The Sun is the Earth’s main energy source. This star emits electromagnetic radiation of different frequency and wavelength in the electromagnetic spectrum. Solar radiation in the atmosphere has a wavelength between 150 nm and 4.000 nm, where 7% corresponds to ultraviolet radiation, 47% to visible radiation, and 46% to infrared radiation (Eraso-Checa et al., 2017; Würfel, 2009). However, the wavelength that reaches the Earth’s surface is attenuated to a range between 380 and 780 nm (Narváez & Hernández, 2013) due to absorption, reflection, and scattering phenomena. This also means that the global radiation on the surface is composed of beam, diffuse, and reflected radiation (Jäguer et al., 2006). Another parameter that attenuates solar radiation is the optical air mass, which is the path length that sunlight follows through the atmosphere (Jäguer et al., 2014).

Figure 1 shows the power received per unit surface exposed to radiation for each wavelength at the outer side of the Earth’s atmosphere. It shows that the visible part of the spectrum has the large area between 400 nm and 700 nm, with a spectral irradiance peak around 2.000 W/m2.nm. Ultraviolet (UV) light occupies the high energetic part of the spectrum.

Figure 1 Source: Würfel (2009)

The integration of spectral irradiance over wavelength corresponds to the irradiance, which is a power per unit area (W/m² or joules/m²sec). This is a magnitude scale that includes the complete wavelength information. Radiation measuring equipment gives the solar radiation response as irradiance (W/m²).

UV radiation is just a part of the solar spectrum at high frequencies (>10¹⁶ Hz). This means that it is very energetic and can ionize atoms by electrically charging them (Casal, 2010). Photon energy ranges from 3,2 eV up to 1,2x10³ eV (Bohorquez-Ballén & Pérez-Mogollón, 2007). UV radiation allows human beings to assimilate vitamin D and, in plants, it makes photosynthesis possible. However, it also has negative effects if there is prolonged exposure, especially in human health (it breaks biological molecules, damages the eyes and skin, it causes cancer, etc.) (Bohorquez-Ballén & Pérez-Mogollón, 2007).

UV radiation is divided into three regions (Ryer, 1988):

UV-A: (315-400 nm): It is the least harmful to human beings, and its intensity reaches the terrestrial surface.

According to Lucas et al. (2006), ambient UV radiation can be measured using a representation of the wavelength variation in the production of skin erythema. The UV index is an instance of this representation (the other are SED and MED). It is defined as the time-weighted average effective UV irradiance multiplied by 40, and it is expressed as power per unit area (W/m²).

Figure 2 shows the behavior of both spectral irradiance and the erythemal action spectrum in the UV region.

Figure 2 Source: National Services Centre (n.d.)

The UV index is categorized as shown in Table 1, and the values range from 0 on. The higher the index value, the greater potential damage to the skin in the less time.

Table 1

UV Index	Danger Category
0-2	Low
3-5	Moderate
6-7	High
8-10	Very High
11+	Extreme

Source: based on National Services Centre (n.d.)

To develop and validate the estimation model, a methodological process was carried out, which was adapted from Eraso-Checa et al. (2018). In this case, the methodology used is composed of a typical process that increases estimation reliability. In Figure 3, the methodological process is shown.

Figure 3 Source: Authors

A set of climatic variables (UV index, temperature, humidity, and solar radiation) was extracted from the Davis Vantage PRO station. In this case, the dataset is made up of approximately 305.000 data for each variable from 2013 to 2018. Since the first data are raw, it was necessary to remove spurious and erroneous data using an inspection algorithm. This process removes not-a-number data, data equal to infinity, and unusual data such as out-of-range values that are physically impossible to reach. Finally, a training dataset of 213.019 data and a validation dataset of 91.696 were obtained.

After that, solar radiation was estimated with following steps: first, the estimation model based on neural networks (NN) was defined, considering estimation structures and inputs; secondly, the structure was trained using the dataset; then, there was an evaluation process using the validation dataset, which calculated some predefined metrics such as mean absolute error (MAE), root mean square error (RMSE), and the coefficient of determination (R²).

Different nonlinear models based on ANNs were used to estimate solar radiation. Mainly, two different ANN structures were used to model the estimators: Neural Network Finite Impulse Response (NNFIR) structure and Neural Network Autoregressive with Exogenous Input (NNARX).

NNFIR is a simple model estimator fed with external excitations to make a prediction. Figure 4 shows, on the left side, a NNFIR that does not have feedback. NNARX is a recurrent neural network. On the right side of Figure 4, the network uses external excitations and past instants to make a prediction.

Figure 4 Source: Authors

As shown in Figure 4, the NNFIR and NNARX models are fed with past instants of one or more inputs (u₁(⋅), … , u_j(⋅)),, and past instants of the estimated output (ŷ(⋅)) in case of the NNARX structure. For more details on NNARX and NNFIR structures, see their description in the referenced literature (Norgaard et al. 2000).

In addition, this paper proposes to use the temperature as part of the estimator, aiming to compensate the lack of information of the reduced band in the UV index sensor. In that way, it is possible to quantify the contribution of the temperature in the estimation of solar radiation. To differentiate the models, models that use UV index and/or estimated radiation are denoted with a at the end, and the estimation models that use temperature in addition to the UV index and/or estimated radiation are denoted with 𝑏 at the end. The definitions of these models are shown below, and they are summarized in Table 2.

