# Modelling and Prediction of Monthly Global Irradiation Using Different Prediction Models

^{1}

^{2}

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^{*}

## Abstract

**:**

^{2}∙day) and 1136 kJ/(m

^{2}∙day), respectively, and predict conveniently for independent stations, 2013 kJ/(m

^{2}∙day) and 2094 kJ/(m

^{2}∙day), respectively. Given the good results obtained, it is convenient to continue with the design of artificial neural networks applied to the analysis of monthly global irradiation.

## 1. Introduction

- Artificial neural networks can be used in food science to model and optimize the extraction of cashew apple juice [19], to optimize an enzymatic approach to obtain modified artichoke pectin and pectic oligosaccharides [20] or to determine the broccoli buds loss green color velocity using hyperspectral camera combined with artificial neural networks [21].

## 2. Related Works

- Multiple linear regression models can be used to predict the net radiation using meteorological data such as global solar radiation, temperature, relative humidity, etc. [29]. The researchers developed 8 different equations to estimate the daily net radiation and the results showed good adjustments and low errors on a daily scale, especially in the models that include the variables of relative humidity of the air, temperature, solar radiation and the inverse of the distance between the earth and the sun. Despite the simplicity of the multiple linear regression models, the authors showed good adjustments compared to the Rn FAO 56 OM model, which allows to conclude that the MLR models developed are an alternative to improve the evapotranspiration estimation.
- According to Diez et al. [1], artificial neural networks have been used to predict the solar irradiation at different time windows (hourly, daily and monthly) from different meteorological variables (temperature, atmospheric pressure, among others) or even including geographical coordinates such as latitude, longitude and altitude. In this sense, these authors developed ANNs to predict the global solar irradiation of the day after using data from one agrometeorological station located in Mansilla Mayor (León, Castilla y León). The authors concluded that artificial neural networks models provide better results compared to classical methods and require less input variables [1]. This kind of models can be used to determine the average monthly, the average weekly and the daily global solar radiation in Fortaleza (Brazilian Northeast region) using 14-year-long data set to train three different ANNs models [3]. ANNs can also be used to determine different parameters such as the global horizontal irradiation (from meteorological data), the global tilted irradiation (from the horizontal global irradiation and others) and to forecast the hourly direct normal and the global horizontal irradiation from one to six hours horizon [6].
- Support vector machines models can be used to generate the daily global solar irradiation using a general (non-locally dependent) model [9]. The model (which used temperatures, wind speed, relative humidity and rainfall, among other variables) presented a high capacity of generalization for the different studied locations and improved, in terms of mean absolute error, the locally trained models in some locations [9]. SVM models can even be used to forecast photovoltaic power (and be compared with other models) [30].
- Random forest can be used to estimate the solar radiation using air pollution index in three different sites [31] or to forecast solar radiation and compared their result with other methods such as multivariate adaptive regression splines (MARS), classification and regression tree (CART) and M5 [32].

- MLR and ANN models can be compared in the estimation of monthly-average daily solar radiation over different locations in Turkey [33]. Different variables (latitude, longitude, altitude, land surface temperature and month) were used as input variables. According to the authors, the results showed that the ANN model could obtain good performance compared to the multiple linear regression model.
- SVM and ANN models were used in a comparative study of different methods carried out by da Silva et al. [34] to estimate the daily global solar irradiation. Four different kinds of architecture combining different input parameters were studied. According to the authors, statistical indicators showed that the SVM technique has better performance than ANN models for the study location (Botucatu/SP/Brazil). Neural models can be compared to random forest models (among other model) to forecast the normal beam, horizontal diffuse and global components [35]. SVM, ANN and deep neural network models can even be used to forecast photovoltaic power [30], to estimate electricity demand (using multiple linear regression, artificial neural network and support vector machine, among other) [12] or to estimate the surface downward longwave radiation (using ANN, SVR and RF, among others) [36]
- Random forest to model the daily variability of solar irradiance can be compared to other methods such as multiple linear regression, obtaining the best results between both [37].

## 3. Materials and Methods

#### 3.1. Study Area

^{2}∙day (3.8 kWh/m

^{2}∙day) and 15.1 MJ/m

^{2}·day (4.2 kWh/m

^{2}∙day) [38]. The selected meteorological stations were: (i) Amiudal in the municipality of Avión, (ii) Serra do Faro in Rodeiro, (iii) Monte Medo in the municipality of Baños de Molgas, (iv) Ourense-Estacións in the city of Ourense and (v) Pazo de Fontefiz in Coles (Figure 1). The meteorological stations were selected taking into account the conditions and the quantity of available data to create useful and accurate models for the prediction of MGI.

