# Global Solar Radiation Forecasting Based on Hybrid Model with Combinations of Meteorological Parameters: Morocco Case Study

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## Abstract

**:**

^{2}), root means square error (RMSE), stand deviation (σ), the slope of best fit (SBF), legate’s coefficient of efficiency (LCE), and Wilmott’s index of agreement (WIA). The best model is selected by using the computed statistical metric, which is present, and the optimal value. The R

^{2}of the forecasted ARIMA, ARMA, FFBP, hybrid ARIMA-FFBP, and ARMA-FFBP models is varying between 0.9472% and 0.9931%. The range value of SPE is varying between 0.8435 and 0.9296. The range value of LCE is 0.8954 and 0.9696 and the range value of WIA is 0.9491 and 0.9945. The outcomes show that the hybrid ARIMA–FFBP and hybrid ARMA–FFBP techniques are more effective than other approaches due to the improved correlation coefficient (R

^{2}).

## 1. Introduction

^{2}of the selected best inputs parameters varies between 0.9860% and 0.9920%, with the range value of MBE (%) being from −0.1076% to −0.5931%, the RMSE between 0.1990 and 0.4580%, the range value of the NRMSE is between 0.0355 and 0.8938, and the lowest value of the MAPE is between 0.0019 and 0.0060%. This technique could be used to predict other parameters for locations where measurement instrumentation is unavailable or costly to obtain. In the meantime, the authors of [16] presented a comparative optimization of daily global solar radiation forecasting with different machine learning and time series methods. The selected methods are compared with the persistence technique and measured data. Several statistical metrics are assessed to obtain the most appropriate method, which presents the lowest value accuracy. The select result is, respectively, the RMSE (%) and MBE (%) values of several models employed in this study, and were computed to be mostly positive. The range value of the selected model measured by RMSE (%) and MBE (%) varied between 4.64% to 8.87% and 6% to 22.93%. Based on all statistical metrics, the lower value of the selected model corresponds to the neural FFBP (6 × 10 × 1) in comparison with the other models. The appropriate one performs well and is close to the measured data. The authors of [17] presented a complete and detailed synthesis of solar radiation modeling, forecasting, and solar radiation data using artificial intelligence methods: ANN, fuzzy logic, genetic algorithm, expert system, and a hybrid method. It is proven that solar radiation is a vital factor in PV system performance and sizing. The same researchers have presented a combination of the above methods for generating horizontal global solar radiation by combining ANN and library of Markov transition metrics (MTM) approaches based on three-parameter coordinates (longitude, altitude, and latitude) and the data were collected from a data basis of 60 stations in Algeria over 9 years. This prediction is comparatively accurate related to the relative Root Mean Square Error (RMSE), which is less than 8.2%. On other hand, López et al. [18] have chosen the automatic relevance determination method (ARD) based on the ANN model to select the relevant input parameters in direct normal solar irradiance forecasting. Clearness index and relative air mass were considered the most important input parameters to the neural network. According to J. Lampinen et al. [19], Penny et al. [20], and B. Belmahdi et al. [8], the ARD and ANN are the priority distribution on the network weights and determine the most relevant input parameters by introducing a hyperparameter for each input unit of the ANN. In ANNs, the important prior distribution of the network weights is controlled by the hyperparameters. Here, the ANN is presented with training data, and posterior weight distribution and hyperparameters are calculated using the Bayes rules.

## 2. Materials and Methods

#### 2.1. Data Collection and Study Sites

_{mean}), maximum (T

_{max}), and minimum (T

_{min}), temperature. The three-temperature dataset describes the timescale distribution of day-to-day temperature variations in the Tetouan and Tangier sites. The annual temperature variation reveals the succession of warm and cold periods over the year. The analysis is conducted for each day. The recorded T

_{max}was observed in the July period (38 °C for Tetouan and 36.6 °C for Tangier), while the T

_{min}was observed in the January period (6 °C for Tetouan and 2 °C for Tangier). The distribution of the T

_{min}is close to normal condition.

