# Selecting Surface Inclination for Maximum Solar Power

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data and Methods

#### 2.1. Data Sets

- (i)
- GHI ≥ 0.19 W/m
^{2}; - (ii)
- GHI ≤ 1.12 × Isc;
- (iii)
- DHI ≤ 1.1 × GHI;
- (iv)
- DHI ≤ 0.8 × Isc, and
- (v)
- BHI ≤ Isc

_{s}

_{c}is the solar constant equal to 1366.1 W/m

^{2}[29]. Solar geometry parameters, such as azimuth angle (Az), zenith angle (Sza) and elevation (El) were estimated for all locations using astronomical calculations.

#### 2.2. Diffuse Irradiance Models

_{βγ}, which could be analyzed to direct DBI

_{βγ}, diffuse DI

_{βγ}and reflected from the ground RGI

_{βγ}components. A graphical representation of the relations between the irradiance components is provided in Figure 2.

_{βγ}, which are estimating the diffuse component. The simplest approach is to assume that the propagation of irradiance is isotropic. Anisotropic models also consider the components of diffuse irradiance coming from circumsolar and the horizon.

_{βγ}. For lower latitudes a constant isotropic model was used, which suggests a constant value of ρ = 0.2. Snow is the main factor that causes significant variations from the constant value. A simple approach in this case is to use an isotropic seasonal model, which changes monthly according to climatological conditions and calculated by the formula found at [38]. We have used this model for cities with latitudes higher than 50°, aiming to reduce the error introduced by snow covered conditions.

## 3. Results

#### 3.1. Model Comparison

^{2}/day. Higher deviation of optimum angle is found in Reykjavik, where Perez model provides a more than 2° higher angle than any other model, and this should be linked with the diffuse irradiance from the horizon, which is only considered at this model. On southern latitudes models’ estimations converge on optimal tilt angle. On the other hand, in terms of absolute energy values at these latitudes the differences are higher than in Northern areas. Perez models seem to constantly provide higher irradiance compared to the others, while Hay model has the lowest values in northern areas and ISO in southern. Reindl model is the one closest to the mean value of all four models, both in terms of mean energy and optimum tilt angle, in most cases, although with very slight differences from Hay. Hence, we will use Reindl model for the rest of the study, clarifying that there is an uncertainty entered by the selection of diffuse irradiance model, in the order of 0.7° in terms of inclination angle and 0.04 kWh/m

^{2}/day in terms of energy output. This uncertainty was calculated by the mean difference of Reindl’s model to other three for all cities.

#### 3.2. Optimum Angle per City

_{β}has been estimated for the whole dataset, for β in the range 1° to 90° with 1° step. Then, the sum of received irradiance at each β for the whole period, was calculated and the β with the higher value for each city was selected as optimal. The results are presented at Table 3, alongside with mean CMF and AOD at 550 nm for each city. The difference between latitude and optimal tilt angle is widen to the northern areas. This behavior should be explained mainly by the increase of cloud coverage at these areas. At southern areas, the tilt angles differences are smaller, because the latitude suggest closer to horizontal tilt and cloud coverage is lower. However, these areas receive more sunlight, hence these smaller variations could lead to significant energy gains. CMF is generally lower as latitude increases, at least in the region of Europe and North Africa. AOD on the other hand has higher values in lower latitudes in this region, which is explained by the dominance of Saharan dust on high aerosol episodes in the area [39,40].

_{β}and GHI, which represents the gain/loss of tilted surface, compared with a horizontal one. Both the cases of tilt angle of latitude and the optimal as calculated above are demonstrated. In addition, monthly values for all 25 cities are implemented at the plot. In wintertime, the relative gain is higher, but also the spread of ratios among cities. In general, areas with very low cloud coverage—at southern areas—have less relative gain, although the absolute values are higher. Areas at the north, where sunshine duration is very low during winter seem to have the biggest relative benefit, since the cloud coverage is almost permanent at this season. In spring and autumn, the relative gains of optimal instead of latitude angle are more obvious. In addition, at these seasons, the relative benefit of higher latitudes is larger than the southern ones. More interesting is that in summer months, almost in all cities, both setups provide less energy than the horizontal surface. This is explained because the Solar Zenith Angle of the periods of peak GHI are closest to 0 than to latitude in all cases, hence the horizontal surface receives more irradiance. The biggest disadvantage of having a fixed angle year-round is that during summer months the solar potential is not fully exploited, but summing the whole year, it is the preferred setup. Previous work (e.g., [8]) have investigated the possibility of 2 or 4 changes of the fixed angle, around the year in order to increase the benefits.

