlab, 2008; Behrang et al., 2010; Tektaş, 2010; Coulson, 2012; Boland et al., 2013; Jafarkazemi et al., 2013; Will et al., 2013; Ramedani et al., 2014b; Choubin et al., 2014; Varzandeh et al., 2014; Mohammadi et al., 2015; Choubin et al., 2016a; Choubin et al., 2016b; Mohammadi et al., 2016b; Despotovic et al., 2016; Kaplanis et al., 2016; Wu and Wang, 2016; Quej et al., 2017; Zou et al., 2017; Almaraashi, 2018b; Halabi et al., 2018; Khosravi et al., 2018; Rafiei-Sardooi et al., 2018). Yadav et al. (2014) have applied the J48 algorithm and WEKA software for selecting significant input parameters such as clearness index, altitude, and longitude for the better prediction of solar radiation in Western Himalayas with ANN (Mani, 2008; Yadav et al., 2014; Yadav and Chandel, 2015). By using the four g-bell input membership function, statistical analysis shows the maximum regression value; R (R = 0.99) in comparison to the other membership function (Bhardwaj et al., 2013a; Ramedani et al., 2013b). M Rizwan et al. (Khan et al., 2008; Rizwan et al., 2012; Chaudhary and Rizwan, 2018; Perveen et al., 2018; Sadhu et al., 2018; Chaudhary and Rizwan, 2019) focused on the GNN model in order to predict global solar energy in India. Parameters used as inputs in this particular model include latitude, longitude, and altitude (Iqbal et al., 2010; Patel and Parekh, 2014; Singh and Rizwan, 2018b; Yadav et al., 2018b; Sujil et al., 2019b; Singh et al., 2019; Vanitha et al., 2019) with temperature ratio, Sunshine/hour whereas the clearness index stands for the output parameter. Solar radiation data set has been prepared for several Indian states for training and performance is evaluated through the mean absolute error. The MAPE during the estimation of global solar energy prediction is found to be nearly 4 percent using GNN but it becomes 6 percent during estimation with the fuzzy logic (Ajil et al., 2010; Chandra et al., 2013; Verma et al., 2019). Joshi (2013) estimated the monthly global solar radiation utilizing the Angstrom model (Angstrom, 1924) forecast solar irradiation with the help of ANN with variables. Thorough comparisons have been carried out between the ANN model and Angstrom model in order to judge the efficiency of the models with mean squared error (MSE) and regression coefficient (R ²). From the learning of MSE and R ² values with Angstrom models they come close to (0.1225 and 0.3965) for Ahmedabad, (0.0059 and 0.0149) for Bangalore, (0.1024 and 0.404) for Dehradun, and (0.0625 and 0.0498) for Kolkata whereas the MSE and R ² values for the ANN model as (0.002 and 0.99) for Ahmedabad, (0.006 and 0.98) for Bangalore, (0.01, 0.90) for Dehradun, and (0.006 and 0.99) for Kolkata. The selected ANN model performs better with less RMSE value with maximum regression value than the empirical model. Kadhambari et al. (2012) proposed a recurrent neural network model to estimate the global solar radiation of the Thiruvallur region. Input parameters utilized in this work are all days of month, all day temperature, humidity (relative), pressure of air, and solar azimuth angle. The RNN-based models are trained by the evolutionary swarm optimization–based algorithm. The performances of these algorithms are verified and compared with each other by calculating the RMSE value. The RMSE value of the evolutionary algorithm comes around 0.0667 which is lower in comparison to the RMSE value of the PSO algorithm. Poudyall khem et al. estimate [GSR] depending on the sunshine duration in the Himalayan region. The performance parameters of the model are investigated on the basis of RMSE value, MBE value, MPE value, and correlation coefficient R ² value. Solar radiation data for a span of 3 years of Indian cities have been studied by Katiyar and monthly daily mean clear sky radiation has been estimated.

A thorough comparison on the basis of (RMSE) and (MBE) has been initiated which shows the percentage of MBE with a new constant for each station vary from 0.22 to 2.09% whereas with RMSE it varies from 2.22 to 10.37%. Krishnaiah et al. (2007b) suggest the neural network approach for modeling which suggests the superiority of the neural network–based model compared to conventional regression models. Premalatha and Arasu (2012) estimated the GSR of India utilizing ANN based on the input parameters such as maximum and minimum ambient temperature with least relative humidity.

