# Predictions of Surface Solar Radiation on Tilted Solar Panels using Machine Learning Models: A Case Study of Tainan City, Taiwan

## Abstract

**:**

^{2}per year when the solar-panel tilt angle was 0° (i.e., the non-tilted position) and peaked at 1.655 million w/m

^{2}per year when the angle was 20–22°. The level of the irradiance was almost the same when the solar-panel tilt angle was 0° as when the angle was 41°. In summary, the optimal solar-panel tilt angle in Tainan was 20–22°.

## 1. Introduction

## 2. Study Site and Data

#### 2.1. Ground Weather Data Set {A}

#### 2.2. Satellite Remote-Sensing Data Set {B}

#### 2.3. Sun Position Data Set {C}

#### 2.3.1. Declination Angle

_{d}is the day as numbered within a solar year (ending with 365 on 31 December).

#### 2.3.2. Hour Angle

#### 2.3.3. Zenith Angle

#### 2.3.4. Elevation Angle

#### 2.3.5. Azimuth Angle

## 3. Methodology and Models

#### 3.1. Procedures of the Methodology

#### 3.2. Machine Learning

#### 3.2.1. Multilayer Perceptron Neural Networks

#### 3.2.2. Random Forests

- Step 1:
- Determine the number of decision trees required to construct RFs.
- Step 2:
- Use bootstrapping to generate new training data in existing ones. Both new and existing training data are equivalent in volume.
- Step 3:
- Use the data derived through bootstrapping to generate classification and regression trees. If the number of independent variables is P during the generation of the trees, then branching is performed on randomly selected nodes whose number of independent variables is lower than $\sqrt{P}$.
- Step 4:
- Repeat Steps 2 and 3 until the number of decision trees as determined in Step 1 is reached.
- Step 5:
- Analyses are performed using all decision trees. If the dependent variables are categorical, and the variable that appears the most frequently in the decision trees is deemed to be the output. If dependent variables are continuous, then the average of the prediction results from all the trees is used as the output.

#### 3.2.3. k-Nearest Neighbor

_{1}, x

_{2}, …, x

_{k}] and y = [y

_{1}, y

_{2}, …, y

_{k}], in a k-dimensional space, and the distance between x and y is expressed as follows:

## 4. Experiments and Modeling

#### 4.1. Data Partition and Combination Cases

- Dataset combination 1: ground weather dataset, denoted by {
**A**}. - Dataset combination 2: ground weather dataset and satellite remote-sensing dataset ({
**A**,**B**}). - Dataset combination 3: ground weather dataset and solar position dataset ({
**A**,**C**}). - Dataset combination 4: ground weather dataset, satellite remote-sensing dataset, and solar position dataset ({
**A**,**B**,**C**}).

**A**} includes parameters of atmospheric pressure, wind speed, precipitation, temperature, relative humidity, and radiation. The subset {

**B**} includes aerosol optical depth, water vapor, cirrus reflectance, and cloud fraction. The subset {

**C**} includes declination angle, hour angle, zenith angle, elevation angle, and azimuth angle.

#### 4.2. Model Parameter Setup and Calibration

**A**} and 0.1 for {

**A**,

**B**}, {

**A**,

**C**}, and {

**A**,

**B**,

**C**} (Figure 4a). After the optimal learning rate was determined for all the dataset combinations, the momentum correction coefficient was estimated on a ten-interval scale ranging from 0 to 1 (Figure 4b). The optimal momentum correction coefficients were 0.2 for {

**A**} and {

**A**,

**C**}, 0.1 for {

**A**,

**B**}, and 0.3 for {

**A**,

**B**,

**C**}.

**A**}, 25 for {

**A**,

**B**}, 45 for {

**A**,

**C**}, and 30 for {

**A**,

**B**,

**C**}.

**A**} and {

**A**,

**C**}, and 25 for {

**A**,

**B**} and {

**A**,

**B**,

**C**}.

#### 4.3. Forecasting Solar Irradiance in t + 1 through the Four Dataset Combinations

#### 4.3.1. Results of Dataset Combinations

**A**,

**C**} outperformed the other dataset combinations in the MAE (and was comparable to {

**A**,

**B**,

**C**}), and that {

**A**,

**B**,

**C**} outperformed the other dataset combinations in the RMSE and r (and was comparable to {

**A**,

**C**}). Accordingly, errors in the prediction results of the models decreased more significantly when {

**A**} as input data was combined with {

**C**} than with {

**B**}; and prediction results from {

**A**,

**B**,

**C**} showed no noticeable improvement in comparison with {

**A**,

**C**}.

**B**}, which can be attributed to the fact that the MODIS is nonsynchronous, delivering atmospheric data only on a daily basis. Thus, the instrument does not observe changes in the Earth’s atmosphere (e.g., cloud shading) on an hourly basis.

