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From video games to solar energy:
3D shading simulation for PV using GPU
Jesús Robledo1, Jonathan Leloux1,2, Eduardo Lorenzo2, Christian A. Gueymard3
1 LuciSun, Sart-Dames-Avelines, Belgium
2 Instituto de Energía Solar – Universidad Politécnica de Madrid (IES-UPM), Madrid, Spain
3 Solar Consulting Services, P.O. Box 392, Colebrook, NH 03576, USA
*Corresponding author: email@example.com
Scientific journal article published in Solar Energy, Elsevier, 2019
Jesús Robledo, Jonathan Leloux, Eduardo Lorenzo, Christian A. Gueymard,
From video games to solar energy: 3D shading simulation for PV using GPU,
Solar Energy, Volume 193, 2019, Pages 962-980, ISSN 0038-092X
Photovoltaic (PV) systems can be affected by complex shading. Software solutions have been
developed over time, offering an ever-increasing set of simulation possibilities to evaluate the
energy losses induced by shading on PV systems. Yet, several practical cases cannot be
satisfactorily solved by means of existing tools. This study explores the possibilities offered by
the powerful graphics processing units (GPUs) that have been developed for the video game
industry. It is shown that complex shading problems applicable to PV systems can be
satisfactorily analyzed, both visually and quantitatively, with a focus on the rasterization
process for an in-depth evaluation of the shading dynamics that affect the direct component of
solar irradiance. This analysis can be conducted at high spatiotemporal resolution for maximum
accuracy. Its application is illustrated based on several practical cases that are typically
encountered in the world of PV systems engineering, such as building-integrated PV (BIPV)
on large and complex buildings, urban PV planning, or PV plants equipped with tracking
systems and installed on uneven ground. Additional advantages are also presented, including
the full integration of the GPU-based shading simulation tool into a Web browser, and the use
of online input information.
Keywords: Solar, PV, shading, GPU, rasterization, WebGL
Shading reduces the power output of any kind of photovoltaic (PV) solar system. In the built
and urban environments, PV installations are surrounded by numerous objects that cast shadows
such as other buildings, antennas, lighting poles or trees (Woyte et al., 2003; Zomer et al., 2013;
Masa-Bote and Caamaño-Martín, 2014; Mulcué-Nieto and Mora-López, 2014, 2015).
Building-integrated photovoltaics (BIPV) present complex geometries that include mutual
shading between the PV elements and from the building itself (Jayathissa et al., 2017; Zomer
et al., 2013), or curved surfaces (Jakica and Zanelli, 2017). Similarly, but in field applications,
PV generators mounted on solar trackers are operated with a wide variety of tracking and back-
tracking strategies (Lorenzo et al., 2011). PV power plants are built on uneven grounds, in the
vicinity of mountains, or are affected by the presence of transmission towers or electricity
pylons (Leloux, 2005; García et al., 2008). Under these conditions, the accurate estimation of
the PV energy losses specifically induced by shading is a difficult, yet important task (Nagy et
al., 2016; Moser et al., 2017). As a consequence, the PV industry has attempted to tackle these
issues in a global effort to improve the prediction of power production and energy yields, thus
resulting in many studies in the literature on this specific aspect of solar energy engineering.
One possibility for modeling how the solar irradiance incident on a solar system is affected by
shading is to reproduce the scene at a smaller scale by physical means and to apply artificial
lighting. This method, referred to as heliodon, has historically been the first to be developed
and continues to be used (Blewett et al., 1997; Zomer and Rüther, 2017). Empirical models can
also be developed on the basis of field data (Brecl and Topič, 2011). Nevertheless, over the last
decades, the market has offered software solutions that have grown into sophisticated tools.
Specialized Computer Assisted Drawing (CAD) software such as AutoCAD (Shumaker et al.,
2018) or SketchUp (Brightman, 2018) allows to model complex 3D scenes and to visually
render shading situations (Alam et al., 2012). Unfortunately, such CAD software cannot directly
translate the shading geometry that is visualized into PV energy losses.