^{2}and 19.99 MJ/m

^{2}with a standard deviation of 1.98 kW·h/m

^{2}and 1.76 kW·h/m

^{2}for the Tetouan and Tangier sites, respectively. In the warm period, the lowest solar radiation deviation was 1.106 kWh/m

^{2}for the Tetouan site and 0.7375 kW·h/m

^{2}for the Tangier site. The highest solar radiation deviation in the cold period was 7.48 kW·h/m

^{2}and 7.413 kW·h/m for Tetouan and Tangier sites, respectively. As for month averages, the highest solar radiation months over the year were from May to July, while the lowest values were from December to January.

_{s}, m/s; average), relative humidity (RH,%), top of atmosphere radiation (TOA radiation), and geographic coordinates (Latitude, Altitude, and Longitude). These are exogenous data, whose periods and characteristics of measurements are summarized in Table 2.

#### 2.2. ARIMA and ARMA Model

_{i}refers to the i-th term autoregressive parameter, θ

_{i}refers to the ith term moving average parameter, c is the constant, e

_{t}is defined as an error at time t, B

_{p}refers to the p-th order backward shift operator, and X(t) is defined as a time series value at time t.

#### 2.3. Artificial Neural Network Model (FFBP)

_{t−1}…. x

_{t−4}could be defined in Equation (3):

_{j}and w

_{i}represent the connection weights, and S

_{1}and S

_{2}are the activation function.

#### 2.4. Hybrid Model

_{t}is the linear pattern obtained by ARMA and ARIMA models and Φ

_{t}is the residual that can be estimated from the feed-forward with the backpropagation algorithm.

_{t}) from the forecasted models:

_{t}and Φ

_{t}are the linear and nonlinear forecasting of the original series.

- -
- Data pre-processing in section one (grey color) involves the collection of meteorological, computational, astronomical, and geographical data. These parameters require many corrections of missing data and outlier removal.
- -
- The application of multiple combinations of several input parameters in order to select the appropriate architecture executed in section two (gold color) was accomplished by splitting data into two steps, which are training data (80%), testing, and validation (20%) data.
- -
- The step of the training (green color) was operating the proposed methodologies. The input parameters were tested by using time series model stationarity (Ljung–Box test). After that, the data stationarity was implemented for ARIMA and ARMA models. In the case of the ARIMA model, that involves the residual generated by the FFBP model, which built the combined ARIMA and FFBP models.
- -
- The models were built and divided into simple (ARMA, ARIMA, and FFBP), and hybrid methods (hybrid ARMA-FFBP and hybrid ARIMA-FFBO models; orange color).
- -
- The obtained result (grey color) was evaluated and interpreted by using various statistical metrics in order to choose the best model, which presents the lowest value of MBE (%), RMSE (% Sd (%), AIC, and BIC and the highest values of R
^{2}, SBF, LCE, WIA.

#### 2.5. Model Selection

#### 2.6. Performance Criterion

^{2}).

^{2}defines the appropriate sequential match among measured and forecasted values and the writing:

## 3. Results and Discussion

^{2}, WIA, SBF, and LCE of the selected ARMA (10.0.0) present a significant accuracy and vary between 0.8074 and 0.9939. In terms of BIC and AIC, the selected model presents the worst values forecasting after the ARIMA (2.1.0) model. The average value of the daily GSR forecasted by the ARMA (10.0.0) model is lower than the average daily GSR measured. In terms of error forecasting, it is seen that the selected model presents various observations with multiple error forecastings, like observation number 63 (EF = 1.878 kW·h/m

^{2}), 87 (EF = 2.378 kW·h/m

^{2}) 102 (EF = 2.818 kW·h/m

^{2}), and 263 (EF = 2.994 kW·h/m

^{2}), respectively.

