2.1 Angstrom-Prescott model

The Angstrom-Prescot empirical model uses sunshine duration to estimate solar radiation using a first order linear correlation which is shown in equation (1); it defines the relationship between solar energy available at the ground level and the sunshine duration (^{Paulescu et al., 2016}).

Where: H: Daily solar irradiation on horizontal surface; H_o: Daily extraterrestrial irradiation on horizontal surface; n: number of hours measured by the sunshine recorder; N: maximum daily sunshine duration; a, b: empirical determined regression constants.

This equation establishes a relationship between the sunshine hours and the solar irradiation on a specific point on Earth; this is an advantage in zones where there is not solar irradiation or solar irradiance data as in the Norte de Santander region in Colombia. The parameters H_o and N can be calculated from the latitude and the day of the year, using the equations reported by ^{Almorox, Benito, and Hontoria (2005)}. The variables H and n are obtained from meteorological stations. The a and b constants can be calculated for a specific place by fitting calculated and measured datapoints of (H/Ho) and (n/N) to equation 1.

With solar estimation models, the calculated solar resource can be used to design photovoltaic (PV) systems; if the solar irradiation data in one geographical point is known, the solar irradiation of surrounding places can be easily calculated by using the constants a and b in equation (1). These constants can be found through the correlation between the solar irradiation and the sunshine hours in a place where solar radiation data is available. After that, the Angström-Prescott equation can be evaluated in another nearby location to estimate solar irradiation. Higher order empirical models have been proposed by ^{Akinoǧlu and Ecevit (1990)}, but the Angström-Prescott model is simpler and faster to calculate.

2.2 Implementation of the Angström-Prescott model in Norte de Santander

This is a quantitative study developed at Francisco de Paula Santander University in Cucuta, with the goal of obtaining information of the solar resource in the Department of Norte de Santander. The study was made using: R programming language, with the integrated development environment for R called RStudio; the XL-Connect package (^{Mirai Solutions GmbH, 2017}), to load xlsx files; the sfsmisc package (^{Maechler et al., 2016}), to calculate the area under the curve or the daily sun peak hours; and the zoo package (^{Zeileis & Grothendieck, 2005}), to calculate missing values.

All the data analyzed in this work was obtained from IDEAM meteorological stations. A few stations measure solar insolation, while most of the stations measure sunshine duration (also known as sunshine hours). In total, 24 stations that measure the duration of sunlight and 6 that measure solar insolation, were reviewed.

After gathering all the information, we applied two filters to organize and prepare the data. In the first filter, we discarded data from stations which had missing values during long periods of time (months or years); in the second filter, the datapoints with solar radiation values greater than the solar extraterrestrial constant (1366 W/m²) were excluded. After applying the filters described above to the databases of different stations, only the solar radiation stations presented in table 1, remained and were used in this work to obtain the regression coefficients.

Table 2 describes the raw data provided by IDEAM. Regarding solar radiation, hourly data for different years were obtained, while monthly average data were available for the sunshine duration.

Despite of the filters applied to these databases, some data points of solar radiation were missing; therefore, we used the zoo package to create a function in R to complete the missing values. This function takes the data of every day in the database and tests if there are missing values. If there are not missing data or if all the data is missing, the function ends. If there are some missing data points, the function calculates the first missing values before reaching one real value then fits these values to zero. Then, the function calculates the missing values after the last real value in the day and fits these values to zero. By the end, the function takes the missing values between the real values in the day and uses the function zoo for interpolation. Figure 1 displays a flowchart which describes this function.

Figure 1 Algorithm to interpolate missing data values. 

With the database full of real values, the next step is to calculate the Angström coefficients in order to generate the solar insolation information and the graphs with the simulated values. To accomplish this goal, the algorithm shown in figure 2 was developed. Once the Angström-Prescott coefficients are calculated, the solar radiation values are determined through equation 1.

Figure 2 Algorithm to fit the Angström-Prescott equation. 

3.1 UFPS meteorological station

Figure 3 shows the regression between the irradiation and the sunshine duration using the Angström-Prescott equation for the city of Cucuta, with the data from the UFPS station.

Figure 3 Solar data regression results on UFPS station. 

The line in figure 3 is defined by equation 2, and represents the Angström-Prescott equation with its respective constants for the city of Cucuta. To calculate the constants, we used the lm() function in R to perform a linear regression with the data points obtained from the calculations of H_o and N from the equations described by ^{Almorox et al. (2005)}, and the processing of the meteorological data from IDEAM.

This regression has the statistic parameters shown in table 3.

The R-squared coefficient of 0.7273 in table 3, indicates a significant linear relationship between the solar irradiation and the sunshine duration. Additionally, the p-value and F-statistic show statistically significant results.