_{opt}= f(φ)

_{opt}= 0.34 × φ + 15.72

^{2}= 0.937 and RMSE = 1.81. Hence, this equation could be in use.

_{op}

_{t}= f(φ)

_{op}

_{t}= 0.65 × φ − 15.72

^{2}= 0.945 and RMSE = 1.71. The quantity φ − β that we parameterized, is the component of the tilt angle that is depended on atmospheric conditions, since the astronomical calculations suggest the latitude as preferred tilt angle for clear skies conditions.

#### 3.3. Cloud Effect

_{opt}= f(CMF)

_{opt}= 30.24 × CMF

^{2}− 112.16 × CMF + 78.12

^{2}= 0.962 and RMSE = 1.48. Thus, we should interpret that this approach is a more accurate representation of the actual state. For cities with lower cloud coverage—which are also at lower latitudes—change of tilt angle is more accurately predicted, except for Tamanrasset, where desert dust aerosols add some variation to the calculations. Bucharest and Ljubljana show lower change of the tilt angle than expected from the regression of CMF of the area. Finally, at Stockholm the actual change is more than 5° greater than predicted from the CMF equation (19.4° instead of 24.7°), where the latitude parameterization provided a more accurate estimation for the area (24.1°).

^{2}/day, for Ljubljana is 0.04 kWh/m

^{2}/day, for Bucharest is 0.03 kWh/m

^{2}/day and for Tamanrasset is 0.01 kWh/m

^{2}/day.

#### 3.4. Energy/Financial Profits

^{2}in Alvsbyn (977 and 1000 kWh/m

^{2}for latitude and optimal angles, respectively). For the estimation of the financial profit, the feed-in tariffs (FIT) retrieved from the European Environment Agency (https://www.eea.europa.eu/ (accessed on 1 April 2022)) were incorporated in the calculations, indicating the policy making dimension into the PV technology promotion and penetration into society. The above statement is reflected in the wide price range which is from 0.019 to 0.19 euro/kWh with the lowest FIT occurring in cities such as Alvsbyn, Stockholm, Copenhagen, Aswan and Cairo (<0.04 euro/kWh) and the highest prices in Madrid, Rome, Valletta, Kyiv, Warsaw and Dublin (>0.12 euro/kWh). Especially in Dublin, the revenue difference is close to 2750 euro for 1 MWp capacity on an annual basis, making the optimal angle identification a key parameter for the financial profit optimization.

## 4. Discussion

## 5. Conclusions

^{2}/day energy output.

^{2}—was found between 0.93 and 0.97. Better statistics for the parameterization using the CMF, but with some regions having considerable differences. These equations could be employed anywhere in the same region to select the optimal tilt angle. In general, installations at higher latitudes could benefit from considerably lower tilt angle than the latitude. Higher cloud coverage also leads to optimal tilt angles closer to the horizontal. At middle latitudes the absolute differences of the tilt angles were smaller but lead to considerable energy benefits.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{0}and α

_{1}are calculated as:

_{1}and F

_{2}are estimated by parameter ε and Δ as:

_{Ζ}.

_{11}, F

_{12}, F

_{13}, F

_{22}, F

_{23}are calculated and finally the diffuse irradiance DI

_{βγPerez}.

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**Figure 2.**Graphical Presentation of the β tilt angle and γ azimuthal angle, alongside of Diffuse, Direct Beam and Reflective components of received solar irradiance.

**Figure 3.**Mean Daily Total Radiation for Athens, Nicosia, Bergen and Reykjavik, for inclinations of 0° to 90° as calculated by different models.

**Figure 4.**Ratio of GI

_{β}on inclined surface to horizontal one, for each month. Mean value for each month for β = Latitude and β = optimum angle, along with error bars representing the 1σ standard deviation of ratios for each month. Small points are all 25 cities ratios for each month.

**Figure 5.**Optimum tilt angle with 1σ against latitude for all cities and corresponding linear regression (

**a**). Difference of Latitude and Optimum tilt angle with 1 σ against latitude for all cities and corresponding linear regression (

**b**).

**Figure 6.**Optimum tilt angle with 1 σ against latitude for all cities and corresponding linear regression.

**Figure 7.**Annual solar energy potential and revenue (for 1 MWp capacity) differences and percentage differences for all cities included in the study, between latitude and optimum tilt angle installations.

**Table 1.**Mean Optimum Angle for test cities as calculated with each model, the mean and standard deviation σ for each city as calculated from model’s prognosis.