#### 3.2. Database

^{2}·day)) were: (i) latitude, (ii) longitude and (iii) altitude (m) of the station, (iv) month order, (v–vii) average, average of the maximum and average of the minimum temperatures (°C); (viii–xi) average, average of the maximum and average of the minimum relative humidities (%) and (xii) precipitation (L/m

^{2}).

#### 3.3. Implementation of Models

#### 3.4. MLR Models

_{0}the constant, β

_{1}–β

_{n}the regression coefficients, x

_{1}–x

_{n}the input variables and ε is the error.

#### 3.5. ANN Models

#### 3.6. SVM Models

^{−15}to 2

^{3}in 18 steps, with a logarithmic scale) and (iii) C (from 2

^{−5}to 2

^{15}in 20 steps, with a logarithmic scale).

#### 3.7. RF Models

#### 3.8. Statistics of the Developed Models

^{2}), the root mean square error (RMSE, Equation (2)) and the average absolute relative error (Error, Equation (3)). The best model was chosen according to the lowest RMSE in the validation phase:

#### 3.9. Equipment and Software Used

^{®}Core™ i7-8700 processor at 3.20 GHz, with 16 GB of RAM). All models were run on Windows 10 Pro 64-bit operating system. Data were collected and processed using the software Microsoft Excel 2016, from Microsoft Office Professional Plus 2016 package, (Microsoft, Albuquerque, NM, USA). MLR, ANN, SVM and RF models were developed using a trial/free version of RapidMiner Studio 9.0.993 software (RapidMiner, Inc., Boston, MA, USA). Figures were made with SigmaPlot v. 13.0 (Systat Software, Inc., San Jose, CA, USA).

## 4. Results and Discussion

#### 4.1. MLR Models

^{2}∙day) for the validation phase, which corresponds to a low r

^{2}(0.468). These bad adjustments for the validation phase are extensible to all phases of the model, training and querying. Thus, for these phases, the RMSE values are 5215 kJ/(m

^{2}∙day) and 6300 kJ/(m

^{2}∙day) which together with the low squared correlation, 0.426 and 0.343, make this model a model that cannot be used for modelling the MGI. The rest of the models present better adjustments than the previous model, with RMSE values for the validation phase, between 2924 kJ/(m

^{2}∙day) and 2411 kJ/(m

^{2}∙day). These models offer for the querying phase some RMSE similar to those provided for the training and/or the validation phase and an average absolute relative error between 18.1% and 19.6%. The best MLR model corresponds to a model with combination type 1 (Table 2), that is, an MLR that uses all the input available variables to model the behaviour of the MGI.

#### 4.2. ANN Models

^{2}∙day) which corresponds to an average absolute relative error of 12.1%. This value is close to the 10% that it is considered as, to our understanding, a good error percentage for this kind of modelling. Nevertheless, some authors suggest that prediction error less than 20% could be good accuracy in terms of solar radiation prediction [28]. The training and querying phase present similar adjustments to the validation phase with squared correlation coefficients of 0.943 and 0.953 for training and querying, respectively. These adjustments make the worst ANN an almost usable method to model the MGI, however, the other developed combination types improve the worst ANN model, presenting RMSE values between 1225 kJ/(m

^{2}∙day) and 1494 kJ/(m

^{2}∙day) for the validation phase. The best ANN model (Table 2) corresponds to a model with combination type 4 (input variables; latitude, longitude, altitude, month and the three humidities).

#### 4.3. SVM Models

^{2}∙day) which corresponds to a good r

^{2}of 0.956. These adjustments for the validation phase are extensible to the training and querying phase where the RMSE are 1525 kJ/(m

^{2}∙day) and 1743 kJ/(m

^{2}∙day) with high squared correlation values, 0.951 and 0.962 which make this SVM a model that could be used for modelling the MGI. The rest of the models present better adjustments being the RMSE value in the validation phase dropped to 1556 kJ/(m

^{2}∙day) for the second-best model. The best SVM model corresponds to a model with combination type 5, that is, an SVM that uses eight input variables to model the MGI response (Table 2).

#### 4.4. RF Models

^{2}∙day) with an average absolute relative error of 15.0%. During the training and the querying phase, the model presents very different adjustments, 925 kJ/(m

^{2}∙day) and 1651 kJ/(m

^{2}∙day), respectively. The other combination types slightly improve this model and a better model is obtained when configuration 5 is used (Table 2).

#### 4.5. Best Models Developed

^{2}∙day) and 2411 kJ/(m

^{2}∙day).

^{2}∙day) and the worst squared correlation coefficient (0.904). This model obtained an average absolute relative error around 19.2%. Regarding the training phase, the model presents lower RMSE value of 2263 kJ/(m

^{2}∙day) compared with the validation phase, nevertheless, the r

^{2}also present lower value (0.892).