^{2}, the ARIMA (2.1.0) is 0.9628, which presents successful forecasting accuracy compared with the ARMA (10.0.0) model. The considering error forecasting of the ARIMA (2.1.0) model was seen in observation number 161 (3.542 kW·h/m

^{2}).

^{2}, the shown model gives the highest accuracy, which is estimated at 0.9890. The lowest value of the FFBP (12.2.1) in terms of the MBE, RMSE, BIC, and AIC is 0.817 kW·h/m

^{2}, 0.5119 kW·h/m

^{2}, 991.3442, and 890.6528, respectively. Unlike the ARMA (10.0.0) and ARIMA (2.1.0) models, the error forecasting of the significant FFBP (12.2.1) model is less in observation number 161(3.542 kW·h/m

^{2}) compared with ARIMA (2.1.0) and ARMA (10.0.0). It can be concluded that the FFBP (12.2.1) model performed better than the ARMA (10.0.0) and ARIMA (2.1.0) models.

^{2}, SBF, LCE, and WIA, the presented value of the hybrid model is 0.9890, 0.9148, 0.9580, and 0.9910, respectively. The R

^{2}value is close to one, which indicates the good agreement between forecasted hybrid ARMA-FFBP and measured data. The other statistical metric of the hybrid ARMA-FFBP shows the lowest values compared with the three previous models. The error forecasting of the proposed hybrid ARMA-FFBP model shows significant and lower values than other models. Among the three previous models, it was seen that the hybrid ARMA-FFBP is the most suitable model to forecast the daily GSR compared with ARMA (10.0.0), ARIMA (2.1.0), and FFBP (12.2.1) models.

^{2}, the shown hybrid ARIMA-FFBP is approximately higher by about 0.41% on the hybrid ARMA- FFBP model, 0. 53% on the FFBP (12.2.1) model, 4.59% on the ARMA (10.0.0) model and around 3.03% on ARIMA (2.1.0) model. The computed MBE (%), RMSE (%) Sd (%), SBF, LCE, WIA, BIC, and AIC showed a very close forecasting success compared with FFBP (12.2.1) and hybrid ARMA-FFBP models. The Hybrid ARMA-FFBP model is close to the hybrid ARIMA-FFBP model, particularly in R

^{2}(%). In term of error forecasting, the proposed hybrid ARIMA-FFBP can be recognized as “the very most suitable model forecasting”, which present the lowest value of statistical performance and the highest values of R

^{2}, SPE, LCE, and WIA. Likewise, particularly in the previous observation number 161, the presented error forecasting of the hybrid ARIMA-FFBP was seen to be very low.

^{2}, SPE, LCE, and WIA for the Tangier site. The range value of the selected model is between 0.8074 and 0.9601. Compared with the Tetouan site, the ARMA (16.0.0) model performed better than the ARMA (10.0.0) model.

^{2}), 88 (3.638 kW·h/m

^{2}), 89 (2.95 kW·h/m

^{2}) 128 (5.368 kW·h/m

^{2}), 131 (3.882 kW·h/m

^{2}), and 360 (0.6491 kW·h/m

^{2}). In terms of MBE, the selected model is the only one that has the lowest value (0.0042 kW·h/m

^{2}) compared to the other models. It can be concluded that the ARIMA (2.2.0) exceeds the ARMA (16.0.0) model and present a significant agreement between the forecasted daily GSR and measured data.

^{2}, the selected model rank third after the combined models. The lowest value of the FFBP (12.2.1) is by about 0.03092 kW·h/m

^{2}for MBE, 0.0517% for MBE (%), and 0.79265 (%) for Sd (%) respectively. The highest value of WIA is about 0.9891 and is nearly close to 1. As a result, the selected FFBP (12.2.1) exceeds the ARIMA (2.2.0) and ARMA (16.0.0) models and illustrates a successful forecast of the daily GSR.