Specialized PV design and simulation software programs, such as PVSyst (Mermoud and
Lejeune, 2010; Eke and Demircan, 2015), PV*SOL (Patarau, 2015), or SAM (Deline et al.,
2013), include the possibility to quantitatively evaluate shading losses. They include a basic
Graphical User Interface (GUI) through which the user can draw a 3D shading scene, and then
evaluate the corresponding PV energy losses. These GUIs are not as powerful or user-friendly
as the ones provided by specialized 3D modeling software, and they are not suited to draw
complex shading scenes. Nevertheless, these GUIs are constantly improving, and some recent
developments include the possibility to integrate CAD software with PV simulation software,
thus offering more advanced simulations of shading losses for PV systems (Cascone et al.,
2011; Melo et al., 2013; Jakica 2018).
Most, if not all, existing software that includes shading analysis is usually based upon one of
two general approaches to simulate the shading geometry. A first type of approach consists in
using either projection algorithms or numerical methods. This approach constitutes an effective
means to simulate a 3D scene inasmuch as it is made of a number of polygons that is kept low
enough (Cascone et al., 2011; Celik et al., 2013). The second approach uses ray tracing
algorithms, which can handle very complex shading scenarios (Whitted, 1980; Fartaria and
Collares-Pereira, 2013; Erdelyi et al., 2014). The system resources that they require depend on
the number of rays that need to be traced, which swiftly increases with the desired accuracy and
the complexity of the scene (Marton and Szirmay-Kalos, 1995; Kovach and Schmid, 1996;
Walter and Shirley, 1997). The complexity of the objects or structures that the two approaches
just described can manage is limited by the power of the Central Processing Unit (CPU) that
handles their execution (Reif et al., 1994). Even though CPU power has increased over time,
some practical cases are still out of reach (Nagy et al., 2016; Jayathissa et al., 2017; Jakica,
A third software approach is now emerging as a viable alternative. Today, virtually all graphics
systems are characterized by a special-purpose Graphics Processing Unit (GPU) that is custom-
tailored to carry out specific graphics functions. The GPU is a specialized electronic circuit
designed to rapidly manipulate and alter the computer memory and accelerate the creation of
images on a display device via a frame buffer. During the last decade, the performance of GPUs
has surged, mainly driven by the ever-increasing demand emanating from the video game
industry. Modern GPUs are very effective at manipulating computer graphics and image
processing, and their highly parallel structure makes them more efficient than general-purpose
CPUs for algorithms that are designed to process large blocks of data in parallel (Nehab et al.,
2007). In a personal computer, a GPU can be present on a video card or can be embedded on
the motherboard. The availability of powerful GPUs opens very promising possibilities for the
simulation of complex 3D shading scenes applied to PV systems because the impacting
shadows can be evaluated with a very high spatial resolution that reaches well beyond the PV
cell level in short calculation times.
Several pioneering studies have already started to investigate the use of GPUs for the simulation
of shading effects on PV systems (Veldhuis and Reinders, 2012, 2015). The present
contribution intends to develop a further step towards the extended use of GPU-based methods
in solar energy simulation software. Some first preliminary results of this work had already
been introduced (Robledo et al., 2014), and this article presents a considerably more advanced
and detailed version of that initial development. To that effect, this study focuses on one of the
main features offered by GPUs, called rasterization (Williams 1978; Kolivand and Sunar,
2011). This specific feature makes it possible to simulate and visualize the shadows that result
in the complete blocking of the direct component of solar irradiance. This effect accounts for
the vast majority of shading-induced PV energy losses. Section 2 describes a step-by-step
rasterization procedure to improve, accelerate, and expand the capabilities of shading
simulations. Section 3 adds results of the method’s application to several practical cases of large
PV systems, which are typically encountered in the current world of solar engineering.
Some additional applications, whose satisfactory analysis becomes possible through the use of
GPUs, are also presented in Section 3. The software solution described here is implemented
using the programming language WebGL (Mattila and Mikkonen, 2013; Dirksen, 2015). This
powerful language is selected because all simulations can be run locally on the user's client-
side (e.g. computer) from a user-friendly and interactive Web browser, without requiring any
local installation from the user. This approach also empowers the user, who can obtain full
benefit from the underlying information already available online, such as 3D object libraries.
Some of the examples provided here illustrate the direct importation into the proposed PV
simulation tool of a complex 3D scene from existing specialized 3D modeling software or from
a 3D object library.