Optimum Angle | Hay | ISO | Reindl | Perez | Mean | σ |
---|---|---|---|---|---|---|

Nicosia | 28.8 | 28.47 | 29.67 | 31.27 | 29.55 | 1.25 |

Athens | 28.33 | 28.27 | 29.13 | 30.87 | 29.15 | 1.21 |

Bergen | 37.07 | 38.13 | 38.8 | 40.40 | 38.60 | 1.40 |

Reykjavik | 33.53 | 35.27 | 35.67 | 37.87 | 35.59 | 1.78 |

**Table 2.**Daily Energy outcome at the suggested optimum tilt angle for test cities as calculated with each model, the mean and standard deviation σ for each city as calculated from model’s prognosis.

kWh/m^{2}/Day | Hay | ISO | Reindl | Perez | Mean | σ |
---|---|---|---|---|---|---|

Nicosia | 5.81 | 5.71 | 5.83 | 5.93 | 5.82 | 0.09 |

Athens | 5.15 | 5.08 | 5.17 | 5.26 | 5.17 | 0.07 |

Bergen | 2.75 | 2.69 | 2.76 | 2.85 | 2.76 | 0.07 |

Reykjavik | 2.20 | 2.16 | 2.21 | 2.28 | 2.21 | 0.05 |

**Table 3.**Latitude and Longitude of Cities used in the study, alongside with mean CMF and AOD and the calculated optimum tilt angle β for the period 2005–2019.

City | Country | Lat | Long | β_{opt} | <CMF> | <AOD> |
---|---|---|---|---|---|---|

Tamanrasset | Algeria | 22.78 | 5.51 | 21 | 0.869 | 0.312 |

Aswan | Egypt | 24.09 | 32.89 | 23 | 0.914 | 0.310 |

Cairo | Egypt | 30.03 | 31.24 | 25 | 0.863 | 0.282 |

Marrakesh | Morocco | 31.63 | −7.98 | 28 | 0.841 | 0.205 |

Nicosia | Cyprus | 35.17 | 33.36 | 29 | 0.840 | 0.245 |

Valletta | Malta | 35.89 | 14.51 | 29 | 0.835 | 0.238 |

Athens | Greece | 37.98 | 23.72 | 29 | 0.784 | 0.222 |

Madrid | Spain | 40.43 | −3.70 | 31 | 0.783 | 0.140 |

Thessaloniki | Greece | 40.65 | 22.92 | 31 | 0.744 | 0.232 |

Rome | Italy | 41.88 | 12.47 | 32 | 0.783 | 0.197 |

Bucharest | Romana | 44.44 | 26.08 | 32 | 0.683 | 0.224 |

Ljubljana | Slovenia | 46.05 | 14.50 | 32 | 0.664 | 0.196 |

Paris | France | 48.94 | 2.41 | 32 | 0.642 | 0.178 |

Kyiv | Ukraine | 50.41 | 3050 | 33 | 0.641 | 0.190 |

Warsaw | Poland | 52.23 | 21.00 | 33 | 0.616 | 0.192 |

Berlin | Germany | 52.52 | 13.37 | 33 | 0.623 | 0.180 |

Dublin | Ireland | 53.34 | −6.28 | 34 | 0.636 | 0.170 |

Copenhagen | Denmark | 55.71 | 12.54 | 34 | 0.626 | 0.161 |

Moscow | Moscow | 55.8 | 37.59 | 34 | 0.578 | 0.176 |

Riga | Lithuania | 56.95 | 24.11 | 35 | 0.625 | 0.158 |

Stockholm | Sweden | 59.27 | 18.02 | 35 | 0.633 | 0.137 |

Helsinki | Finland | 60.19 | 24.93 | 36 | 0.565 | 0.137 |

Bergen | Norway | 60.37 | 5.32 | 37 | 0.587 | 0.145 |

Reykjavik | Iceland | 64.16 | −21.95 | 37 | 0.511 | 0.138 |

Alvsbyn | Norway | 65.68 | 20.99 | 40 | 0.593 | 0.101 |

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**MDPI and ACS Style**

Raptis, I.-P.; Moustaka, A.; Kosmopoulos, P.; Kazadzis, S.
Selecting Surface Inclination for Maximum Solar Power. *Energies* **2022**, *15*, 4784.
https://doi.org/10.3390/en15134784

**AMA Style**

Raptis I-P, Moustaka A, Kosmopoulos P, Kazadzis S.
Selecting Surface Inclination for Maximum Solar Power. *Energies*. 2022; 15(13):4784.
https://doi.org/10.3390/en15134784

**Chicago/Turabian Style**

Raptis, Ioannis-Panagiotis, Anna Moustaka, Panagiotis Kosmopoulos, and Stelios Kazadzis.
2022. "Selecting Surface Inclination for Maximum Solar Power" *Energies* 15, no. 13: 4784.
https://doi.org/10.3390/en15134784