# Influence of Reflectivity and Cloud Cover on the Optimal TiltAngle of Solar Panels

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Optimal Tilt Angle

#### 1.2. Influence of Weather

#### 1.3. Outline of Paper

## 2. Mathematical Model

Symbol | Description |
---|---|

δ | Angle of Declination |

α | Solar elevation angle |

β | Angle of solar panel with respect to horizontal |

γ | Angle of rotation of solar panel from due north |

ρ | Surface reflectivity |

θ | Angle between sun and solar panel normal |

${\theta}_{z}$ | Zenith angle = ${90}^{\xb0}-\alpha $ |

ϕ | Latitude |

ω | Hour angle |

A | Solar azimuth angle measured from due north |

N | Day of the year |

Asphalt | Soil | Concrete (Weathered) | Grass | Sand | Old Snow | New Snow |
---|---|---|---|---|---|---|

0.15 | 0.17 | 0.20 | 0.25 | 0.36 | 0.45–0.70 | 0.80–0.90 |

## 3. The Influence of Reflectivity on the Optimal Tilt Angle

**Figure 1.**Contour plot of irradiance $\frac{W}{{m}^{2}}$ as a function of day and time of day at a 30° tilt and a 30° latitude.

Reflectivity | Optimal Angle(β) | kWh/m^{2} |
---|---|---|

0 | $34.{2}^{\xb0}$ | 2496 |

0.2 | $37.{3}^{\xb0}$ | 2536 |

0.4 | $41.{0}^{\xb0}$ | 2583 |

0.6 | $45.{3}^{\xb0}$ | 2639 |

0.8 | $50.{4}^{\xb0}$ | 2708 |

^{2}. The optimal angle for the second study is 41° with an annual yield of 2581 kWh/m

^{2}. The difference is only 1.8% in the annual yield at the different optimal angles. However, for comparison, we note that solar irradiation increases 4.1% if panel tilt angles are adjusted twice a year [7].

Reflectivity | Optimal Angle(β) | kWh/m^{2} |
---|---|---|

0 | $26.{1}^{\xb0}$ | 2628 |

0.2 | $28.{8}^{\xb0}$ | 2655 |

0.4 | $32.{0}^{\xb0}$ | 2688 |

0.6 | $36.{0}^{\xb0}$ | 2728 |

0.8 | $40.{8}^{\xb0}$ | 2779 |

Reflectivity | Optimal Angle (β) | kWh/m^{2} |
---|---|---|

0 | $17.{7}^{\xb0}$ | 2713 |

0.2 | $19.{7}^{\xb0}$ | 2727 |

0.4 | $22.{1}^{\xb0}$ | 2743 |

0.6 | $25.{2}^{\xb0}$ | 2765 |

0.8 | $29.{2}^{\xb0}$ | 2794 |

a | b | c | d | e | f |
---|---|---|---|---|---|

–4.6230 | 1.2063 | 4.8992 | –0.00574 | 0.20679 | 8.0612 |

**Table 7.**Coefficients for multiple regression correlation for maximum energy per unit area at optimal angle.

$\overline{a}$ | $\overline{b}$ | $\overline{c}$ | $\overline{d}$ | $\overline{e}$ | $\overline{f}$ |
---|---|---|---|---|---|

2666.94 | 8.4470 | –113.25 | –0.31756 | 7.0728 | 103.85 |

## 4. The Influence of Cloud Cover on the Optimal Tilt Angle

#### 4.1. Power Variability at PNM Prosperity Site

**Figure 6.**(

**a**) Aerial view of Prosperity. The upper left corner shows a closer view of the panel array; (

**b**) Power signal generated by Prosperity on 1 November 2013.

**Figure 8.**(

**a**) Average energy production, December 2012–November 2013; and (

**b**) corresponding coefficient of variation.

#### 4.2. Sensitivity to Cloud Cover at Prosperity 35° Latitude

^{2}to a range from 1581 to 1607 kWh/m

^{2}. However, there is little difference in the optimal angles 34.2°–34.6° in the ten simulations. Moreover, the optimal cloudless angle (33.2°) fares well in producing the near optimal energy over the cloudy year compared to the optimal angles which account for cloud cover. The difference between the integrated yearly irradiance (kWh/m

^{2}) at the optimal angle (considering cloud angle) and the integrated yearly irradiance at the optimal cloudless angle is less than 0.03% in all ten simulations.

Simulation | Optimal Angle(β) | kWh/m^{2} at Optimal Angle | kWh/m^{2} at Optimal Cloudless Angle $33.{2}^{\xb0}$ |
---|---|---|---|

Simulation 1 | $34.{4}^{\xb0}$ | 1606.6 | 1606.3 |

Simulation 2 | $34.{6}^{\xb0}$ | 1598.2 | 1597.9 |

Simulation 3 | $34.{4}^{\xb0}$ | 1607.1 | 1606.8 |

Simulation 4 | $34.{4}^{\xb0}$ | 1581.8 | 1581.5 |

Simulation 5 | $34.{4}^{\xb0}$ | 1601.9 | 1601.6 |

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Torres, D.J.; Crichigno, J. Influence of Reflectivity and Cloud Cover on the Optimal TiltAngle of Solar Panels. *Resources* **2015**, *4*, 736-750.
https://doi.org/10.3390/resources4040736

**AMA Style**

Torres DJ, Crichigno J. Influence of Reflectivity and Cloud Cover on the Optimal TiltAngle of Solar Panels. *Resources*. 2015; 4(4):736-750.
https://doi.org/10.3390/resources4040736

**Chicago/Turabian Style**

Torres, David J., and Jorge Crichigno. 2015. "Influence of Reflectivity and Cloud Cover on the Optimal TiltAngle of Solar Panels" *Resources* 4, no. 4: 736-750.
https://doi.org/10.3390/resources4040736