Evaluation of Empirical Daily Solar Radiation Models for the Northeast Coast of the Iberian Peninsula
Abstract
:1. Introduction
- Sunshine-based models. These models are based on daily sunshine duration. Although they allow accurate predictions of solar radiation, these models require data from specific equipment not present at many meteorological stations. Despotovic et al. [12] carried out an exhaustive analysis using 101 different sunshine-based models using data from 924 sites around the world. The authors concluded that although they could be used anywhere on the planet, their versatility was hindered by their limited accuracy.
- Cloud-based models. These models are based on daily measurements of average cloudiness. As with the sunshine-based models, many meteorological stations are not equipped with specific apparatus to measure cloudiness, and furthermore, such instruments are usually regarded as highly subjective. Ahamed et al. [13] made an exhaustive review of this type of model, concluding that they could be a good alternative for estimating solar radiation when sunshine hour data are not available.
- Temperature-based models. These models are usually based on daily maximum and minimum temperature values. Their major advantage is that these two quantities are measured by most meteorological stations.
- Other meteorological-parameter-based models. These models use a combination of several meteorological variables that often include the ratio of daily and maximum daily sunshine duration, relative humidity, air water content, average temperature, precipitation, etc.
- Day-of-year-based models. In addition to the four previous groups, there is a family of models based only on the day of the year. As they do not require meteorological data to characterize the atmospheric conditions, they are readily usable, although they offer predictions of limited accuracy compared to other model types.
2. Materials and Methods
2.1. Study Area and Data
2.2. Data Quality
3. Modeling of Global Solar Radiation
3.1. Extraterrestrial Daily Global Solar Radiation
3.2. Empirical Models for Estimation of Solar Radiation
3.2.1. Day-of-the-Year-Based Models (DYBs)
3.2.2. Sunshine-Based Models (SBMs)
3.2.3. Cloud-Based Models (CBMs)
3.2.4. Temperature-Based Models (TBMs)
3.2.5. Other Meteorological-Parameter-Based Models (OPMs)
3.3. Statistical Performance Validation
- Mean absolute error or MAE (MJ m−2 day−1) is a statistical indicator used to determine how close the calculated values are to the measurements. It is the sum of the absolute value of the differences between the measured values and the calculated values, divided by the number of measures. Performance increases as this metric tends toward zero.
- 2.
- Root mean square error or RMSE (MJ m−2 day−1) is often used in the literature, although some authors have expressed concerns about its suitability for this type of analysis given the large impact that a small set of large-discrepancy measurements can have on this metric, which uses the square of the difference between observed and predicted values [46].
- 3.
- The mean absolute relative error or MARE is calculated as the sum of the absolute value of the relative differences between the measured and calculated data. Some authors express it as a percentage. The lower this value is, the better the model performance.
- 4.
- Uncertainty at 95% or U95 puts the emphasis on model deviation. For the standard normal distribution, a value of 1.96 implies that there is a 95% probability that a standard normal variable will fall between −1.96 and 1.96 [47]. SD is the standard deviation of the difference between calculated and measured solar radiation data.
- 5.
- Root mean squared relative error or RMSRE is often preferred over the RMSE, although it is more sensitive to very low observation values [48]. This metric score decreases as model performance improves.
- 6.
- 7.
- Mean bias error or MBE is used to quantify the tendency of the model to overestimate or underestimate the measured values. This indicator may not be an appropriate metric when simultaneously overestimated and underestimated values can cancel each other out.
- 8.
- The coefficient of determination or R2 is frequently used in statistics to estimate how well model predictions capture trends in the observed data. Bounded between 0 and 1, the larger the value of R2 the better the model performance.
- 9.
- The maximum absolute relative error or errMAX uses the largest relative difference between predicted and observed values.