The PV energy output losses caused by shading are evaluated through a procedure that follows
four main steps. First, a 3D scene is created, where the objects are defined by geometrical
meshes. Second, the PV array, defined here as the ensemble of PV modules that compose the
PV system, is described, based on its geometry and its electrical connections. Third, the
shadows cast on the PV array are evaluated. Fourth, the shading geometry is converted into
energy losses at the PV system level.
2.1 STEP 1: Define the 3D scene
A 3D scene consists of an ensemble of objects defined by triangular meshes. There are several
ways to add an object to a scene. Objects can be imported from an object library, such as 3D
Warehouse (Fisher and Hanrahan, 2010) or Google Earth (Gorelick et al., 2017). Doing so
makes the user benefit from the vast amount of 3D objects and geo-information already
available online. Such geo-information can be imported from topographical maps or near-
horizon profiles. Objects can alternatively be modeled separately, using specialized 3D
modeling software, such as SketchUp or AutoCAD, and then imported. This opens the
possibility to draw the objects with a user-friendly and powerful GUI. Finally, the basic
volumetric information can be generated from 3D models of buildings, obstacles, etc., or from
aerial photographs, using photogrammetry techniques.
Once imported by one of the methods just described, the objects can be attributed a dynamic
behavior. For instance, a tracking and back-tracking strategy can be defined for a tracking PV
system, or the growth rate and seasonal phenology of trees can be considered.
Fig. 1 shows a scene composed of a house with chimneys and surrounding objects, such as a
TV antenna, a tree, and a light pole. All these objects were obtained online from the 3D
Fig. 1: 3D scene composed of a house with chimneys and surrounding objects such as a TV antenna, a tree and a
light pole. All 3D objects were obtained online from the 3D Warehouse library.
2.2 STEP 2: Define the PV array
The PV array is defined by its modules, their location in the 3D scene, the electrical connections
between them, their architecture up to the cell level, their by-pass diodes, and their arrangement
into specific strings.
Fig. 2 illustrates a PV array located on the roof of the house shown in Fig. 1. It is made of 24
PV modules that are distributed in 4 rows of 6 modules, and installed in portrait mode. The roof
is facing the equator and has a tilt angle of 33º. The PV modules are interconnected as two
strings of 12 modules. The two bottom rows of 6 modules each form the first string, and the
two top rows form the other. Each string thus consists of 12 modules. The two strings are
connected in parallel to a single inverter.
State-of-the-art PV modules are assumed, with a nominal power of 250 Wp. Each module
contains 60 crystalline silicon PV cells. These cells are all connected in series and are
distributed along 6 files of 10 cells. Each one of the 3 groups of 20 consecutive cells (two
consecutive files of 10 cells) is protected by a by-pass diode.
Fig. 2: PV array located on the roof of the house shown in Fig. 1. It is made of 24 PV modules that are distributed
in 4 rows of 6 modules, installed in portrait mode. The roof is facing the equator and has a tilt angle of 33º.
2.3 STEP 3: Evaluate the shadows cast on the PV array
For each moment (or solar position) of the simulation time frame, the simulation requires an
evaluation of the shadows cast by each one of the objects on each part of the PV array. An
element of an object casts a shadow on a part of the PV array during periods when the sun
appears to be situated behind it, relatively to the PV array. The shading evaluation is directly
carried out from the built-in z-coordinate feature of the GPU. The GPU converts each geometric
entity into pixel colors and locations in the frame buffer through rasterization. One of the
functionalities of the GPU during rasterization is to evaluate the depth of each pixel to be
displayed for a specified field of view, defined by its source point and angle of view. This
information is stored as a z-coordinate in the depth buffer, where a lower z-value corresponds
to a lower depth (closer to the observer).
