# Analysis of the Influence of Terrain Orientation on the Design of PV Facilities with Single-Axis Trackers

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Astronomical Bases and Irradiance Model

_{B}, the second term refers to the diffuse irradiance I

_{D}and the third term corresponds to the reflected component, taking into account the visible soil fraction and the albedo, $\rho $. I

_{B}and I

_{D}are determined by the Collares–Pereira model [38].

#### 2.2. Optimisation of Collector Orientation

#### 2.3. Backtracking

#### 2.4. Software Applications for Analysis

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$\overrightarrow{a}$ | projection vector of the solar panel being studied for shading |

$\overrightarrow{d}$ | distance between the centers of two adjacent solar collectors |

$\overrightarrow{e}$ | unit vector contained in the collector rotation axis |

$GCR$ | ground cover ratio |

$h$ | collector width |

$\overrightarrow{i},\overrightarrow{j},\overrightarrow{k}$ | unit vectors associated with a local Cartesian system |

$I$ | global solar irradiance on the tilted collector |

${I}_{B}$ | direct solar irradiance on the horizontal plane |

${I}_{D}$ | diffuse solar irradiance |

${I}_{OH}$ | extraterrestrial irradiance |

$\overrightarrow{n}$ | normal vector to the surface |

$\overrightarrow{{n}^{\prime}}$ | optimal normal vector to the surface |

${{n}^{\prime}}_{x},{{n}^{\prime}}_{y},{{n}^{\prime}}_{z}$ | components of optimal normal vector to the surface |

$\overrightarrow{{n}_{T}}$ | normal terrain vector |

$\overrightarrow{p}$ | unit vector perpendicular to ground normal vector $\overrightarrow{e}$ |

$\overrightarrow{q}$ | perpendicular vector to $\overrightarrow{{n}_{T}}$ and $\overrightarrow{e}$ |

$\overrightarrow{s}$ | solar vector |

${s}_{x},{s}_{y},{s}_{z}$ | components of solar vector |

$\overrightarrow{{s}^{\prime}}$ | optimal solar vector |

${{s}^{\prime}}_{x},s{{s}^{\prime}}_{y},{{s}^{\prime}}_{z}$ | components of optimal solar vector |