4. Results and Discussion
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Nomenclature
a–g | Model regression coefficients |
CC | Cloud cover index |
The inverse of the relative distance between the sun and the earth | |
Solar constant (0.0820 MJ·m−2·min−1) | |
H | Daily global solar radiation (MJ m−2 day−1) |
Hm | Measured daily global solar radiation (MJ m−2 day−1) |
Hc | Calculated daily global solar radiation (MJ m−2 day−1) |
Ho | Daily extraterrestrial radiation (MJ m−2 day−1) |
J | Day of the year starting 1 January |
KT | Daily sky clearness index |
PPT | Accumulated precipitation (mm) |
R2 | Coefficient of determination |
RH | Relative humidity |
S | Daily sunshine duration (h) |
So | Maximum possible daily sunshine duration (h) |
Tmax | Daily maximum temperature |
Tmin | Daily minimum temperature |
Tmean | Daily mean temperature |
U95 | Uncertainty at 95% (MJ m−2 day−1) |
W | Atmospheric precipitable water per unit volume of air |
Sunset hour angle (rad) | |
Greek Symbols | |
Solar declination (rad) | |
Latitude (rad) | |
Acronyms | |
ECMWF | European Centre for Medium-Range Weather Forecasts |
errMAX | Maximum absolute relative error (MJ m−2 day−1) |
GPI | Global Performance Indicator |
MAE | Mean absolute error (MJ m−2 day−1) |
MARE | Mean absolute relative error |
MBE | Mean bias error. (MJ m−2 day−1) |
RMSE | Root mean square error (MJ m−2 day−1) |
RMSRE | Root mean squared relative error |
RRMSE | Relative root mean squared error (%) |
SD | Standard deviation |
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Station Name | Station ID | Latitude | Longitude | Elevation (m) |
---|---|---|---|---|
GIRONA/COSTA BRAVA | S01 | 41.91167 | 2.763056 | 143 |
LLEIDA | S02 | 41.62556 | 0.595 | 192 |
LLEIDA-AJUNTAMENT | S03 | 41.61639 | 0.583056 | 169 |
BARCELONA/AIRPORT | S04 | 41.29278 | 2.069722 | 4 |
REUS/AIRPORT | S05 | 41.14944 | 1.178889 | 71 |
TORTOSA—OBSERVATORIO DEL EBRO | S06 | 40.82056 | 0.491389 | 44 |
CASTELLON DE LA PLANA | S07 | 39.98917 | 0.040556 | 25 |
CASTELLON | S08 | 39.95 | 0.071389 | 35 |
MENORCA/MAO | S09 | 39.85444 | 4.215556 | 91 |
VALENCIA/AEROPUERTO | S10 | 39.48667 | 0.473056 | 69 |
VALENCIA | S11 | 39.48056 | 0.366389 | 11 |
ALICANTE | S12 | 38.3725 | 0.494167 | 81 |
ALICANTE EL ALTET | S13 | 38.28278 | 0.570556 | 43 |
PALMA DE MALLORCA/SON SAN JUAN | S14 | 39.56056 | 2.736667 | 8 |
PALMA DE MALLORCA CMT | S15 | 39.55556 | 2.626389 | 3 |
IBIZA/ESCODOLA | S16 | 38.87639 | 1.384167 | 6 |
