# Tilt Angle Adjustment for Incident Solar Energy Increase: A Case Study for Europe

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background

_{y}), towards the ground, which is considered to provide the best performances over the entire year. However, using β

_{y}will not ensure the highest amount of incident solar irradiation on the considered surface throughout the various seasons across the year. Consequently, one possible solution to increase the total amount of incident solar irradiation would be the possibility to manually adjust the tilt angle of the surface, considering other optimum tilt angles, such as bi-annual optimum tilt angles (β

_{b}), seasonal optimum tilt angles (β

_{s}) or even monthly optimum tilt angles (β

_{m}).

_{s}determined for Lahore, Pakistan. In [16], the authors have determined that adjusting the PV panels’ tilt angle according to β

_{b}, β

_{s}, and β

_{m}will increase the annual incident radiation with 10.5%, 10.7% and 11.7%. An 8% increase in incident radiation was determined when adjusting the tilt angle two times per year in Zahedan, Iran [17], and twelve times per year (monthly) in Madinah, Saudi Arabia [18]. In [19], the electricity output of a 1 MW sample PV plant was evaluated for various locations in Turkey, considering the manual adjustment of the PV panels’ tilt angle according to the bi-annual, seasonal, and monthly optimum tilt angles, and the increase in the energy output was 3.21–3.71% (for β

_{b}angles), 3.64–4.26% (for β

_{s}angles), and of 4.53–5.3% (for β

_{m}angles), respectively. Although these gains may seem insignificant, the authors have determined that the monthly adjustment of the tilt angle will increase the internal return rate by 0.7–0.9% and reduce the discounted payback period by 8 to 10 months [20]. A monthly optimization of the PV panels’ tilt angles using the levelized cost of energy (LCOE) criteria was performed in [21] for the cities of Tripoli (Lebanon), Belfort (France), and Tantan (Morocco), and it was found that the LCOE has decreased by 4.32%, 3.73%, and 4.35% when a monthly adjustment of the PV panels’ tilt angle was performed in those specific locations.

#### 1.2. Objectives and Paper Structure

_{b}and β

_{s}angles, two different scenarios for the bi-annual and seasonal time intervals will be considered, calendar-based and astronomical-based, respectively.

## 2. Materials and Methods

_{b}), four times per year (using seasonal optimum tilt angles—β

_{s}), and twelve times per year (using monthly optimum tilt angles—β

_{m}) in terms of total solar irradiation incident on a surface. To determine these various optimum tilt angles, several mathematical models from the literature were selected; the mathematical expressions of each model are presented in Table 1.

_{spring}) and autumn angles (β

_{autumn}), while for the summer and winter optimum angles (β

_{summer}, β

_{winter}), the maximum value of the Earth’s declination (δ = 23.45) should be subtracted or added from the considered location latitude.

- Read input data: time, location (latitude and longitude), daily total radiation on a horizontal surface (H), and constant values (G
_{sc}, ρ); - Set the surface’s tilt angle (β = 0°);
- Compute the Earth’s declination angle (δ) and solar geometry (ω
_{s}, ω’_{s}); - Determine the solar irradiation components (beam, diffuse, and reflected) incident to the surface;
- Compute and store the value of the total incident radiation to the surface (H
_{t}); - Increase the tilt angle with 1° and run steps 3 ÷ 5 again until β = 90°;
- Compare the values of the H
_{t}and select β_{opt}as the angle for which H_{t}has the highest value.

_{b}), diffuse (H

_{d}), and reflected (H

_{r}), considering the equation proposed by Liu and Jordan in [26], and presented in Table 1. In this model, an important parameter is the ratio between the average values of the diffuse component of the irradiation and the daily total irradiation on a horizontal (H

_{d}/$H$) surface, which depends on the clearness index (K). Several expressions were proposed for this correlation, but for this study, the authors opted for the multi-location model proposed for Europe by Bortolini in [31], which is described by Equation (1).