#### 4.3.2. Evaluation

_{i}

_{,j}is the MAE for model j in dataset i, MAE

_{MAX}is the maximum MAE (the negative ideal solution) for all models in all dataset combinations, MAE

_{MIN}is the minimum MAE (the positive ideal solution) for all models in all dataset combinations.

_{i}

_{,j}is the RMSE for model j in dataset i, RMSE

_{MAX}is the maximum RMSE for all models in all dataset combinations, RMSE

_{MIN}is the minimum RMSE for all models in all dataset combinations, r

_{i}

_{,j}is the r for model j in dataset i, r

_{MAX}is the maximum r for all models in all dataset combinations, and r

_{MIN}is the maximum r for all models in all dataset combinations.

**A**,

**C**} was previously determined as the optimal dataset, the improvement rates for the four models with {

**A**,

**C**} (which is represented by gray histograms in Figure 6) were analyzed, and the following conclusions were reached.

- Regarding ${\mathrm{I}}_{i,j}^{\mathrm{MAE}}$ (Figure 6a), the improvement rate was the highest in RF (100%), followed by kNN (93.6%), MLP (87.7%), and LR (23.1%).
- Regarding ${\mathrm{I}}_{i,j}^{\mathrm{RMSE}}$ (Figure 6b), the improvement rate was the highest in RF (99.9%), followed by MLP (95.4%), kNN (89.4%), and LR (44.6%).
- Regarding ${\mathrm{I}}_{i,j}^{\mathrm{r}}$ (Figure 6c), the improvement rate was the highest in RF (98.4%), followed by MLP (94.3%), kNN (91%), and LR (47.3%).

#### 4.4. Forecasting Solar Irradiance across Different Forecast Horizons

**A**,

**C**} was used to establish all forecasting models. The training set was applied for model training, and the validation set was used to verify the training results. Prediction results are described as follows.

#### 4.4.1. Prediction Results as Represented by MAE, RMSE, and r

#### 4.4.2. Predicted vs. Observed Changes in Solar Irradiance

#### 4.4.3. Prediction Errors across Seasons

^{2}and 57.8 w/m

^{2}in winter, 34.1 w/m

^{2}and 70.8 w/m

^{2}in autumn, 37.4 w/m

^{2}and 73.8 w/m

^{2}in spring, and 47.6 w/m

^{2}and 93.1 w/m

^{2}in summer. In the same model, the rMAE and rRMSE were the lowest in spring (0.198 and 0.391, respectively), relatively low in autumn (0.205 and 0.426) and winter (0.207 and 0.421), and highest in summer (0.217 and 0.426). Overall, the MAE and rMAE were the highest in summer, indicating changes in cloud thickness or in the atmosphere (e.g., the occurrence of typhoons or convective rains).

^{2}and 104.1 w/m

^{2}in winter, 63.7 w/m

^{2}and 123.7 w/m

^{2}in autumn, 68.5 w/m

^{2}and 129.7 w/m

^{2}in spring, and 78.7 w/m

^{2}and 140.1 w/m

^{2}in summer. Furthermore, the rMAE and rRMSE within t + 6 were the lowest in summer and the highest in winter. In RF, they were 0.360 and 0.640 respectively in summer; 0.363 and 0.687 in spring; 0.384 and 0.745 in autumn; and 0.392 and 0.758 in winter. Overall, the respective values of the MAE and RMSE within t + 1 were ranked in the same order across seasons as those within t + 6 (that is, they peaked in summer, became lower in spring and autumn, and reached their lowest in winter). However, the respective values of the rMAE and rRMSE within t + 1 (peaked in summer, became lower in winter and autumn, and reached their lowest in spring) were ranked in a different order across seasons than those within t + 6 (peaked in winter, became lower in autumn and spring, and reached their lowest in summer).