Station ID | T Max (°C) | T Min (°C) | T Mean (°C) | Precipitation (mm) | Humidity (%) | Daily Solar Radiation (MJ m−2 Day−1) | Sunshine Hours (h) |
---|---|---|---|---|---|---|---|
S01 | 37.0 | −5.9 | 15.3 | 405.5 | 71.3 | 14.8 | 7.2 |
S02 | 38.8 | −5.6 | 16.2 | 227.2 | 66.5 | 17.0 | 8.4 |
S03 | 38.0 | −5.6 | 15.9 | 217.6 | --- | 17.0 | 8.4 |
S04 | 33.1 | −0.1 | 17.1 | 315.8 | 68.1 | 15.8 | 6.8 |
S05 | 35.8 | −4.0 | 16.6 | 269.4 | 66.5 | 16.0 | 7.7 |
S06 | 38.4 | −0.8 | 18.5 | 315.8 | 64.6 | 15.9 | 7.9 |
S07 | 35.8 | 1.5 | 18.2 | 263.9 | --- | 16.7 | 8.2 |
S08 | 35.8 | 1.5 | 18.2 | 261.1 | 63.4 | 16.7 | 8.2 |
S09 | 34.3 | 1.9 | 17.6 | 412.9 | 74.1 | 15.9 | 7.4 |
S10 | 38.5 | −1.8 | 18.1 | 239.1 | 63.0 | 16.9 | 8.2 |
S11 | 36.7 | 1.9 | 18.8 | 246.6 | 65.3 | 16.5 | 7.8 |
S12 | 36.5 | 0.6 | 18.7 | 212.7 | 67.4 | 17.7 | 8.5 |
S13 | 36.9 | 0.2 | 18.5 | 202.2 | 62.9 | 17.4 | 8.1 |
S14 | 37.7 | −2.2 | 17.3 | 298.6 | 73.2 | 16.4 | 7.8 |
S15 | 35.0 | 3.2 | 18.8 | 342.1 | 73.6 | 16.6 | 8.0 |
S16 | 34.2 | 1.5 | 18.4 | 279.6 | 72.6 | 16.8 | 7.8 |
Mean | 36.4 | −0.8 | 17.6 | 281.9 | 68.0 | 16.5 | 7.9 |
Model ID | Model Equations | a | b | c | d | e | f | g |
---|---|---|---|---|---|---|---|---|
SBM1 | 0.283 | 0.457 | ||||||
SBM2 | 0.233 | 0.714 | −0.238 | |||||
SBM3 | 0.223 | 0.829 | −0.508 | 0.173 | ||||
SBM4 | −0.378 | 0.333 | ||||||
SBM5 | 0.670 | 0.147 | ||||||
TBM1 | 0.182 | |||||||
TBM2 | 0.174 | 0.986 | ||||||
TBM3 | 0.019 | −0.015 | 0.339 | |||||
TBM4 | 0.026 | −0.013 | 4.14 | |||||
TBM5 | 0.673 | 0.220 | 0.990 | |||||
DYB1 | 16.60 | −8.970 | 0.159 | |||||
DYB2 | 16.08 | −9.259 | −0.95 | −5.03 | 0.21 | −11.03 | 7.88 | |
DYB3 | 31.16 | −140.20 | 164.15 | 164.06 | 134.50 | 165.93 | −131.34 | |
DYB4 | 6.16 | 18.69 | ||||||
DYB5 | 16.57 | −8.862 | 1.509 | −0.356 | 0.503 | |||
CBM1 | 0.724 | −0.042 | ||||||
CBM2 | 0.675 | 0.00305 | −0.00621 | |||||
CBM3 | 0.673 | 0.00751 | −0.00777 | 0.00014 | ||||
OPM1 | 1.007 | 0.324 | −0.094 | |||||
OPM2 | 0.321 | 0.445 | −0.00046 | |||||
OPM3 | 0.285 | 0.453 | −0.00032 | |||||
OPM4 | 0.148 | 0.184 | −0.290 | 0.002 | −0.694 | |||
OPM5 | Parameters do not need to be adjusted and are a function of Tmean and PPT |
MAE | RMSE | MARE | U95 | MRSRE | RRMSE | MBE | R2 | errMax | GPI | |
---|---|---|---|---|---|---|---|---|---|---|
SBM1 | 1.23 | 2.32 | 0.101 | 6.42 | 0.268 | 13.78 | 0.050 | 0.909 | 6.264 | 4.814 |
SBM2 | 1.20 | 2.31 | 0.093 | 6.40 | 0.260 | 13.75 | 0.120 | 0.909 | 5.061 | 5.002 |