^{2}, Sd (%), and Sd. Unlike the ARMA (16.0.0) and ARIMA (2.2.0) models, the error forecasting of the significant hybrid ARMA-FFBP model is less in observation numbers 73 (1.98 kW·h/m

^{2}), 88 (3.638 kW·h/m

^{2}), 89 (2.95 kW·h/m

^{2}) 128 (5.368 kW·h/m

^{2}), 131 (3.882 kW·h/m

^{2}), and 360 (0.6491 kW·h/m

^{2}).

^{2}, SPE, LCE, and WIA. In terms of R

^{2}, the shown hybrid ARIMA-FFBP is approximately higher by about 1.57% on the ARMA (16.0.0) model, 3% on the ARIMA (2.2.0) model, 0.67% on the FFBP (12.2.1) model, and around 0.13% on the hybrid ARMA-FFBP (2.1.0) model. All computed statistical performance metrics showed a very close forecasting success compared with the FFBP (12.2.1) and hybrid ARMA-FFBP models. In terms of error forecasting, the proposed hybrid ARIMA-FFBP can be recognized as “the very most suitable model forecasting”. Likewise, in the previous observation number, the presented error forecasting of the hybrid ARIMA-FFBP was seen to be very low.

^{2}), the root means square error (RMSE), and the standard deviation (Sd) in a polar (two-dimensional) diagram. The main objective of this illustration is to closely inspect the forecasted results and the measured data on a particular day. Figure 11 and Figure 12 compare the performance of the most appropriate inputs, and graphs based on the statistical error metric. The figures illustrate the accuracies of the 16 relevant models, which have relatively lower errors in terms of standard deviation (Sd) and RMSE (value between 0.1 and 0.8 kWh/m

^{2}). In addition, the, highest value of R

^{2}(99.31%) presents the accuracy relationship between the measured and predicted values.

^{2}, which is varying between 16.6421~15.6709 and 0.9472~0.9628, respectively. This model is less than the forecasted FFBP (12.2.1) model and revealed the optimum RMSE (0.5119%) and Sd (9.9852%). The forecasted combined hybrid ARMA-FFBP and hybrid-FFBP models resulted in the lowest value of RMSE than the simple models.

^{2}and the lowest value in terms of RMSE compared with previous models. In this case study, the Taylor correlation increased by about 10% to 15% compared with the Tetouan site.

^{2}) values of the best model for the Tetouan and Tangier sites is close to 1, which explains the good relationship between the forecasted and measured data.

^{2}) of the forecasted ARIMA, ARMA, FFBP, and hybrid models varies between 0.9472% and 0.9931% depending on the study location and the trained methods. The range value of the slope of the best-fit line (SPE) varies between 0.8435 and 0.9296. The range value of the legate’s coefficient of efficiency (LCE) is 0.8954, 0.9696 and the range value of Willmott’s index of agreement (WIA) is 0.9491 and 0.9945. These results show that the hybrid ARIMA-FFBP is more reliable in the forecasting of the daily GSR for the Tetouan and Tangier sites. In this context, the obtained performance will be compared and discussed by considering Table 6 as the reference. Further, the results obtained from the hybrid ARIMA-FFBP model compared with single and combined models have exposed the highest correlation coefficient of 0.9901% for Tetouan city and 0.9831% for Tangier city. In addition, the values of MBE (%), RMSE (%), Sd (%), Akaike information criterion (AIC), and Bayesian information criterion (BIC) for both cities are 0.0297 (%), 0.02101 (%), 9.6917 (%), 9.06742 (%), 8.67911 (%), 6.87613 (%), 792.8625, 765.091 and 756.3418, 504.816, respectively. Eventually, the results have defined that the hybrid ARIMA-FFBP model is more accurate and suitable compared with the other methods to predict the daily global solar radiation for any location with the same weather conditions.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**The architecture of the proposed hybrid models is the Auto-Regressive Integrated Moving Average-Feed Forward back-propagation algorithm (ARIMA-FFBP) and Auto Regressive Moving Average-Feed Forward back-propagation algorithm (ARMA-FFBP).