$\overrightarrow{u}$ | irradiance gradient |

Greek Letters | |

α | elevation angle of the collector |

$\beta $ | inclination angle of the terrain |

γ | azimuth angle of the collector rotation axis |

$\delta $ | solar declination |

$\xi $ | inclination angle of the collector |

$\theta $ | angle of incidence of sunbeams on the inclined plane |

${\theta}_{z}$ | solar zenith angle |

$\lambda ,\mu ,v$ | Lagrange multipliers |

$\rho $ | albedo |

$\tau $ | scalar multiplying solar vector to accomplish parallelogram rule |

$\phi $ | latitude |

$\Phi $ | Lagrange function |

$\chi $ | azimuth of the terrain |

$\Omega $ | Earth’s rotation speed |

## References

- Carballo, J.A.; Bonilla, J.; Roca, L.; Berenguel, M. New low-cost solar tracking system based on open source hardware for educational purposes. Sol. Energy
**2018**, 174, 826–836. [Google Scholar] [CrossRef] - United Nations. The Sustainable Development Goals Report 2019; United Nations Publications (Department of Economic and Social Affairs): Herndon, VA, USA, 2019. [Google Scholar]
- Ribó-Pérez, D.; Van der Weijde, A.H.; Álvarez-Bel, C. Effects of self-generation in imperfectly competitive electricity markets: The case of Spain. Energy Policy
**2019**, 133, 110920. [Google Scholar] [CrossRef] - International Renewable Energy Agency. Renewable Power Generation Costs in 2019; International Renewable Energy Agency: Abu Dhabi, UAE, 2020; ISBN 978-92-9260-244-4. [Google Scholar]
- Kavlak, G.; McNerney, J.; Trancik, J.E. Evaluating the causes of cost reduction in photovoltaic modules. Energy Policy
**2018**, 123, 700–710. [Google Scholar] [CrossRef] [Green Version] - Hua, Z.; Ma, C.; Lian, J.; Pang, X.; Yang, W. Optimal capacity allocation of multiple solar trackers and storage capacity for utility-scale photovoltaic plants considering output characteristics and complementary demand. Appl. Energy
**2019**, 238, 721–733. [Google Scholar] [CrossRef] - Hafez, A.Z.; Yousef, A.M.; Harag, N.M. Solar tracking systems: Technologies and trackers drive types—A review. Renew. Sustain. Energy Rev.
**2018**, 91, 754–782. [Google Scholar] [CrossRef] - Nsengiyumva, W.; Chen, S.G.; Hu, L.; Chen, X. Recent advancements and challenges in Solar Tracking Systems (STS): A review. Renew. Sustain. Energy Rev.
**2018**, 81, 250–279. [Google Scholar] [CrossRef] - Koussa, M.; Cheknane, A.; Hadji, S.; Haddadi, M.; Noureddine, S. Measured and modelled improvement in solar energy yield from flat plate photovoltaic systems utilizing different tracking systems and under a range of environmental conditions. Appl. Energy
**2011**, 88, 1756–1771. [Google Scholar] [CrossRef] - Bahrami, A.; Okoye, C.O.; Atikol, U. The effect of latitude on the performance of different solar trackers in Europe and Africa. Appl. Energy
**2016**, 177, 896–906. [Google Scholar] [CrossRef] - Abdallah, S.; Nijmeh, S. Two axes sun tracking system with PLC control. Energy Convers. Manag.
**2004**, 45, 1931–1939. [Google Scholar] [CrossRef] - Braun, J.E.; Mitchell, J.C. Solar geometry for fixed and tracking surfaces. Sol. Energy
**1983**, 31, 439–444. [Google Scholar] [CrossRef] - Duffie, J.A.; Beckman, W.A. Solar Engineering of Thermal Processes; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2013; ISBN 9781118671603. [Google Scholar]
- Narvarte, L.; Lorenzo, E. Tracking and ground cover ratio. Prog. Photovolt. Res. Appl.
**2008**, 16, 703–714. [Google Scholar] [CrossRef] - Parkin, R.E. Solar angles revisited using a general vector approach. Sol. Energy
**2010**, 84, 912–916. [Google Scholar] [CrossRef] - Sproul, A.B. Derivation of the solar geometric relationships using vector analysis. Renew. Energy
**2007**, 32, 1187–1205. [Google Scholar] [CrossRef] - Fan, X.; Deng, F.; Chen, J. Voltage band analysis for maximum power point tracking of stand-alone PV systems. Sol. Energy
**2017**, 144, 221–231. [Google Scholar] [CrossRef] - Satpathy, P.R.; Sharma, R. Diffusion charge compensation strategy for power balancing in capacitor-less photovoltaic modules during partial shading. Appl. Energy
**2019**, 255. [Google Scholar] [CrossRef] - Seyedmahmoudian, M.; Horan, B.; Soon, T.K.; Rahmani, R.; Than Oo, A.M.; Mekhilef, S.; Stojcevski, A. State of the art artificial intelligence-based MPPT techniques for mitigating partial shading effects on PV systems—A review. Renew. Sustain. Energy Rev.
**2016**, 64, 435–455. [Google Scholar] [CrossRef] - Belhachat, F.; Larbes, C. Modeling, analysis and comparison of solar photovoltaic array configurations under partial shading conditions. Sol. Energy
**2015**, 120, 399–418. [Google Scholar] [CrossRef] - Saint-Drenan, Y.M.; Barbier, T. Data-analysis and modelling of the effect of inter-row shading on the power production of photovoltaic plants. Sol. Energy
**2019**, 184, 127–147. [Google Scholar] [CrossRef] [Green Version] - Perpiñán, O. Cost of energy and mutual shadows in a two-axis tracking PV system. Renew. Energy
**2012**, 43, 331–342. [Google Scholar] [CrossRef] [Green Version] - Deline, C.; Dobos, A.; Janzou, S.; Meydbray, J.; Donovan, M. A simplified model of uniform shading in large photovoltaic arrays. Sol. Energy
**2013**, 96, 274–282. [Google Scholar] [CrossRef] - Martínez-Moreno, F.; Muñoz, J.; Lorenzo, E. Experimental model to estimate shading losses on PV arrays. Sol. Energy Mater. Sol. Cells
**2010**, 94, 2298–2303. [Google Scholar] [CrossRef] [Green Version] - Panico, D.; Garvison, P.; Wenger, H.; Shugar, D. Backtracking: A novel strategy for tracking PV systems. In Proceedings of the Conference Record of the Twenty-Second IEEE Photovoltaic Specialists Conference, Las Vegas, NV, USA, 7–11 October 1991; Volume 1, pp. 668–673. [Google Scholar] [CrossRef]
- Antonanzas, J.; Urraca, R.; Martinez-de-Pison, F.J.; Antonanzas, F. Optimal solar tracking strategy to increase irradiance in the plane of array under cloudy conditions: A study across Europe. Sol. Energy
**2018**, 163, 122–130. [Google Scholar] [CrossRef] - Fernández-Ahumada, L.M.; Ramírez-Faz, J.; López-Luque, R.; Varo-Martínez, M.; Moreno-García, I.M.; Casares de la Torre, F. A novel backtracking approach for two-axis solar PV tracking plants. Renew. Energy
**2020**, 145, 1214–1221. [Google Scholar] [CrossRef] - Kelly, N.A.; Gibson, T.L. Improved photovoltaic energy output for cloudy conditions with a solar tracking system. Sol. Energy
**2009**, 83, 2092–2102. [Google Scholar] [CrossRef] - Quesada, G.; Guillon, L.; Rousse, D.R.; Mehrtash, M.; Dutil, Y.; Paradis, P.-L. Tracking strategy for photovoltaic solar systems in high latitudes. Energy Convers. Manag.
**2015**, 103, 147–156. [Google Scholar] [CrossRef] - Fernández-Ahumada, L.M.; Casares, F.J.; Ramírez-Faz, J.; López-Luque, R. Mathematical study of the movement of solar tracking systems based on rational models. Sol. Energy
**2017**, 150, 20–29. [Google Scholar] [CrossRef] - Hay, J.E. Calculating solar radiation for inclined surfaces: Practical approaches. Renew. Energy
**1993**, 3, 373–380. [Google Scholar] [CrossRef] - Mousazadeh, H.; Keyhani, A.; Javadi, A.; Mobli, H.; Abrinia, K.; Sharifi, A. A review of principle and sun-tracking methods for maximizing solar systems output. Renew. Sustain. Energy Rev.
**2009**, 13, 1800–1818. [Google Scholar] [CrossRef] - Muneer, T. Solar radiation model for Europe. Build. Serv. Eng. Res. Technol.
**1990**, 11, 153–163. [Google Scholar] [CrossRef] - Perez, R.; Ineichen, P.; Seals, R.; Michalsky, J.; Stewart, R. Modeling daylight availability and irradiance components from direct and global irradiance. Sol. Energy
**1990**, 44, 271–289. [Google Scholar] [CrossRef] [Green Version] - Diez-Mediavilla, M.; De Miguel, A.; Bilbao, J. Measurement and comparison of diffuse solar irradiance models on inclined surfaces in Valladolid (Spain). Energy Convers. Manag.
**2005**, 46, 2075–2092. [Google Scholar] [CrossRef] - Loutzenhiser, P.G.; Manz, H.; Felsmann, C.; Strachan, P.A.; Frank, T.; Maxwell, G.M. Empirical validation of models to compute solar irradiance on inclined surfaces for building energy simulation. Sol. Energy
**2007**, 81, 254–267. [Google Scholar] [CrossRef] [Green Version] - Mubarak, R.; Hofmann, M.; Riechelmann, S.; Seckmeyer, G. Comparison of modelled and measured tilted solar irradiance for photovoltaic applications. Energies
**2017**, 10, 1688. [Google Scholar] [CrossRef] [Green Version] - Collares-Pereira, M.; Rabl, A. The average distribution of solar radiation-correlations between diffuse and hemispherical and between daily and hourly insolation values. Sol. Energy
**1979**. [Google Scholar] [CrossRef]