## 3. Results and Discussions

#### 3.1. Optimum Tilt Angles for Europe

- a.
- Yearly optimum tilt angles (β
_{y})

_{y}estimation: the empirical model proposed by Patko (PTK) [23], the regression model proposed by C. Martin (CM) [24], the regression model proposed by Modarresi (MOD) [25] and the radiation search-based method (SBM). The yearly optimum tilt angles obtained are presented in Table 2.

_{y}, of all four models. A quite good correlation can be observed in values provided by the model proposed by C. Martin and the search-based method.

- b.
- Bi-annual optimum tilt angles (β
_{b})

- Calendar-based: the warm season extends from 1 March to 30 September, while the cold season extends from 1 October to 28 February.
- Astronomical definition: the warm season extends between the spring and autumn equinox (from 20 March to 22 September), while the cold season extends from 23 September to 19 March.

- c.
- Seasonal optimum tilt angles (β
_{s})

_{s}) is determined for each season.

_{s}for each season using the mathematical expressions previously presented in Table 1.

- Calendar-based: Spring—1 March to 30 May; Summer—1 June to 31 August; Autumn—1 September to 30 November; Winter—1 December to 28 February;
- Astronomical definition: Spring—20 March to 20 June, Summer—21 June to 22 September; Autumn—23 September to 20 December; Winter—21 December to 19 March.

_{s}are presented in Table 4 for the considered latitudes.

_{s}in spring, and the lowest values in autumn. Changing how seasons are delimited has a significant impact on the optimum tilt values. As one can notice in Table 4, when using the astronomical definition β

_{s}, there will be lower values in winter and spring, and higher values in summer and autumn, compared to those obtained under the calendar-based scenario.

- d.
- Monthly optimum tilt angles (β
_{m})

_{m}are presented in Table 5 for latitudes between 35° and 45°, and in Table 6 for the other latitudes.

#### 3.2. Efficiency of Using Bi-Annual, Seasonal, and Monthly Optimum Tilt Angles

^{2}, apart from the model proposed by Patko, which underperforms at latitudes higher than 40°, as the differences in terms of incident irradiation grow to over 15 kWh/m

^{2}, the maximum difference being over 32 kWh/m

^{2}at 55° latitude. A very good correlation can be observed between the results obtained using the angle determined by SBM and the regression model proposed by C. Martin.

^{2}) are reported at 35° and 40° latitude, with a maximum of 157.2 kWh/m

^{2}at 35°, while at lower and higher latitudes, the gains of incident radiation are less important.

^{2}, at 35° latitude and even smaller in the rest.

#### 3.3. Analysis of the Impact of How the Seasons Are Defined

_{ba}) will ensure a higher total incident radiation even when the calendar-based seasonal optimum tilt angles (β

_{sc}) are used, as shown in Figure 3.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Differences in incident irradiation between the use of SBM angles and those determined by PTK, CM, and MOD models.

**Figure 2.**Relative incident irradiation gains for bi-annual, seasonal, and monthly tilt angle optimization.

**Figure 3.**Comparison between the relative increase in incident radiation when β

_{ba}and β

_{sc}angles are used.

Optimum Angle | Mathematical Expression | Author | |||
---|---|---|---|---|---|

Yearly—β_{y} | ${\beta}_{y}=\varphi $ | Patko [23] | |||

${\beta}_{y}=-0.007209\cdot {\varphi}^{2}+1.096\cdot \varphi +2.373$ | C. Martin [24] | ||||

0.83 ϕ + 0.62 | Modarressi [25] | ||||

${H}_{t}={H}_{b}\cdot {R}_{b}+{H}_{d}\cdot {R}_{d}+\rho H\cdot {R}_{r}$ | SBM–Liu & Jordan [26] | ||||

Bi-annual—β_{b} | ${H}_{t}={H}_{b}\cdot {R}_{b}+{H}_{d}\cdot {R}_{d}+\rho H\cdot {R}_{r}$ | SBM–Liu & Jordan [26] | |||

Seasonal—β_{s} | ${\beta}_{spring}={\beta}_{autumn}=\varphi $ ${\beta}_{summer}=\varphi -23.45$$,{\beta}_{w\mathrm{int}er}=\varphi +23.45$ | Patko [23] | |||