**Figure 8.**(

**A**,

**C**) autocorrelation function (ACF) and (

**B**,

**D**) partial autocorrelation function (PACF) of the daily GSR from two cities.

**Figure 9.**(

**A**–

**E**) Comparison between predicted and measured values of daily global solar radiation from ARMA, FFBP, ARIMA, and hybrid models.

**Figure 10.**(

**A**–

**E**) Comparison between predicted and measured values of daily global solar radiation from ARMA, FFBP, ARIMA, and hybrid models.

**Figure 11.**Taylor diagram of several (

**A**) ARMA, (

**B**) ARIMA, (

**C**) FFBP, and (

**D**,

**E**) hybrid models of Tetouan city.

**Figure 12.**Taylor diagram of several (

**A**) ARMA, (

**B**) ARIMA, (

**C**) FFBP, and (

**D**,

**E**) hybrid models of Tangier city.

**Figure 13.**(

**A**,

**B**) Correlation coefficient (R

^{2}) of daily GSR vs. measured data of best five models (regression plot).

References | Simple and Combined Modeling for Short-Term and Long-Term Prediction of Solar Radiation |
---|---|

[28] | Seasonal ARIMA (0, 1, 2) (1, 0, 1) 30 was found to be a suitable model for predicting daily solar radiation at Reese Research Centre of Lubbock, Texas |

[29] | ARIMA (1, 0, 0) was found reasonable in capturing the autocorrelative structures of the daily average of solar irradiance in Awali, Kingdom of Bahrain. |

[30] | Non-seasonal ARIMA (2, 1, 3) was trained to predict day-ahead hourly global horizontal irradiance (GHI) in Abu Dhabi. |

[8] | Hybrid ARIMA-backed propagation does not outperform ARIMA for hourly solar irradiance from National Solar Radiation Database (NRSDB) site from 2008 to 2009. |

[31] | ARMA (2, 0) and ARMA (4, 0) were identified as appropriate models combined with ANN for the prediction of daily global solar radiation. |

[32] | ARIMA (2, 1, 1) was developed for the prediction of the daily clearness index In Abu Dhabi. |

[33] | Employed ARMA, which revealed that the residuals were best estimated by non-seasonal ARMA (2, 0) for daily solar radiation data over four locations in Malaysia. |

[34] | Employed ANN-BP neural network and multilayered feed-forward neural network |

Cities | TAO (KWh/m^{2}/Day) | Kt | T_{mean} (°C) | T_{max} (°C) | T_{min} (°C) | $\mathbf{\Delta}\mathit{T}(\xb0\mathbf{C})$ | T_{ratio} (°C) | ${\mathit{R}}_{\mathit{h}}(\%)$ | ${\mathit{W}}_{\mathit{s}}(\mathbf{m}/\mathbf{s})$ | ${\mathit{D}}_{\mathit{l}\mathit{e}\mathit{n}\mathit{g}\mathit{t}\mathit{h}}$ | Longitude (Degree) | Latitude (Degree) | Altitude (Degree) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Tangier | 5.92 | 0.681 | 17.429 | 21.8 | 13.3 | 8.6 | 1.5878 | 73.542 | 4.708 | 12.338 | −5.9 | 35.733 | 21 |

Tetouan | 4.909 | 0.653 | 18.671 | 22.4 | 15.5 | 7.1 | 1.423 | 70.08 | 4.263 | 12.306 | −5.33 | 35.58 | 10 |

**Table 3.**Estimation of ARMA model parameters for daily global solar radiation prediction for both cities.

Cities | ARMA Models | Parameters | Estimation | Standard Error | TS Statistic | p-Value |
---|---|---|---|---|---|---|

