# The Effect of Azimuth and Tilt Angle Changes on the Energy Balance of Photovoltaic System Installed in the Southern Slovakia Region

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## Featured Application

**The research results are beneficial for precise prediction of the photovoltaic systems energy production. Presented mathematical models are very valuable for practice, which provide a platform for optimalisation of photovoltaic systems operating conditions in the regions of Central Europe. The research results are focused on the detailed analyzation of the tilt and azimuth angle influence on the photovoltaic energy production. The results are used easily for creation of application for smart devices, because simplified mathematical model equations require the entry of a minimum number of parameters by the user.**

## Abstract

## 1. Introduction

^{−2}[2]. The energy consumption is five times less than the amount of energy captured from the Sun. Based on the presented facts, it is clear that the solar energy can be transformed into electric and thermal energy with positive energetic and economical effect. Nowadays one of the most important reasons for installation of solar systems is their positive ecological aspect and sustainability [3]. Authors [4,5] observed that solar energy offers one of the best solutions to the problem of climate change.

## 2. Materials and Methods

_{A}is the angle between the horizontal and the line to the Sun (0° ≤ α

_{A}≤ 90°). The complement of this angle is the zenith angle (α

_{Z}). It is defined by the vertical and the line to the Sun (i.e., the angle of incidence of beam radiation on a horizontal surface); β is the tilt angle and γ is the azimuth angle which determine the azimuth orientation of the solar module.

_{S}: angular displacement from South of the projection of beam radiation on the horizontal plane. (In general, α

_{S}= 0 is South, α

_{S}< 0 is East, and α

_{S}> 0 is West).

^{2}(Figure 2).

^{®}version R2015b. All data were analysed using analysis of variance (ANOVA). The comparison of the averages was carried out by Duncan’s test with a 95% confidence level. The arithmetic averages, medians, and standard error of the arithmetic average were computed from the data. The depth data analysis and the data extraction were applied on the data files obtained from real PV system for creating a mathematical model.

## 3. Results

^{2}= 0.9999.

^{®}was used for detection of the optimal tilt angle for examined PV system installed in southern Slovakia region. The maximum of energy production was obtained for the tilt angle 34.5°.

^{2}, standard deviations σ, standard error of arithmetic average $\overline{\mathsf{\delta}}$(t), and the median were identified. Selected summary results are presented in Table 2.

^{2}= 0.998.

^{®}version R 2015a was used for the mathematical model creation. Three-dimensional dependencies were created by mathematical software. The suitable polynomial approximation was applied to the relation of data files. For the three-dimensional relation, selected statistical parameters and regression coefficients of Equation (13) were calculated, which are summarized in Table 3.

## 4. Discussion

^{®}.

_{G}is solar radiation (W.m

^{−2}), A is area of PV system (m

^{2}), eff is efficiency of PV system (%), and T

_{m}is temperature of PV module (°C) [62]. The total amount of energy PV system production was identified as a product of the PV system power by using Equation (14) and selected time range.

_{type}coefficient for PV module type, k

_{const}is coefficient for PV system construction, k

_{tilt}coefficient for PV module tilt angle, k

_{azim}is coefficient for PV module azimuth angle, k

_{tcell}is coefficient for PV cell temperature, k

_{loss}coefficient for PV system loss, f

_{G}(t) is the function for solar radiation intensity in kW∙m

^{−2}, and P

_{inst}is installed power of PV system in kW

_{p.}

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Illustration to the definition of the tilt angle, the solar azimuth angle. and the azimuth orientation.

**Figure 3.**(

**a**) Monitoring device: The Solar-Log 2000; (

**b**) Voltage converters type Schüco Central Inverter type SGI 33.

**Figure 5.**(

**a**) The azimuth orientation of the model PV system; (

**b**) The recommended orientation (azimuth angle marked in blue color) for southern Slovakia.

**Figure 8.**Comparison of the experimental curve and the curve of the analytical model expressed by the transfer function.

**Figure 9.**Three-dimensional dependence before polynomial approximation—the influence of tilt angle on the electric energy production of the PV system.

**Figure 10.**Three-dimensional dependence after polynomial approximation—the influence of tilt angle on the electric energy production of PV system for calendar months.

**Figure 11.**Dependencies of energy produced by the PV system for azimuth angle from −90° to 0° in different calendar months.

**Figure 12.**Dependencies of energy produced by the PV system for azimuth angle from 0° to 90° in different calendar months.

**Figure 14.**Comparison of relations between the energy and azimuth angle with 90% confidence for azimuth angle range (−90–90°) and detail for range (−40–40°).

**Figure 15.**Three-dimensional dependence after polynomial approximation—the influence of azimuth angle on the electric energy production of PV system for calendar months.

**Table 1.**The statistical and regression characteristics of approximated polynomial function for three-dimensional dependence on Figure 10.