${\beta}_{spring}=0.80\varphi -15.67$ ${\beta}_{summer}=0.79\varphi -21.99$ ${\beta}_{autumn}=0.86\varphi +16.78$ ${\beta}_{winter}=0.87\varphi +23.46$ | Modarressi [25] | ||||

${H}_{t}={H}_{b}\cdot {R}_{b}+{H}_{d}\cdot {R}_{d}+\rho H\cdot {R}_{r}$ | SBM–Liu & Jordan [26] | ||||

Monthly—β_{m} | ${\beta}_{m1-3}=60.00012+1.49986M-3.49996{M}^{2}+$ $+\left(\varphi -30\right)\left(0.7901+0.01749M+0.0165{M}^{2}\right)$ | El-Kassaby [27] | |||

${\beta}_{m4-6}=216.0786-72.03219M+6.00312{M}^{2}+$ $\left(\varphi -40\right)\left(1.07515+0.11244M-0.03749{M}^{2}\right)$ | |||||

${\beta}_{m7-9}=29.11831-20.52981M+2.50186{M}^{2}+$ $+\left(\varphi -50\right)\left(-11.17256+2.70569M-0.15035{M}^{2}\right)$ | |||||

${\beta}_{m10-12}=-441.2385+84.54322M-3.50196{M}^{2}+$ $+\left(\varphi -40\right)\left(4.2137-054834M+0.0223{M}^{2}\right)$ | |||||

January | 0.88 ϕ + 27.61 | July | 0.78 ϕ–26.7 | Modarressi [25] | |

February | 0.86 ϕ + 17.88 | August | 0.80 ϕ–16.66 | ||

March | 0.84 ϕ + 3.83 | September | 0.82 ϕ–2.08 | ||

April | 0.81 ϕ–11.52 | October | 0.85 ϕ + 13.23 | ||

May | 0.78 ϕ–23.61 | November | 0.88 ϕ + 25.14 | ||

June | 0.77 ϕ–29.15 | December | 0.89 ϕ + 30.45 | ||

${H}_{t}={H}_{b}\cdot {R}_{b}+{H}_{d}\cdot {R}_{d}+\rho H\cdot {R}_{r}$ | SBM–Liu&Jordan [26] |