For the UFPS meteorological station, figure 4 displays the monthly solar insolation data from IDEAM and the estimated values obtained with the Angström-Prescott equation. The estimated data points follow the measured data very close, which is consistent with the high correlation coefficient of the regression.

Figure 4 Real vs estimated irradiation data in UFPS station. 

3.2 Unipamplona meteorological station

Next, the regression analysis was performed with data from Unipamplona meteorological station. Equation 3 corresponds to the line shown in figure 5; this line represents the regression between the solar irradiation and the sunshine duration as established by the Angström-Prescott equation. The statistics parameters of the regression, are presented in table 4.

Figure 5 Solar data regression results on Unipamplona station. 

From table 4, the R-squared coefficient of 0.7818 demonstrates a strong linear correlation between the solar irradiation and the sunshine duration. Furthermore, the p-value and the F-statistic provide statistical significance to the regression.

Figure 6 is a plot of the monthly solar radiation, where empirical data is confronted with the estimated data from the Angström-Prescott equation. In the plot, the expected and observed data points are very close to each other, which confirms the strong linear correlation between solar irradiation and sunshine hours.

Figure 6 Real vs estimated data in Unipamplona Station. 

3.3 Herran meteorological station

The regression process with data from the Herran meteorological station, generated the results shown in figure 7. Unlike the previous two analyses, the data points are dispersed from the regression line.

Figure 7 Solar data regression results on Herran meteorological station. 

The Angström-Prescott model for Herran, is presented in equation 4. Table 5 displays the statistic parameters of the regression.

Although the R-squared coefficient reflects a moderate fit, the p-value has statistical significance at an alpha value of 0.05. In other words, the model is still valid with lower accuracy than the models obtained for Cucuta and Pamplona. The lower R-squared for the Herran model, can be explained by the different time period of the data for solar radiation and sunshine duration (see Table 2). Another reason is that the solar radiation data for Herran has the largest amount of missing data compared to the other two models.

A plot of the monthly solar radiation showing observed and estimated data for Herran, is presented in figure 8. The greatest deviations occur in the months of March, April, July, August, and September.

Figure 8 Real vs estimated data in Herran meteorological station. 

3.4 Comparison with other works in the literature

Our results can be compared with other studies reported in the literature. The R-squared coefficients that we obtained were: 0.73, 0.78 and 0.42, for Cucuta, Pamplona, and Herran, respectively. ^{Vélez-Pereira et al. (2015}), modeled linear regressions of solar radiation with different meteorological variables, and their highest R-squared was 0.14 when they included all the available variables in a multiple linear regression model. On the other hand, ^{Noriega-Angarita et al. (2016)}, used a sophisticated model based or artificial neural networks that require a high computation processing; they obtained an R-squared of 0.77. Similarly, ^{Quej et al. (2016)}, employed 13 empirical models to estimate solar radiation in multiple sites in Mexico; they reported an R-squared between 0.62 and 0.71. Our results for Pamplona outperformed all the other studies, even the one that uses machine learning techniques. The model for Cucuta also has high accuracy, when compared to the other cited works. Finally, the results for Herran only outperforms the single variable linear regressions done by Vélez-Pereira et al. (2015). Therefore, the Angström-Prescott model has a high dependence on specific locations as well as the quality of the available data.

3.5 Extrapolation of the Angström-Prescott results to other subregions in Norte de Santander

With the coefficients of the Angström-Prescott equation for each subregion of Norte de Santander, the next step is to apply these equations to the sunshine duration data in other regions of the Department, in which there is no information about solar radiation.

The Department of Norte de Santander in Colombia is divided into 6 sub-regions, namely: Western, Northern, Eastern, South-West, Center, and South-East. We applied the regression results to the corresponding sub-region and also to the other adjacent sub-regions. For instance, equation 2 corresponds to the city of Cucuta which is located in the Eastern sub-region. Hence, we use that model for the municipalities in the Eastern and Center sub-regions.

Similarly, equation 3 was obtained with data from Pamplona that belongs to the South-West sub-region. Therefore, that model was used for the South-West and Northern sub-regions. Finally, equation 4 represents the Angström-Prescott model for Herran in the South-East sub-region. That model was applied in the South-East and Western sub-regions.

The results of extrapolating the model defined by equation 2 to the Eastern and Center sub-regions are presented in table 6. The estimated insolation data for theses sub-region using the described empirical method have a mean above 3800 Wh/m² for each station.

For the West and South-East results (shown in table 7) calculated with equation 3, the mean of each station is above 3700 Wh/m², with some places having a high solar potential such as Aeropuerto Agua Clara Station with a solar irradiation mean of 4415 Wh/m².