^{2}∙day)) and the adjustments, in terms of squared correlation, was the lowest for the three phases (0.885).

^{2}∙day). This value is improved in the model’s training phase (948 kJ/(m

^{2}∙day)). In both phases, the RF model improves the MLR model, both in RMSE values and in its squared correlation values (0.982 and 0.962 vs. 0.892 and 0.904, for the training and validation phase, respectively). Besides this, the model presents good behaviour in terms of average absolute relative error.

^{2}∙day)) although the average absolute relative error remains at similar levels to those of the validation phase (10.7%).

^{2}∙day) compared to 1595 kJ/(m

^{2}∙day) for the RF model and the squared correlation values are the same (0.961 vs. 0962 for SVM and RF, respectively). The same happens with the error, which remains for both around 11%. For the training phase, a slight worsening of the fit for the SVM model reaching an RMSE of 1056 kJ/(m

^{2}∙day) is observed.

^{2}values remain close to the RMSE of the validation phase.

^{2}∙day) that corresponds with the highest squared correlation coefficient (0.975) for all validation phases. Regarding the training phase, the RMSE value is around 1271 kJ/(m

^{2}∙day) which supposes an absolute average relative error of 7.3%.

^{2}∙day)) and corresponds with a squared correlation of 0.980 (the highest for all the models in this phase).

#### 4.6. ANN Generalization to Different Locations

^{2}∙day)) that the error presented by the best-selected model (ANN); while for the Ourense-Estacións station, the error (8079 kJ/(m

^{2}∙day)) is almost four times greater than that presented by the best model (ANN). As expected, these high errors affect the average relative absolute error presented by each station, so the Pazo de Fontefiz station presents an error of 24.8%, being overcome by the error obtained in Ourense-Estacións, 47.2%. The SVM model presents good adjustments in terms of squared correlation (upper than 0.940); however, taking into account the adjustments of the root mean square error and the average absolute relative error it can be concluded that the SVM model is not a suitable model for the MGI modelling.

^{2}∙day) and 2334 kJ/(m

^{2}∙day) for the stations of Pazo de Fontefiz and Ourense-Estacións, respectively (Table 3). Compared to the SVM model, this model improves its adjustments in terms of RMSE and error, although not in terms of squared correlation. The errors of this model are around 19% for each station. According to this error level, we can say that the model shows good behaviour, but shows a higher error percentage than desired, especially for the Pazo de Fontefiz station.

^{2}for the Pazo de Fontefiz station), reporting errors in terms of RMSE, around 2013 kJ/(m

^{2}∙day) and 2094 kJ/(m

^{2}∙day) for the station of Pazo de Fontefiz and Ourense-Estacións, respectively. Likewise, for this model, the squared correlation is high (0.935 and 0.971) and the average absolute relative error remains lower than 15% error for each stations (which is considered as a good error percentage). These good adjustments are reflected in Figure 3. The first thing to note is the different size in the database between the two stations. The Pazo de Fontefiz station has data from July 2012 to August 2018 (a total of 73 months), while the Ourense-Estacións station has data from June 2014 to August 2018 (a total of 49 monthly measurements). Figure 3 shows the time series for the real MGI values (olive colour) and the values predicted by the ANN model.

^{2}∙day) to 26,050 kJ/(m

^{2}∙day)). The ANN predictions are shown in the Figure 3A as a black line. It can be seen how the modellings fit, almost perfectly, to the real-time series, which means (as we have already seen in the adjustments) that the ANN model can accurately predict the behaviour of the MGI for Pazo de Fontefiz station. It can be seen how for the low-value areas of MGI the prediction overestimates the values while the model behaves, in general, well for high-value areas of MGI (although also some underestimation is observed). Given the good adjustments, and taking into account the Figure 3A modelling time series, it can be said that this model is capable of generalizing the knowledge of the previous phase to other nearby geographical stations (Pazo de Fontefiz).

^{2}∙day) to 26,470 kJ/(m

^{2}∙day). It can be seen how the modelling fits the real-time series, however in this case the adjustments show a worse behaviour than in the case of the Pazo de Fontefiz station. Again, it can be seen how for the low areas of MGI the prediction is overestimated, in general, the MGI values (can even see how this behaviour is observed in some measurement in the maximum area) although for high MGI values the ANN model generally predict well. Taking into account the adjustments and the Figure 3B, it can be said that the ANN model is usable on other nearby geographical stations.

## 5. Conclusions

^{2}∙day) and 1136 kJ/(m

^{2}∙day), respectively, and predict conveniently for Coles and Ourense station 2013 kJ/(m

^{2}∙day) and 2094 kJ/(m

^{2}∙day), respectively. These good RMSE values are reinforced by the low percentage error obtained during the prediction phase at the two stations reserved for this purpose.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Sketch of the Iberian Peninsula with the approximate location of the stations used in this research. Derivative work from ME2000raster 2020 CC-BY 4.0 ign.es [39].