## 5. Deriving Equations for Solar Irradiance Received by a Tilted Solar Panel

#### 5.1. Estimating Theoretical Clear-Sky Solar Irradiance

#### 5.1.1. Theoretical Values of G_{C}, I_{C}, and D_{C}

_{C}). Solar irradiance is higher outside the atmosphere, because within the atmosphere radiation is diffused by clouds, water vapor, and particulate matter. G

_{C}is the sum of clear-sky direct irradiance (I

_{C}·cosθ) and diffuse horizontal irradiance (D

_{C}) [28]. The respective equations for G

_{C}, I

_{C}, and D

_{C}are as follows:

_{s}and R

_{s}are constant coefficients (that are Q

_{0}= 1.105, Q

_{1}= −1.435, Q

_{2}= −1.072, Q

_{3}= 6.685, Q

_{4}= −13.899, Q

_{5}= 13.080, Q

_{6}= −4.463, R

_{0}= 0.986, R

_{1}= −0.200, R

_{2}= −1.188, R

_{3}= 3.371, R

_{4}= −5.767, R

_{5}= 3.721, and R

_{6}= −0.922).

#### 5.1.2. Solar Incident Angle (Θ) and Global Irradiance with Tilted Solar Panels (G_{tilt})

_{tilt}) was estimated by:

#### 5.2. Estimating Observed and Predicted Solar Irradiance with Tilted Solar Panels

#### 5.2.1. Estimating the Observed and Predicted Direct Irradiance and Diffuse Horizontal Irradiance

^{obs}computed by observed irradiance ${G}_{C}^{obs}$ and the parameter $\widehat{p}$ computed by predicted irradiance ${\widehat{G}}_{C}$, both of which are expressed by:

#### 5.2.2. Estimating the Observed and Predicted Global Irradiance with the Solar Panels Set at a Tilted Position

## 6. Estimating Solar Irradiance with the Solar Panels Set at a Tilted Position

#### 6.1. Estimating the Observed and Predicted Values of Global Horizontal Irradiance and Diffuse Horizontal Irradiance

#### 6.2. Estimating the Observed and Predicted Global Irradiance with Solar Panels set at Different Tilt Angles

#### 6.3. Total Annual Global Irradiance in Relation to Different Solar-Panel Tilt Angles

#### 6.3.1. Total Annual Global Irradiance and Its Increase Rate

^{2}with a β′ of 0° (i.e., when the solar panels were placed in the non-tilted position) and peaked at 1.655 MWh/m

^{2}with a β′ ranging from 20° to 22°. Moreover, the increase rate of the total annual global irradiance was calculated, with the total annual global irradiance observed at a β′ of 0° as the basic value. The increase rate is estimated as follows:

#### 6.3.2. Total Annual Global Irradiance at Different Solar-panel Tilt Angles

## 7. Conclusions

^{2}with a β′ of 0° and peaked at 1.655 MWh/m

^{2}with a β′ of 20–22°, and its level was almost the same with a β′ of 0° and 41°. Moreover, the irradiance was lower with a β′ of 42° than with a β′ of 0°, suggesting that the optimal tilt angle for the solar panels was 20–22°.

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Calibration of model parameters: (

**a**) learning rate for MLP; (

**b**) momentum correction for MLP; (

**c**) size of each bag for RF; and (

**d**) number of neighbors for kNN.

**Figure 5.**Performance of dataset combinations for solar irradiance at t + 1: (

**a**) MAE, (

**b**) RMSE, and (

**c**) r.

**Figure 6.**Improvement rates for MLP, RF, kNN, and LR with all dataset combinations: (

**a**) MAE, (

**b**) RMSE, and (

**c**) r.

**Figure 7.**Prediction errors by all models over a 12-h forecast horizon in the year 2016: (

**a**) MAE, (

**b**) RMSE, and (

**c**) r.

**Figure 8.**Observed and predicted changes in solar irradiance for the 4 consecutive days within 1 h (t + 1), 3 h (t + 3), 6 h (t + 6), and 12 h (t + 12) starting from: (