Table 2

Model	Mathematical definition
NNFIRa	𝑅(𝑘) = 𝑓(𝑈𝑉(𝑘))
NNFIRb	𝑅(𝑘)= 𝑓(𝑈𝑉(𝑘), 𝑇(𝑘))
NNARX111a	𝑅(𝑘)= 𝑓(𝑅(𝑘−1),𝑈𝑉(𝑘), 𝑈𝑉(𝑘−1))
NNARX111b	𝑅(𝑘)= 𝑓(𝑅(𝑘−1), 𝑈𝑉(𝑘), 𝑈𝑉(𝑘−1),𝑇(𝑘), 𝑇(𝑘−1))
NNARX211a	𝑅(𝑘)= 𝑓(𝑅(𝑘−1), 𝑅(𝑘−2), 𝑈𝑉(𝑘), 𝑈𝑉(𝑘−1))
NNARX211b	𝑅(𝑘)= 𝑓(𝑅(𝑘−1), 𝑅(𝑘−2), 𝑈𝑉(𝑘), 𝑈𝑉(𝑘−1), 𝑇(𝑘), 𝑇(𝑘−1))
NNARX221a	𝑅(𝑘)= 𝑓(𝑅(𝑘−1), 𝑅(𝑘−2), 𝑈𝑉(𝑘), 𝑈𝑉(𝑘−1), 𝑈𝑉(𝑘−2))
NNARX221b	𝑅(𝑘)= 𝑓(𝑅(𝑘−1), 𝑅(𝑘−2), 𝑈𝑉(𝑘), 𝑈𝑉(𝑘−1), 𝑈𝑉(𝑘−2), 𝑇(𝑘), ??(𝑘−1), 𝑇(𝑘−2))

Source: Authors

𝑅(⋅) is the global solar radiation, 𝑈𝑉(⋅) is the UV index, 𝑇(⋅) is the ambient temperature, and k is the time instant. NNFIR and NNARX structures are configured as follows: one hidden layer with 20 neurons, sigmoidal activation function, and Bayesian Regularization as the training algorithm.

Solar radiation, the UV index, temperature, and humidity data were collected from November 2013 to December 2018 using the DAVIS Vantage Pro 2.0 meteorological station. This equipment uses the 6450 solar radiation sensor (Davis Instruments, 2014a) and the 6490 UV sensor that measures the global solar UV index (Davis Instruments, 2014b).

After applying the methodological process to 305.172 data, a training dataset of 213.019 elements and a validation set of 91.696 data for each variable (30% of total data) were obtained. There were 457 spurious and erroneous data, as shown in Figure 5.

Figure 5 Source: Authors

The NN MATLAB toolbox was used to train the eight proposed estimation models. Each model was trained using its corresponding dataset (inputs, outputs, and validation dataset).

The following Figures show the solar radiation estimation results using a random day from the validation data set.

As shown in Figure 6, the radiation estimation with the NNFIRa and NNFIRb structures was able to follow the general behavior of the measured radiation. However, the error increases due to instant radiation changes. These models have an estimation error of and , respectively.

Figure 6 Source: Authors

The NNARX111a and NNARX111b structures improve the estimation of solar radiation reducing the error to and , respectively (Figure 7). This optimization occurs because these structures make a prediction using past and present data. Therefore, they have a measurement radiation rate that is used to predict the next radiation value.

Figure 7 Source: Authors

Figures 8 and 9 show the NNARX211a, NNARX211b, NNARX221a, and NNARX221b structures. These kind of structures reduce the error between and . This is posssible because the network makes a prediction from two past instants. In the same way, predictions with structures that include more than two past instants were developed. Nevertheless, the error does not reduce its value significantly.

Figure 8 Source: Authors

Figure 9 Source: Authors

From Figures 7-9, it can be noticed that the differences between type a and b models are not significant. Despite this, type b models use other variables (temperature, humidity) to complement the estimation in contrast with type a models, which only use the input variable and the past instants. These additional variables do not significantly reduce the error. In contrast, the error increases in some cases.

According to the numerical results presented in Table 3, considering the validation data set (91.696 elements), type α models have a similar behavior to type b ones. Type b models have a smaller RMSE and a bigger R² coeficient in comparison with type α models. In contrast, type α models have a smaller MAE than type b models. However, the differences between models α and b are not significant.

Table 3

Structure/Metric	RMSE	MAE	R2
NNFIRa	87,855	42,526	0,92294
NNFIRb	86,863	41,301	0,92489
NNARX111a	55,835	18,624	0,96805
NNARX111b	55,532	18,643	0,96838
NNARX211a	55,066	18,314	0,96895
NNARX211b	54,809	18,347	0,96925
NNARX221a	54,710	17,905	0,96927
NNARX221b	54,321	18,069	0,96971

Source: Authors

Also, NNARX221a and NNARX221b have the best performance in terms of RMSE, MAE, and R² compared to other estimation models.

As seen in the Figure 10 and Table 3, the differences between estimation structures NNARX221a and NNARX221b are not significant. Hence, the contribution of the auxiliary variables (temperature and humidity) in the estimation of solar radiation is not relevant. In this sense, the NNARX221a model has the best performance (based on R²) only using past and present instants of solar radiation and the UV index.

Figure 10 Source: Authors