The monthly global solar radiation in 31 districts of Tamil Nadu, India was predicted by using A N F I S in Verma et al. (2019). Considering the input parameters such as solar radiance, ambient temperature collector, tilt angle, and working fluid mass flow rate, the flat plate collector efficiency was predicted using the MLP and ANFIS model (Verma et al., 2019). This specific model M L P utilizes the Levenberg–Marquardt algorithm with logistic sigmoid function. Comparative analysis proves A N F I S model’s superiority over normal controlling architectures. After the introduction of unglazed flat plate solar collectors, analytical and experimental studies have been carried out on a solar-assisted heat pump water heating system (Chandra et al., 2013). Mohammad Hossein et al. (2014) use the W N N (wavelet neural network) and A N F I S algorithm for the prediction of meteorological station in Tehran, Iran. The results establish better performance in the field of solar radiation estimation and wind short-term solar radiation velocity time series. The analysis in terms of R ² and R M S E establish that with lower R M S E and higher R ² values a perfect model can be achieved. Dushyant Patel and Falguni Parekh (Chandra et al., 2013) used A N F I S for forecasting the flood of the Dharoi dam on the Sabarmati river in Mehsana in the state of Gujarat in India. In this case, statistical indices such as R M S E , correlation coefficient (R), and discrepancy ratio (D) are used. The evaluation of the model for forecasting has been carried out by comparing the A N F I S model and statistical method such as the log-Pearson type III method. The comparison indicates that the A N F I S model accurately addresses the forecasting of flood. In another work, rainfall forecasting with A N F I S has been developed by Jignesh Patel and Falguni Parekh (Krishnaiah et al., 2007a; Ajil et al., 2010; Premalatha and Arasu, 2012; Chandra et al., 2013; Joshi, 2013; Awasthi and Poudyal, 2018) for Gandhi Nagar station. Eight models based on different membership functions and climatic parameters such as temperature, relative humidity, and wind speed are developed. In this case, a generalized bell-shaped membership function has been chosen. The outcome of the hybrid model with seven membership functions and three inputs produces better results with a correlation factor of 0.99 for training and 0.92 for validation. The application of A N F I S for wind energy short-term forecasting was first developed by Pousinho et al. (Krishnaiah et al., 2007a). Experiments established the efficiency of neuro-fuzzy inference system and proved its performance regarding M A P E & E r r o r variance in comparison to A R I M A & N N .

The ANFIS-based PV model predicts the I-V and P-V characteristics of the PV modules in a given environmental setting. For a certain irradiance and temperature combination of the PV cell, the voltage of PV array (from zero to open-circuit voltage) can be determined on the manufacturing datasheet. The corresponding anticipated current set is obtained from the proposed PV estimation model. The equivalent circuit of the PV is described as follows: I = I p h − I o { exp ( V + I R s / n s V t ) − 1 } − ( V + I R s / R s h ) , (8) V t = k T A / q , (9) I p h = ( G / G S T C ) I p h ( S T C ) ( 1 + k i ( T − T S T C ) ) , (10) I M P = I p h − I 0 { exp ( I s c R S / n s V t ) } − ( I s c R S / R S h ) , (11) I M P = I p h − I o { exp ( V M P + I M P I R S / n s V t ) } − ( V M P + I M P I R S / R s h ) , (12) I O C = 0 = I p h − I 0 { exp ( V o c / n s V t ) } − ( V o c / R s h ) , (13) | d p / d v | V = V M P , I = I M P = 0 , (14) | d i / d v | V = 0 , I = I M P = − 1 / R s h , (15) V t = ( ( I M P R s + V M P − V O C ) / n s ⁡ log   B ) , (16)

where I = output current.

I_Ph = photo current or generated current under given insolation.

I_O = diode reverse saturation current.

η = ideality factor of PV cell.

R_S = series loss resistance.

R_Sh = shunt loss resistance.

V_t = thermal voltage.

V t h = k T q , where k is Boltzmann’s constant = 1.3806 X 10 − 23 J / K .

Input data = solar irradiance G.

Ambient temp = T.

Operating voltage = V.

Output current = I.

During the training and testing of the model, only one input is considered. The performance outcome of both training and testing are presented in (Table 4) considering the sunshine input as the most favorable parameter.

The performance evaluation of different models with respect to the number of inputs is described in Table 4.

Here, two input parameters are combined for the purpose of training and testing. In total, six models have been created. The output obtained has been shown in Table 6. The statistical result shows for state Bhubaneswar model 1 with input temperature and sunshine duration produces improved results in comparison to other input combinations. But in the case of Visakhapatnam and Kolkata, model two with inputs (sunshine duration and Relative Humidity) will give a superior outcome compared to other input combinations.

Table 6 shows the statistical analysis with ANFIS models for cities Bhubaneswar, Kolkata, and Visakhapatnam.

Considering three vital input parameters, models have been formulated. Furthermore, all the statistical results have been verified providing better outcome with inputs such as sunshine duration, temperature, and humidity. The outputs after training and testing have been presented in Table 7. Proper knowledge about the inputs helps a lot in forecasting solar radiation at any particular place.