Model | Latitude (°) | |||||
---|---|---|---|---|---|---|

35 | 40 | 45 | 50 | 55 | 60 | |

PTK | 35 | 40 | 45 | 50 | 55 | 60 |

CM | 31.9 | 34.7 | 37.1 | 39.2 | 40.9 | 42.2 |

MOD | 29.7 | 33.8 | 38 | 42.1 | 46.3 | 50.4 |

SBM | 31 | 31 | 34 | 37 | 39 | 45 |

Season | Model | Latitude (°) | |||||
---|---|---|---|---|---|---|---|

35 | 40 | 45 | 50 | 55 | 60 | ||

Warm | cSBM | 10 | 13 | 17 | 20 | 23 | 26 |

aSBM | 11 | 16 | 20 | 24 | 28 | 33 | |

Cold | cSBM | 56 | 59 | 64 | 67 | 72 | 78 |

aSBM | 55 | 57 | 62 | 65 | 69 | 76 |

Season | Model | Latitude (°) | |||||
---|---|---|---|---|---|---|---|

35 | 40 | 45 | 50 | 55 | 60 | ||

Spring | PTK | 35.0 | 40.0 | 45.0 | 50.0 | 55.0 | 60.0 |

MOD | 12.3 | 16.3 | 20.3 | 24.3 | 28.3 | 32.3 | |

cSBM | 19.0 | 21.0 | 25.0 | 28.0 | 31.0 | 33.0 | |

aSBM | 8.0 | 11.0 | 15.0 | 18.0 | 21.0 | 22.0 | |

Summer | PTK | 11.5 | 16.5 | 21.5 | 26.5 | 31.5 | 36.5 |

MOD | 5.7 | 9.6 | 13.6 | 17.5 | 21.5 | 25.4 | |

cSBM | 5.0 | 10.0 | 16.0 | 20.0 | 25.0 | 30.0 | |

aSBM | 14.0 | 19.0 | 26.0 | 31.0 | 36.0 | 42.0 | |

Autumn | PTK | 35.0 | 40.0 | 45.0 | 50.0 | 55.0 | 60.0 |

MOD | 46.9 | 51.2 | 55.5 | 59.8 | 64.1 | 68.4 | |

cSBM | 51.0 | 54.0 | 60.0 | 64.0 | 70.0 | 77.0 | |

aSBM | 57.0 | 60.0 | 65.0 | 68.0 | 73.0 | 80.0 | |

Winter | PTK | 58.5 | 63.5 | 68.5 | 73.5 | 78.5 | 83.5 |

MOD | 53.9 | 58.3 | 62.6 | 67.0 | 71.3 | 75.7 | |

cSBM | 58.0 | 60.0 | 63.0 | 66.0 | 69.0 | 74.0 | |

aSBM | 54.0 | 55.0 | 59.0 | 61.0 | 64.0 | 69.0 |

Month | Latitude (°) | ||||||||
---|---|---|---|---|---|---|---|---|---|

35 | 40 | 45 | |||||||

KSB | MOD | SBM | KSB | MOD | SBM | KSB | MOD | SBM | |

January | 62.1 | 58.4 | 61 | 66.2 | 62.8 | 63 | 70.4 | 67.2 | 67 |

February | 53.5 | 48.0 | 52 | 57.9 | 52.3 | 55 | 62.4 | 56.6 | 59 |

March | 38.0 | 33.2 | 38 | 42.9 | 37.4 | 41 | 47.9 | 41.6 | 46 |

April | 19.4 | 16.8 | 19 | 24 | 20.9 | 23 | 28.6 | 24.9 | 28 |

May | 2.5 | 3.7 | 3 | 6 | 7.6 | 8 | 9.5 | 11.5 | 12 |

June | 0 | 0 | 0 | 0 | 1.7 | 0 | 2 | 5.5 | 2 |

July | 2 | 0.6 | 0 | 4 | 4.5 | 3 | 6 | 8.4 | 8 |

August | 12.2 | 11.3 | 13 | 16.5 | 15.3 | 18 | 20.7 | 19.3 | 23 |

September | 32.0 | 26.6 | 31 | 37 | 30.7 | 35 | 42 | 34.8 | 41 |

October | 49.2 | 43.0 | 48 | 54 | 47.2 | 51 | 58.8 | 51.5 | 56 |

November | 60.6 | 55.9 | 59 | 65 | 60.3 | 61 | 69.4 | 64.7 | 65 |

December | 64.8 | 61.6 | 63 | 69 | 66.1 | 65 | 73.2 | 70.5 | 69 |

Month | Latitude (°) | ||||||||
---|---|---|---|---|---|---|---|---|---|

50 | 55 | 60 | |||||||

KSB | MOD | SBM | KSB | MOD | SBM | KSB | MOD | SBM | |

January | 74.5 | 71.6 | 69 | 78.6 | 76 | 73 | 82.7 | 80.4 | 79 |

February | 66.8 | 60.9 | 62 | 71.3 | 65.2 | 66 | 75.7 | 69.5 | 73 |

March | 52.8 | 45.8 | 50 | 57.8 | 50 | 54 | 62.7 | 54.2 | 61 |

April | 33.3 | 29.0 | 32 | 37.9 | 33 | 37 | 42.5 | 37.1 | 41 |

May | 13 | 15.4 | 16 | 16.5 | 19.3 | 20 | 20 | 23.2 | 24 |