Tetouan | ARMA (10 0 0) | AR{1} | 0.39838 | 0.044196 | 9.0139 | 1.989410^{−19} |

AR{2} | −0.14705 | 0.054537 | −2.6963 | 0.0070107 | ||

AR{3} | 0.10918 | 0.054265 | 2.012 | 0.044216 | ||

AR{4} | 0.034376 | 0.059844 | 0.57443 | 0.56568 | ||

AR{5} | 0.10994 | 0.059868 | 1.8364 | 0.066304 | ||

AR{6} | 0.096692 | 0.056285 | 1.7179 | 0.085814 | ||

AR{7} | 0.16186 | 0.055111 | 2.9371 | 0.0033132 | ||

AR{8} | 0.053943 | 0.049207 | 1.0962 | 0.27298 | ||

AR{9} | 0.098247 | 0.054629 | 1.7985 | 0.072105 | ||

AR{10} | 0.050805 | 0.051959 | 0.97779 | 0.32818 | ||

Tangier | ARMA (16 0 0) | AR{1} | 0.38961 | 0.043978 | 8.8593 | 8.053110^{−19} |

AR{2} | 0.06638 | 0.057461 | 1.1552 | 0.248 | ||

AR{3} | 0.23474 | 0.053071 | 4.4231 | 9.73110^{−6} | ||

AR{4} | −0.0063764 | 0.062151 | −0.1026 | 0.91828 | ||

AR{5} | 0.061862 | 0.061564 | 1.0048 | 0.31498 | ||

AR{6} | 0.083309 | 0.054007 | 1.5426 | 0.12293 | ||

AR{7} | −0.00041644 | 0.06213 | −0.006702 | 0.99465 | ||

AR{8} | 0.034286 | 0.066352 | 0.51674 | 0.60534 | ||

AR{9} | 0.0060834 | 0.055003 | 0.1106 | 0.91193 | ||

AR{10} | −0.05637 | 0.054918 | −1.0264 | 0.30469 | ||

AR{11} | −0.046987 | 0.059012 | −0.79623 | 0.4259 | ||

AR{12} | 0.10782 | 0.047812 | 2.255 | 0.024135 | ||

AR{13} | −0.04943 | 0.049427 | −1.0001 | 0.31728 | ||

AR{14} | 0.0078379 | 0.052373 | 0.14966 | 0.88104 | ||

AR{15} | −0.0036433 | 0.054602 | −0.066725 | 0.9468 | ||

AR{16} | 0.16003 | 0.048865 | 3.2749 | 0.0010569 |

**Table 4.**Estimation of ARIMA model parameters for the prediction of daily global solar radiation of the two cities (Tetouan and Tangier).

Cities | ARMA Models | Parameters | Estimation | Standard Error | TS Statistic | p-Value |
---|---|---|---|---|---|---|

Tetouan | ARIMA (2.1.0) | AR{1} | −0.03912 | 0.009215 | −4.2452 | 0.21838 |

AR{2} | −0.15594 | 0.012313 | −12.6654 | 0.92005 | ||

Tangier | ARIMA (2.2.0) | AR{1} | −0.58945 | 0.009849 | −59.8438 | 0.16258 |

AR{2} | −0.33481 | 0.010903 | −30.7077 | 0.44859 |

Cities | Measured Data | FFBP Architecture | Coefficient of Variation (CV) | RMSE (%) |
---|---|---|---|---|

Tetouan | ${K}_{t}$ | FFBP (1 × 2 × 1) | 0.575 | 0.5957 |

${K}_{t},TOA$ | FFBP (2 × 2 × 1) | 0.571 | 0.5119 | |

${K}_{t},TOA,{T}_{max}$ | FFBP (3 × 2 × 1) | 0.562 | 0.5045 | |

${K}_{t},TOA,{T}_{max},{T}_{ratio}$ | FFBP (4 × 2 × 1) | 0.555 | 0.5002 | |

${K}_{t},TOA,{T}_{max},{T}_{ratio},\Delta T$ | FFBP (5 × 2 × 1) | 0.526 | 0.5002 | |