**a**–

**d**) the vernal equinox on 20 March 2016, and (

**e**–

**h**) the summer solstice on 21 June 2016.

**Figure 9.**Observed and predicted changes in solar irradiance for the 4 consecutive days within 1 h (t + 1), 3 h (t + 3), 6 h (t + 6), and 12 h (t + 12) starting from: (

**a**–

**d**) the autumnal equinox on 22 September 2016, and (

**e**–

**h**) the winter solstice on 21 December 2016.

**Figure 10.**Performance of all models over a 12-h forecast horizon: (

**a**–

**c**) in summer, and (

**d**–

**f**) in winter.

**Figure 11.**Observed versus predicted changes in the direct irradiance within t + 1, t + 3, t + 6, and t + 12 for 4 consecutive days starting from: (

**a**–

**d**) the summer solstice, and (

**e**–

**h**) the winter solstice.

**Figure 12.**Observed values versus predicted changes in the diffuse horizontal irradiance within t + 1, t + 3, t + 6, and t + 12 for four consecutive days starting from: (

**a**–

**d**) the summer solstice, and (

**e**–

**h**) the winter solstice.

**Figure 13.**Hourly changes in the observed and predicted global irradiance with a β′ of 23° for the four consecutive days starting from: (

**a**–

**d**) the summer solstice, and (

**e**–

**h**) the winter solstice.

**Figure 14.**Hourly changes in the observed and predicted global irradiance with a β′ of 33° for the four consecutive days starting from: (

**a**–

**d**) the summer solstice, and (

**e**–

**h**) the winter solstice.

**Figure 15.**Results with a β′ of 0–50°: (

**a**) amount of total annual global irradiance, and (

**b**) increase rate of the total annual global irradiance.

**Figure 16.**Observed and predicted values of the total annual global irradiance with a β′ of 0–41° within (

**a**) t + 1, (

**b**) t + 3, (

**c**) t + 6, and (

**d**) t + 12.

**Figure 17.**Relative error of the predicted total annual global irradiance within (

**a**) t + 1, (

**b**) t + 3, (

**c**) t + 6, and (

**d**) t + 12.

Data Set | Attribute | Unit | Min–Max | Mean | Standard Deviation |
---|---|---|---|---|---|

Ground weather | Atmospheric pressure | hPa | 973.8–1031.5 | 1011.2 | 5.72 |

Wind speed | m/s | 0–18.4 | 2.83 | 1.70 | |

Precipitation | mm | 0–95 | 0.23 | 1.91 | |

Temperature | °C | 5.6–35.9 | 24.3 | 5.38 | |

Relative humidity | % | 23–100 | 74.0 | 10.17 | |

Radiation | w/m^{2} | 0–1125.00 | 162.04 | 249.38 | |

Satellite Remote-sensing | Aerosol optical depth | - | 0.18–10.74 | 2.80 | 1.56 |

Water vapor | cm | 0.15–77.21 | 37.47 | 13.72 | |

Cirrus reflectance | - | 0.25–67.42 | 2.92 | 4.99 | |

Cloud fraction | - | 0.93–100 | 68.50 | 28.76 | |

Sun Position | Declination angle | Deg. | −23.45–23.45 | −0.01 | 16.58 |

Hour angle | Deg. | −165.00–80.00 | 7.50 | 103.83 | |

Zenith angle | Deg. | 0.01–179.99 | 90.00 | 43.83 | |

Elevation angle | Deg. | −89.99–89.99 | 0.00 | 43.83 | |

Azimuth angle | Deg. | −90.00–90.00 | 0.00 | 65.09 |

Model Case | MLP | RF | kNN | |
---|---|---|---|---|

Learning Rate | Momentum | Size of Each Bag | Number of Neighbors | |

Dataset {A} | 0.5 | 0.2 | 40 | 30 |

Dataset {A, B} | 0.1 | 0.1 | 25 | 25 |

Dataset {A, C} | 0.1 | 0.2 | 45 | 30 |

Dataset {A, B, C} | 0.1 | 0.3 | 30 | 25 |

Performance | Case {A} | Case {A,B} | Case {A,C} | Case {A,B,C} |
---|---|---|---|---|