SBM3 | 1.19 | 2.31 | 0.092 | 6.39 | 0.259 | 13.73 | 0.110 | 0.910 | 4.827 | 5.048 |
SBM4 | 1.46 | 2.40 | 0.108 | 6.62 | 0.258 | 14.30 | 0.369 | 0.902 | 3.393 | 4.673 |
SBM5 | 1.70 | 2.61 | 0.134 | 7.22 | 0.293 | 15.54 | 0.287 | 0.884 | 3.737 | 3.822 |
TBM1 | 3.01 | 3.98 | 0.226 | 10.97 | 0.399 | 23.68 | 0.568 | 0.731 | 13.410 | −1.752 |
TBM2 | 2.97 | 3.94 | 0.233 | 10.93 | 0.422 | 23.50 | 2.64 × 10−1 | 0.735 | 13.620 | −1.722 |
TBM3 | 2.74 | 3.62 | 0.224 | 10.03 | 0.421 | 21.59 | 0.304 | 0.776 | 13.734 | −0.827 |
TBM4 | 2.88 | 3.83 | 0.232 | 10.61 | 0.425 | 22.80 | 1.28 × 10−1 | 0.751 | 13.584 | −1.353 |
TBM5 | 2.80 | 3.69 | 0.223 | 10.17 | 0.419 | 21.97 | 0.548 | 0.769 | 13.991 | −1.080 |
DYB1 | 3.13 | 3.99 | 0.261 | 11.01 | 0.489 | 23.73 | 5.10 × 10−1 | 0.729 | 14.093 | −2.437 |
DYB2 | 3.13 | 4.00 | 0.261 | 11.05 | 0.489 | 23.82 | 5.14 × 10−1 | 0.728 | 14.123 | −2.478 |
DYB3 | 3.15 | 4.03 | 0.264 | 11.12 | 0.495 | 23.95 | 0.478 | 0.724 | 13.922 | −2.556 |
DYB4 | 3.20 | 4.07 | 0.267 | 11.24 | 0.492 | 24.23 | 5.34 × 10−1 | 0.718 | 15.072 | −2.813 |
DYB5 | 3.12 | 4.00 | 0.260 | 11.05 | 0.490 | 23.80 | 4.79 × 10−1 | 0.728 | 14.583 | −2.494 |
CBM1 | 1.90 | 2.87 | 0.157 | 8.19 | 0.335 | 19.58 | 0.385 | 0.841 | 11.011 | 1.821 |
CBM2 | 1.86 | 2.85 | 0.149 | 8.10 | 0.318 | 19.42 | 0.472 | 0.844 | 10.910 | 2.000 |
CBM3 | 1.86 | 2.85 | 0.149 | 8.10 | 0.318 | 19.42 | 0.470 | 0.844 | 10.872 | 2.000 |
OPM1 | 1.30 | 2.28 | 0.097 | 6.50 | 0.245 | 15.48 | 0.297 | 0.899 | 3.365 | 4.855 |
OPM2 | 1.12 | 2.22 | 0.092 | 6.36 | 0.258 | 15.06 | 0.043 | 0.904 | 6.263 | 4.881 |
OPM3 | 1.13 | 2.22 | 0.093 | 6.37 | 0.259 | 15.10 | 0.061 | 0.904 | 6.293 | 4.847 |
OPM4 | 2.83 | 3.77 | 0.220 | 10.46 | 0.386 | 22.48 | 1.80 × 10−1 | 0.758 | 7.444 | −0.479 |
OPM5 | 2.84 | 3.65 | 0.209 | 8.86 | 0.356 | 21.73 | −2.486 | 0.774 | 4.792 | 1.348 |
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Vernet, A.; Fabregat, A. Evaluation of Empirical Daily Solar Radiation Models for the Northeast Coast of the Iberian Peninsula. Energies 2023, 16, 2560. https://doi.org/10.3390/en16062560
Vernet A, Fabregat A. Evaluation of Empirical Daily Solar Radiation Models for the Northeast Coast of the Iberian Peninsula. Energies. 2023; 16(6):2560. https://doi.org/10.3390/en16062560
Chicago/Turabian StyleVernet, Anton, and Alexandre Fabregat. 2023. "Evaluation of Empirical Daily Solar Radiation Models for the Northeast Coast of the Iberian Peninsula" Energies 16, no. 6: 2560. https://doi.org/10.3390/en16062560