${K}_{t},TOA,{T}_{max},{T}_{ratio},\Delta T,{T}_{Average}$ | FFBP (6 × 2 × 1) | 0.519 | 0.4975 | |

${K}_{t},TOA,{T}_{max},{T}_{ratio},\Delta T,{T}_{Average},{T}_{min}$ | FFBP (7 × 2 × 1) | 0.492 | 0.4966 | |

${K}_{t},TOA,{T}_{max},{T}_{ratio},\Delta T,{T}_{Average},{T}_{min},Long$ | FFBP (8 × 2 × 1) | 0.473 | 0.4935 | |

${K}_{t},TOA,{T}_{max},{T}_{ratio},\Delta T,{T}_{Average},{T}_{min},Long,Alt$ | FFBP (9 × 2 × 1) | 0.457 | 0.4928 | |

${K}_{t},TOA,{T}_{max},{T}_{ratio},\Delta T,{T}_{Average},{T}_{min},Long,Alt,Lat$ | FFBP (10 × 2 × 1) | 0.440 | 0.4915 | |

${K}_{t},TOA,{T}_{max},{T}_{ratio},\Delta T,{T}_{Average},{T}_{min},Long,Alt,Lat,\delta $ | FFBP (11 × 2 × 1) | 0.435 | 0.4901 | |

${K}_{t},TOA,{T}_{max},{T}_{ratio},\Delta T,{T}_{Average},{T}_{min},Alt,Long,Lat,\delta ,{D}_{length}$ | FFBP (12 × 2 × 1) | 0.426 | 0.489 | |

Tangier | ${K}_{t}$ | FFBP (1 × 2 × 1) | 0.467 | 0.5119 |

${K}_{t},TOA$ | FFBP (2 × 2 × 1) | 0.453 | 0.5045 | |

${K}_{t},TOA,{T}_{max}$ | FFBP (3 × 2 × 1) | 0.448 | 0.5002 | |

${K}_{t},TOA,{T}_{max},{T}_{ratio}$ | FFBP (4 × 2 × 1) | 0.442 | 0.5002 | |

${K}_{t},TOA,{T}_{max},{T}_{ratio},\Delta T$ | FFBP (5 × 2 × 1) | 0.434 | 0.4975 | |

${K}_{t},TOA,{T}_{max},{T}_{ratio},\Delta T,{T}_{Average}$ | FFBP (6 × 2 × 1) | 0.426 | 0.4966 | |

${K}_{t},TOA,{T}_{max},{T}_{ratio},\Delta T,{T}_{Average},{T}_{min}$ | FFBP (7 × 2 × 1) | 0.426 | 0.4957 | |

${K}_{t},TOA,{T}_{max},{T}_{ratio},\Delta T,{T}_{Average},{T}_{min},Long$ | FFBP (8 × 2 × 1) | 0.419 | 0.4935 | |

${K}_{t},TOA,{T}_{max},{T}_{ratio},\Delta T,{T}_{Average},{T}_{min},Long,Alt$ | FFBP (9 × 2 × 1) | 0.410 | 0.4928 | |

${K}_{t},TOA,{T}_{max},{T}_{ratio},\Delta T,{T}_{Average},{T}_{min},Long,Alt,Lat$ | FFBP (10 × 2 × 1) | 0.409 | 0.4895 | |

${K}_{t},TOA,{T}_{max},{T}_{ratio},\Delta T,{T}_{Average},{T}_{min},Long,Alt,Lat,\delta $ | FFBP (11 × 2 × 1) | 0.399 | 0.4395 | |

${K}_{t},TOA,{T}_{max},{T}_{ratio},\Delta T,{T}_{Average},{T}_{min},Alt,Long,Lat,\delta ,{D}_{length}$ | FFBP (12 × 2 × 1) | 0.382 | 0.406 |