MAE | 62.52 | 63.23 | 39.33 | 39.42 |

RMSE | 104.45 | 104.67 | 76.78 | 76.75 |

r | 0.924 | 0.923 | 0.960 | 0.961 |

Season | Performance | t + 1 | t + 6 | ||||||
---|---|---|---|---|---|---|---|---|---|

MLP | RF | kNN | LR | MLP | RF | kNN | LR | ||

Spring | MAE (w/m^{2}) | 39.4 | 37.4 | 41.1 | 57.9 | 76.0 | 68.5 | 70.6 | 123.1 |

rMAE | 0.209 | 0.198 | 0.218 | 0.307 | 0.403 | 0.363 | 0.374 | 0.652 | |

RMSE (w/m^{2}) | 74.6 | 73.8 | 80.0 | 89.0 | 125.6 | 129.7 | 135.2 | 184.5 | |

rRMSE | 0.395 | 0.391 | 0.424 | 0.472 | 0.665 | 0.687 | 0.716 | 0.978 | |

r | 0.964 | 0.965 | 0.959 | 0.950 | 0.896 | 0.888 | 0.876 | 0.767 | |

Summer | MAE (w/m^{2}) | 50.7 | 47.6 | 48.8 | 67.8 | 84.9 | 78.7 | 75.2 | 134.7 |

rMAE | 0.232 | 0.217 | 0.223 | 0.310 | 0.388 | 0.360 | 0.344 | 0.616 | |

RMSE (w/m^{2}) | 96.5 | 93.1 | 95.9 | 110.6 | 134.0 | 140.1 | 134.4 | 201.6 | |

rRMSE | 0.441 | 0.426 | 0.438 | 0.506 | 0.613 | 0.640 | 0.614 | 0.922 | |

r | 0.950 | 0.954 | 0.951 | 0.933 | 0.906 | 0.901 | 0.906 | 0.805 | |

Autumn | MAE (w/m^{2}) | 36.2 | 34.1 | 36.8 | 56.0 | 72.4 | 63.7 | 63.8 | 107.6 |

rMAE | 0.218 | 0.205 | 0.222 | 0.338 | 0.436 | 0.384 | 0.384 | 0.648 | |

RMSE (w/m^{2}) | 72.1 | 70.8 | 73.4 | 89.3 | 122.2 | 123.7 | 121.3 | 170.1 | |

rRMSE | 0.434 | 0.426 | 0.442 | 0.538 | 0.736 | 0.745 | 0.731 | 1.025 | |

r | 0.962 | 0.964 | 0.962 | 0.941 | 0.892 | 0.892 | 0.892 | 0.781 | |

Winter | MAE (w/m^{2}) | 29.4 | 28.5 | 30.7 | 52.4 | 64.6 | 53.8 | 57.1 | 104.9 |

rMAE | 0.214 | 0.207 | 0.223 | 0.382 | 0.470 | 0.392 | 0.416 | 0.764 | |

RMSE (w/m^{2}) | 59.4 | 57.8 | 61.5 | 79.2 | 108.4 | 104.1 | 109.1 | 157.9 | |

rRMSE | 0.432 | 0.421 | 0.448 | 0.577 | 0.790 | 0.758 | 0.795 | 1.150 | |

r | 0.968 | 0.969 | 0.966 | 0.940 | 0.884 | 0.893 | 0.880 | 0.725 |

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## Share and Cite

**MDPI and ACS Style**

Wei, C.-C.
Predictions of Surface Solar Radiation on Tilted Solar Panels using Machine Learning Models: A Case Study of Tainan City, Taiwan. *Energies* **2017**, *10*, 1660.
https://doi.org/10.3390/en10101660

**AMA Style**

Wei C-C.
Predictions of Surface Solar Radiation on Tilted Solar Panels using Machine Learning Models: A Case Study of Tainan City, Taiwan. *Energies*. 2017; 10(10):1660.
https://doi.org/10.3390/en10101660

**Chicago/Turabian Style**

Wei, Chih-Chiang.
2017. "Predictions of Surface Solar Radiation on Tilted Solar Panels using Machine Learning Models: A Case Study of Tainan City, Taiwan" *Energies* 10, no. 10: 1660.
https://doi.org/10.3390/en10101660