In the past, several initiatives have been taken in India regarding solar radiation data forecasting with conventional empirical models. Three important cities of Eastern India have been taken as case studies to carry out analysis for solar radiation prediction purposes. Furthermore, input parameters are fixed such as proportion of surface air pressure P / P O , temperature T / T O , sunshine span S / S O , and relative humidity R / R O . Few statistical tests such as M B E , M S E , and correlation coefficient [R] have been calculated from the measured and predicted output ( G S R ). This can be carried out after utilizing the dissimilar input–output membership function of A N F I S . Furthermore, this A N F I S model utilizes the grid participating method and follows dual output membership functions such as constant and linear membership functions. Adaptive neuro-fuzzy (Krishnaiah et al., 2007a; Awasthi and Poudyal, 2018) system has been utilized to identify most pertinent parameters for the forecasting of daily GSR. Different cities of central and southern Iran are considered for case studies. This work discussed (Kadhambari et al., 2012; Premalatha and Arasu, 2012; Bhardwaj et al., 2013b; Citakoglu, 2015; Meenal and Selvakumar, 2018; Sobri et al., 2018) the accuracy and performances of different soft computing techniques such as ANFIS, ANN, and SVM for the forecasting of daily horizontal GSR. The performance of the model is assessed from statistical indicators such as (RMSE, MAE, and coefficient of determination (R ²). Authors in this particular work (Choubin et al., 2018b) have advocated standalone ANFIS and hybrid models to predict global solar radiation using several meteorological parameters such as sunshine duration, air temperature, and optimization techniques such as PSO and genetic algorithms are used. Monthly solar radiation values (Melin and Castillo, 2005; Pousinho et al., 2011; Melin et al., 2012; Pérez et al., 2012; Choubin et al., 2017; Choubin et al., 2018a) have been modeled with the help of ANN, ANFIS, and empirical equations. Input variables such as meteorological data and month numbers are used as input variables. Authors in this work have emphasized the accuracy (Aguilar et al., 2003; Mohanty et al., 2016a; Mohanty et al., 2017a) of SVM, ANN, and empirical solar radiation models with different combinations of input parameters such as month, latitude, longitude, bright sunshine hours, day length, relative humidity, and maximum and minimum temperature. Four novel empirical models have been introduced and validated with experimental data. Authors have proposed (Choubin et al., 2018b) several models such as ANFIS, E-IBCM, and IYHM and evaluated in order to predict global solar irradiance whereas improved empirical models have been found to be better than other original models for solar radiation forecasting. The ANFIS model produces the best global solar irradiance capability in China among the three models. Algorithms such as MLP, ANFIS, and SVMs have been used. The models have been divided into four groups including sunshine, temperature, and other meteorological parameters. The first network uses five inputs to predict the solar irradiance (N₁) while the second network is the time-series prediction of solar radiation (N₂). MLFFNN, RBFNN, FIS, and SVR models are developed for N₁. MLFFNN, SVR, FIS, and three ANFIS models are developed for N₂. Authors have (Choubin et al., 2018a) compared the neuro-fuzzy model with that of the time-series model for the modeling of the drought. Research studies (Mohanty et al., 2016b; Mohanty et al., 2020) have focused on the novel application of classification and regression tree–based (CART) model. (Mohanty et al., 2016a; Mohanty et al., 2016b; Mohanty et al., 2020). After the process of training and testing, monthly average solar radiation data and statistical analysis have been attempted for three cities Kolkata, Bhubaneswar, and Visakhapatnam (Tables 5–7, Figures 11, 12).

Figures 13A–C show the surface plot for the optimal combination of inputs of three cities at the time of training and testing. The significant combination of parameters having one, two, and three inputs of three cities of Eastern zone of India during training and testing are shown in Figures 14A–C. The measured average monthly GSR was compared with measured values of Kolkata, Bhubaneswar, and Visakhapatnam (Tables 8, 9). Due to recurrent cyclonic effects, it is essential to devise new computational methods such as soft computing–based algorithms and their applications. ANFIS predicts better outcomes with calculated values (Figure 15).

In spite of its efficiency and computational ability, additional inputs are always needed for enhanced accuracy and robustness. More emphasis should be given to the input factors where error in computation of training data is found. Further data fluctuations are taken care of because of ANFIS’s robustness and computational skill. In this regard, a lot of work(Krishnaiah et al., 2007a; Ajil et al., 2010; Kadhambari et al., 2012; Premalatha and Arasu, 2012; Chandra et al., 2013; Joshi, 2013; Citakoglu, 2015; Awasthi and Poudyal, 2018; Verma et al., 2019)has been done in recent days emphasizing the combination of different models and input parameters for better forecasting studies (Table 10).