Cities | Models | MBE | MBE (%) | RMSE | RMSE (%) | Sd | Sd (%) | R^{2} | SBF | LCE | WIA | BIC | AIC |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Tetouan | ARIMA (2, 1, 0) | 0.0817 | 0.0839 | 0.80540 | 16.6421 | 0.7554 | 12.6704 | 0.9628 | 0.8998 | 0.9253 | 0.9491 | 1038.213 | 991.7475 |

ARMA (10, 0, 0) | 0.1665 | 0.1098 | 1.0671 | 20.1083 | 0.9642 | 15.6709 | 0.9472 | 0.8915 | 0.9169 | 0.9689 | 1298.657 | 1051.867 | |

FFBP (12, 2, 1) | 0.0529 | 0.0364 | 0.5119 | 10.0253 | 0.5098 | 9.98521 | 0.9878 | 0.9098 | 0.9498 | 0.9887 | 991.3442 | 890.6528 | |

Hybrid ARMA–FFBP | 0.0376 | 0.0301 | 0.4871 | 9.98512 | 0.5001 | 9.10862 | 0.9890 | 0.9148 | 0.9580 | 0.9910 | 862.0175 | 810.6171 | |

Hybrid ARIMA–FFBP | 0.0298 | 0.0297 | 0.4091 | 9.6917 | 0.4678 | 8.67911 | 0.9931 | 0.9163 | 0.9641 | 0.9945 | 792.8625 | 756.3418 | |

Tangier | ARIMA (2, 2, 0) | 0.0042 | 0.06301 | 0.606335 | 17.41963 | 0.90689 | 12.42982 | 0.9744 | 0.8435 | 0.8954 | 0.9686 | 857.8941 | 788.5028 |

ARMA (16, 0, 0) | 0.0709 | 0.10072 | 0.89561 | 23.0964 | 0.99875 | 16.6418 | 0.9601 | 0.8074 | 0.8638 | 0.9487 | 1096.819 | 976.183 | |

FFBP (12, 2, 1) | 0.0517 | 0.03092 | 0.47834 | 10.3863 | 0.79265 | 7.40331 | 0.9834 | 0.8671 | 0.9145 | 0.9891 | 835.265 | 645.765 | |

Hybrid ARMA–FFBP | 0.0401 | 0.02564 | 0.39876 | 9.68745 | 0.71563 | 7.01577 | 0.9888 | 0.9188 | 0.9615 | 0.9981 | 803.465 | 598.615 | |

Hybrid ARIMA–FFBP | 0.0222 | 0.02101 | 0.30762 | 9.06742 | 0.69426 | 6.87613 | 0.9901 | 0.9296 | 0.9696 | 0.9939 | 765.091 | 504.816 |

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## Share and Cite

**MDPI and ACS Style**

Belmahdi, B.; Louzazni, M.; Marzband, M.; El Bouardi, A.
Global Solar Radiation Forecasting Based on Hybrid Model with Combinations of Meteorological Parameters: Morocco Case Study. *Forecasting* **2023**, *5*, 172-195.
https://doi.org/10.3390/forecast5010009

**AMA Style**

Belmahdi B, Louzazni M, Marzband M, El Bouardi A.
Global Solar Radiation Forecasting Based on Hybrid Model with Combinations of Meteorological Parameters: Morocco Case Study. *Forecasting*. 2023; 5(1):172-195.
https://doi.org/10.3390/forecast5010009

**Chicago/Turabian Style**

Belmahdi, Brahim, Mohamed Louzazni, Mousa Marzband, and Abdelmajid El Bouardi.
2023. "Global Solar Radiation Forecasting Based on Hybrid Model with Combinations of Meteorological Parameters: Morocco Case Study" *Forecasting* 5, no. 1: 172-195.
https://doi.org/10.3390/forecast5010009