June | 4 | 9.4 | 6 | 6 | 13.2 | 10 | 8 | 17.1 | 14 |

July | 8 | 12.3 | 11 | 10 | 16.2 | 15 | 12 | 20.1 | 18 |

August | 25 | 23.3 | 26 | 29.3 | 27.3 | 30 | 33.5 | 31.3 | 33 |

September | 47 | 38.9 | 44 | 52 | 43 | 49 | 57 | 47.1 | 54 |

October | 63.6 | 55.7 | 60 | 68.4 | 60 | 65 | 73.2 | 64.2 | 70 |

November | 73.8 | 69.1 | 68 | 78.2 | 73.5 | 73 | 82.6 | 77.9 | 79 |

December | 77.4 | 75 | 72 | 81.7 | 79.4 | 75 | 85.9 | 83.9 | 81 |

Lat. (°) | H_{t} (kWh/m^{2}) | |||||
---|---|---|---|---|---|---|

Yearly Optimum Angles | Bi-Annual Angles | |||||

PTK | CM | MOD | SBM | cSBM | aSBM | |

35 | 2186.44 | 2190.52 | 2190.45 | 2190.79 | 2302.63 | 2320.92 |

40 | 1765.49 | 1779.80 | 1781.07 | 1783.19 | 1858.03 | 1871.08 |

45 | 1666.12 | 1686.01 | 1684.95 | 1687.43 | 1747.52 | 1763.53 |

50 | 1317.05 | 1341.76 | 1338.35 | 1342.65 | 1382.49 | 1395.53 |

55 | 1165.98 | 1197.42 | 1191.35 | 1197.78 | 1228.83 | 1242.14 |

60 | 1076.49 | 1103.97 | 1100.60 | 1104.63 | 1130.65 | 1149.15 |

Lat. (°) | H_{t} (kWh/m^{2}) | ||||||
---|---|---|---|---|---|---|---|

Seasonal Angles | Monthly Angles | ||||||

PTK | MOD | cSBM | aSBM | KSB | MOD | SBM | |

35 | 2291.47 | 2314.15 | 2318.21 | 2319.33 | 2347.50 | 2345.40 | 2348.01 |

40 | 1846.50 | 1868.01 | 1868.76 | 1869.56 | 1890.84 | 1890.67 | 1891.98 |

45 | 1740.10 | 1758.84 | 1758.21 | 1759.80 | 1782.51 | 1781.63 | 1783.68 |

50 | 1372.88 | 1391.60 | 1390.50 | 1391.23 | 1409.10 | 1409.27 | 1410.68 |

55 | 1217.63 | 1238.24 | 1235.39 | 1238.97 | 1253.77 | 1254.12 | 1255.44 |

60 | 1125.91 | 1142.66 | 1137.19 | 1141.12 | 1162.10 | 1161.24 | 1163.55 |

Angle | ΔH_{t} (kWh/m^{2}) | |||||
---|---|---|---|---|---|---|

Latitude (°) | ||||||

35 | 40 | 45 | 50 | 55 | 60 | |

β_{bc} | 111.8 | 74.8 | 60.1 | 39.8 | 31.1 | 26.0 |

β_{ba} | 130.1 | 87.9 | 76.1 | 52.9 | 44.4 | 44.5 |

β_{sc} | 127.4 | 85.6 | 70.8 | 47.8 | 37.6 | 32.6 |

β_{sa} | 128.5 | 86.4 | 72.4 | 48.6 | 41.2 | 36.5 |

β_{m} | 157.2 | 108.8 | 96.2 | 68.0 | 57.7 | 58.9 |

_{bc}, β

_{ba}—bi-annual optimum tilt angle for calendar-based seasons and astronomical-based seasons. β

_{sc}, β

_{sa}—seasonal optimum tilt angle for calendar-based seasons and astronomical-based seasons. β

_{m}—monthly optimum tilt angle.

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**MDPI and ACS Style**

Machidon, D.; Istrate, M.
Tilt Angle Adjustment for Incident Solar Energy Increase: A Case Study for Europe. *Sustainability* **2023**, *15*, 7015.
https://doi.org/10.3390/su15087015

**AMA Style**

Machidon D, Istrate M.
Tilt Angle Adjustment for Incident Solar Energy Increase: A Case Study for Europe. *Sustainability*. 2023; 15(8):7015.
https://doi.org/10.3390/su15087015

**Chicago/Turabian Style**

Machidon, Dragos, and Marcel Istrate.
2023. "Tilt Angle Adjustment for Incident Solar Energy Increase: A Case Study for Europe" *Sustainability* 15, no. 8: 7015.
https://doi.org/10.3390/su15087015