Sum of Squares | 1.735 × 10^{8} | ||
---|---|---|---|

Coefficient of Determination | 0.8723 | ||

Standard error of arithmeticaverage | 1.491 × 10^{3} | ||

Regression Equation | ${E}_{m}\left(t,\beta \right)=A+B\xb7t+C\xb7\beta +D\xb7{t}^{2}+E\xb7t\xb7\beta +F\xb7{\beta}^{2}$ | ||

Coefficient | Value | Minimum of the Range | Maximum of the Range |

A (kWh) | −1438 | −3063 | −185.8 |

B (kWh∙month^{−1}) | 56.33 | 12.4 | 100.3 |

C (kWh) | 4218 | 3785 | 4652 |

D (kWh∙month^{−2}) | −0.8993 | −1.315 | −0.4835 |

E (kWh∙month^{−1}) | 0.8507 | −2.278 | 3.979 |

F (kWh) | −340.9 | −371.6 | −310.2 |

**Table 2.**Average amount of energy ($\overline{E}$ ) produced by PV system with different azimuth orientation and tilt angle 35°, average energy produced per month (${\overline{E}}_{m}$ ).

Azimuth angle | −90° | −80° | −70° | −60° | −55° | −50° | −45° | −40° | −35° | −30° | −25° |

$\overline{E}$ (kWh) | 7190 | 7480 | 7750 | 7990 | 8090 | 8190 | 8280 | 8350 | 8420 | 8470 | 8520 |

Median (kWh) | 7415 | 7855 | 8275 | 8670 | 8840 | 8995 | 9150 | 9290 | 9415 | 9525 | 9605 |

Azimuth angle | −20° | −15° | −10° | −5° | 0° | 5° | 10° | 15° | 20° | 25° | 30° |

$\overline{E}$ (kWh) | 8560 | 8580 | 8590 | 8590 | 8580 | 8560 | 8540 | 8500 | 8450 | 8380 | 8310 |

Median (kWh) | 9675 | 9730 | 9775 | 9785 | 9780 | 9755 | 9715 | 9650 | 9565 | 9460 | 9345 |

Azimuth angle | 35° | 40° | 45° | 50° | 55° | 60° | 70° | 80° | 90° | ||

$\overline{E}$ (kWh) | 8230 | 8150 | 8050 | 7950 | 7830 | 7710 | 7450 | 7170 | 6870 | ||

Median (kWh) | 9220 | 9065 | 8905 | 8735 | 8555 | 8365 | 7965 | 7530 | 7080 | ||

Month | January | February | March | April | May | June | |||||

${\overline{E}}_{m}$ (kWh) | 2200.32 | 4094.19 | 8758.71 | 12280.65 | 12567.74 | 12793.55 | |||||

$\overline{\mathsf{\delta}}$(P) (kWh) | 52.38 | 84.16 | 143.67 | 129.42 | 69.67 | 52.78 | |||||

Month | July | August | September | October | November | December | |||||

${\overline{E}}_{m}$ (kWh) | 13225.81 | 12325.81 | 9221.29 | 5361.61 | 2778.71 | 1868.39 | |||||

$\overline{\mathsf{\delta}}$(P) (kWh) | 72.67 | 104.48 | 101.07 | 97.96 | 64.6 | 51.17 |

**Table 3.**The statistical and regression characteristics of approximated polynomial function for three-dimensional dependence in Figure 14.

Sum of Squares | 4.975 × 10^{8} | ||
---|---|---|---|

Coefficient of Determination | 0.9314 | ||

Standard error of arithmetic average | 1.1680 × 10^{3} | ||

Regression equation | ${E}_{m}\left(t,\gamma \right)=A+B\xb7t+C\xb7\gamma +\mathrm{D}\xb7{t}^{2}+E\xb7t\xb7\gamma +F{\gamma}^{2}$ | ||

Coefficient | Value | Minimum of the Range | Minimum of the Range |

A (kWh) | −2215 | −2656 | |

B (kWh∙month^{−1}) | −2.104 | −7.389 | 3.18 |

C (kWh) | 4965 | 4815 | 5116 |

D (kWh∙month^{−2}) | −0.1949 | −0.2453 | −0.1446 |

E (kWh∙month^{−1}) | −0.009663 | −0.7277 | 0.7084 |

F (kWh) | −396.7 | −408 | −385.4 |

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**MDPI and ACS Style**

Božiková, M.; Bilčík, M.; Madola, V.; Szabóová, T.; Kubík, Ľ.; Lendelová, J.; Cviklovič, V.
The Effect of Azimuth and Tilt Angle Changes on the Energy Balance of Photovoltaic System Installed in the Southern Slovakia Region. *Appl. Sci.* **2021**, *11*, 8998.
https://doi.org/10.3390/app11198998

**AMA Style**

Božiková M, Bilčík M, Madola V, Szabóová T, Kubík Ľ, Lendelová J, Cviklovič V.
The Effect of Azimuth and Tilt Angle Changes on the Energy Balance of Photovoltaic System Installed in the Southern Slovakia Region. *Applied Sciences*. 2021; 11(19):8998.
https://doi.org/10.3390/app11198998

**Chicago/Turabian Style**

Božiková, Monika, Matúš Bilčík, Vladimír Madola, Tímea Szabóová, Ľubomír Kubík, Jana Lendelová, and Vladimír Cviklovič.
2021. "The Effect of Azimuth and Tilt Angle Changes on the Energy Balance of Photovoltaic System Installed in the Southern Slovakia Region" *Applied Sciences* 11, no. 19: 8998.
https://doi.org/10.3390/app11198998