# Shading on Photovoltaic Collectors on Rooftops
## 1 Introduction  
Distributed photovoltaics (PV) currently constitute roughly half of the global PV solar energy installed capacity [1]. The advantage of distributed energy is the generation of electricity right at the consumer location. Distributed PV is a general term for locally generated electricity in an urban environment, and building-integrated photovoltaics (BIPV) are a subcategory. Rooftops may provide a large amount of empty space that can reduce the use of dedicated land for large PV plant installations [2]. PV fields are usually designed with multiple collector rows. The row-to-row distance is an important parameter of the PV field design. The distance may be determined by maintenance requirements, by the amount of allowed percent of shading, or by a customary rule.  
Rooftop and ground PV systems usually adopt different row-to-row distances, because rooftops are usually limited in area and these systems are therefore densely designed. Photovoltaic collectors in the second and subsequent rows, in a multiple-row deployment of PV fields, are subject to mutual shading, and hence generate reduced electric energy. The amount of shading depends on roof and collector geometric parameters. Shading causes uneven distribution of the incident solar radiation on the PV modules, resulting in the formation of steps across the I-V characteristics. Articles that include analytical expressions for shading on PV collectors in solar fields are mentioned in [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19]. Shadow analysis and its effect on solar PV systems for flat-plate collectors was started early by [3] and followed by [6; 7; 8]. References [10; 11; 12; 13; 14] also calculate the shading losses. With the wide broadening of PV plants across the world, authors recently returned to deal with the shading losses of PV systems [15; 16; 17; 18; 19].  
Mathematical expressions were developed in [15] for the distance between PV rows on horizontal and sloping grounds, facing north-south and east-west directions. A method to calculate the minimum distance between the rows to avoid the effect of shadings is developed in [16]. In [17] the coordinates of the shadow on solar collectors are determined for horizontal, step-like, and inclined solar fields. The calculation of row spacing was developed in [18] using a vector analysis method for PV systems on horizontal and non-horizontal planes, and then comparing this method with a ray tracing shadowsoftware package. No equations for shadow height and length, nor shadow losses, are explicitly mentioned in this reference. The distance from a building object to the PV array was calculated in [19], based on the Solo Pro software simulation tool; however, no mathematical expressions for the shading were specified. The references [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18] deal with analytical expressions for the spacing between rows for PV systems deployed on the ground, the exception is reference [19].  
The main objective of the present study is to develop mathematical expressions for distances from objects, like walls and fences encountered on rooftops, to PV collectors deployed on horizontal, sloped, and saw-tooth roofs affecting shading losses. The expressions for distances may be used for predesigns of PV systems, for designing PV systems, and for analyzing systems that are already in operation. The last section studies the optimal inclination and azimuth angles on triangular sloped rooftops. The literature on shading on PV systems concentrates on distances between collector rows deployed on the ground, but it does not address shading and distances from protruding objects encountered on rooftops. The aim of the present study is to deal with this subject.