**Figure 2.**Relevant geometry and vectors of an Earth reference system for current study. (

**a**) Perspective view; (

**b**) orthogonal view.

**Figure 3.**Representation of the plane formed by the vectors $\overrightarrow{u}$, $\overrightarrow{n}$ and $\overrightarrow{e}$.

**Figure 8.**Results for a case study for inclination 15°: (

**a**) maximum radiation; (

**b**) radiation variation for 30° azimuth; (

**c**) maximum radiation loss for 30° azimuth; (

**d**) axis azimuth for optimal terrain radiation.

**Figure 9.**Variation of the maximum radiation affecting the collector with respect to the inclination of the terrain.

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**MDPI and ACS Style**

Gómez-Uceda, F.J.; Moreno-Garcia, I.M.; Jiménez-Martínez, J.M.; López-Luque, R.; Fernández-Ahumada, L.M.
Analysis of the Influence of Terrain Orientation on the Design of PV Facilities with Single-Axis Trackers. *Appl. Sci.* **2020**, *10*, 8531.
https://doi.org/10.3390/app10238531

**AMA Style**

Gómez-Uceda FJ, Moreno-Garcia IM, Jiménez-Martínez JM, López-Luque R, Fernández-Ahumada LM.
Analysis of the Influence of Terrain Orientation on the Design of PV Facilities with Single-Axis Trackers. *Applied Sciences*. 2020; 10(23):8531.
https://doi.org/10.3390/app10238531

**Chicago/Turabian Style**

Gómez-Uceda, Francisco J., Isabel M. Moreno-Garcia, José M. Jiménez-Martínez, Rafael López-Luque, and Luis M. Fernández-Ahumada.
2020. "Analysis of the Influence of Terrain Orientation on the Design of PV Facilities with Single-Axis Trackers" *Applied Sciences* 10, no. 23: 8531.
https://doi.org/10.3390/app10238531