**Figure 2.**Graphical representation for the real and modelled values of MGI during the training phase (white dots), validation phase (black dots) and querying phase (turquoise triangles) for each select model: (

**A**) multiple linear regression -MLR-, (

**B**) random forest -RF-, (

**C**) support vector machine -SVM- and (

**D**) artificial neural network -ANN-. Redline is the line with slope one.

**Figure 3.**Real and modelled time series for (

**A**) Pazo de Fontefiz and (

**B**) Ourense-Estacións stations. The olive shade corresponds to the actual values, and the black line corresponds to the values modelled by the ANN model.

**Table 1.**Variables, and their combination, used to develop the different models: (i) latitude (Lat), (ii) longitude (Long), (iii) altitude (Alt), (iv) month, (v-vii) average (T

_{av}), average of the maximum (T

_{av-max}) and the average of the minimum temperature (T

_{av-min}); (viii-xi) average (RH

_{av}), an average of the maximum (RH

_{av-max}) and the average of the minimum (RH

_{av-min}) relative humidity and (xii) precipitation (P).

Combination Type | Lat | Long | Alt | Month | T_{av} | T_{av-max} | T_{av-min} | RH_{av} | RH_{av-max} | RH_{av-min} | P |
---|---|---|---|---|---|---|---|---|---|---|---|

Type 1 | |||||||||||

Type 2 | |||||||||||

Type 3 | |||||||||||

Type 4 | |||||||||||

Type 5 | |||||||||||

Type 6 | |||||||||||

Type 7 |

**Table 2.**Adjustment parameters for each best approximation model developed according to its selected input variables. Latitude (Lat), longitude (Long), altitude (Alt), month, average (T

_{av}), average of the maximum (T

_{av-max}) and the average of the minimum temperature (T

_{av-min}), average (RH

_{av}), the average of the maximum (RH

_{av-max}) and the average of the minimum (RH

_{av-min}) relative humidity and precipitation (P). RMSE is the root mean square error (10 kJ/(m

^{2}∙day)) and r

^{2}is the squared correlation coefficient.

T | V | Q | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Combination Type | Model | Lat | Long | Alt | Month | T_{av} | T_{av-max} | T_{av-min} | RH_{av} | RH_{av-max} | RH_{av-min} | P | RMSE | r^{2} | RMSE | r^{2} | RMSE | r^{2} |

Type 1 | MLR | 226.3 | 0.892 | 241.1 | 0.904 | 245.8 | 0.885 | |||||||||||

Type 4 | ANN | 127.1 | 0.967 | 122.6 | 0.975 | 113.6 | 0.980 | |||||||||||

Type 5 | SVM | 105.6 | 0.977 | 153.1 | 0.961 | 156.7 | 0.967 | |||||||||||

Type 5 | RF | 94.8 | 0.982 | 159.5 | 0.962 | 227.9 | 0.933 |

**Table 3.**Adjustment parameters for each of the best models applied to the stations of Pazo de Fontefiz and Ourense-Estacións. RMSE is the root mean square error (10 kJ/(m

^{2}∙day)), Error is the average absolute relative error (%) and r

^{2}is the squared correlation coefficient.

Q_{PF} | Q_{Ou} | |||||
---|---|---|---|---|---|---|

Model | RMSE | Error | r^{2} | RMSE | Error | r^{2} |

MLR | 285.2 | 19.5 | 0.865 | 233.4 | 18.1 | 0.915 |

ANN | 201.3 | 13.1 | 0.935 | 209.4 | 14.7 | 0.971 |

SVM | 402.9 | 24.8 | 0.949 | 807.9 | 47.2 | 0.971 |

RF | 246.1 | 21.2 | 0.920 | 216.5 | 19.6 | 0.950 |

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## Share and Cite

**MDPI and ACS Style**

Martinez-Castillo, C.; Astray, G.; Mejuto, J.C.
Modelling and Prediction of Monthly Global Irradiation Using Different Prediction Models. *Energies* **2021**, *14*, 2332.
https://doi.org/10.3390/en14082332

**AMA Style**

Martinez-Castillo C, Astray G, Mejuto JC.
Modelling and Prediction of Monthly Global Irradiation Using Different Prediction Models. *Energies*. 2021; 14(8):2332.
https://doi.org/10.3390/en14082332

**Chicago/Turabian Style**

Martinez-Castillo, Cecilia, Gonzalo Astray, and Juan Carlos Mejuto.
2021. "Modelling and Prediction of Monthly Global Irradiation Using Different Prediction Models" *Energies* 14, no. 8: 2332.
https://doi.org/10.3390/en14082332