For any specific date, time and location, the solar position is unequivocally defined; see, e.g.,
Blanc and Wald, 2012. Using the hypothetical PV installation illustrated in Fig. 2, an example
of shading calculation is detailed below, assuming the system is located in Golden, Colorado
(39.74º N, 105.18º W). An orthographic projection of the 3D scene can be drawn as a
heliocentric projection (from the point of view of the sun), with a field of view limited to the
area of interest where the shadows are evaluated (Fig. 3). Under this projection, the same pixel
can contain several objects that share the same (x,y) coordinates, but that are located at different
z-coordinates. The depth buffer of the GPU stores the z-coordinate values for each pixel and
for each object of the 3D scene. This depth buffer covers a range from 0 to 1, for a predefined
field of view located contained between two planes (hereafter, the near plane and the far plane).
As the view is generated from the apparent Sun, the near plane is the one closer to it (where the
last vertex of the 3D scenario is included), and the far plane is the one where the last vertex of
the modules is contained. These different z-coordinates are visualized through a grayscale,
where a darker grey color corresponds to a lower z-coordinate (closer to the apparent sun, i.e.
closer to 0, the near plane), and lighter when closer to 1 (far plane). For a given pixel shared by
the PV array and other objects, any object whose z-coordinate is lower than the one of the PV
array is casting a shadow onto it. It is then possible to evaluate the fraction of the surface of the
PV array that is shaded, leading to the definition of a geometric shading factor (FGS).
Fig. 3: Orthographic projection of the 3D scene from the point of view of the sun. The GPU stores the z-coordinate
values for each pixel and for each object of the 3D scene. These different z-coordinates are visualized through a
grayscale, where a darker grey color corresponds to a lower z-coordinate (closer to the apparent sun).
The spatial resolution of the 3D scene can be adapted at will by varying the area of the object
that is covered by one pixel. The shadow evaluation on the PV array can be carried out on
elements that are much smaller than a PV cell. Each of these elements is evaluated as totally
shaded when the z-coordinate of the central pixel of this element is diagnosed as shaded and is
otherwise considered as totally free from shading. Fig. 4 shows the impact of varying the spatial
resolution of the shadow evaluation from one element per PV cell to four (2 x 2) elements, nine
(3 x 3) elements, or one hundred (10x10) elements.
Fig. 4: Shadow evaluation for elements that are smaller than a PV cell, e.g. for one element per PV cell (a), four
(2 x 2) elements (b), nine (3 x 3) elements (c), or one hundred (10 x 10) elements (d).
Increasing the number of elements per PV cell automatically increases the accuracy of the
shadow evaluation, but also increases the calculation time. For most cases, using one element
per PV cell is sufficient to provide reasonable accuracy. For cases where the 3D scene is very
complex and/or high accuracy is required, or in the high-end case of specialized research topics
(Sinapis et al., 2015, 2016; Pannebakker et al., 2017), it can be advantageous to use more than
one element per cell. In any case, using nine elements per PV cell is normally more than enough.
Even though it remains possible to carry out the shadow evaluation using more elements per
PV cell (e.g. 16, 25 or more), up to the pixel level, experience shows that this leads to very little
gain in accuracy, so that the increase in computation time is not justified. A more detailed
analysis is presented in the next step.
Fig. 5 shows the application of the shadow evaluation on the PV array installed on the roof of
the house pictured in Fig. 2, using nine elements per PV cell, and for one given time moment.
The unshaded elements are colored in green, whereas the ones that are affected by shading
appear in red.
Fig. 5: Application of the shadow evaluation on the PV array installed on the roof of the house in Fig. 2, using
nine elements per PV cell. The unshaded elements are in green, whereas the shaded ones are in red.
Fig. 6 shows the fraction of the day that any PV cell of the array is affected by shading, at
different key periods of the year: summer solstice (Fig. 6a), autumn equinox (Fig. 6b), winter
solstice (Fig. 6c), and daily average over the whole year (Fig. 6d).
Fig. 6: Fraction of the day that a PV cell from the PV system illustrated in Fig. 2 is affected by shading, at different
key moments of the year: summer solstice (6a), autumn equinox (6b), winter solstice (6c), and average over the
whole year (6d).
2.4 STEP 4: Convert the shading into energy losses
The electrical energy losses induced by shading are not directly proportional to the fraction of
the surface of the PV array that is shaded. These losses also depend on the electrical connection
between PV cells and modules, including the by-pass diodes. Therefore, it is necessary to
quantify the shading losses through a combination of the geometric shading factor and its
electrical impact on the PV array, which leads to an effective shading factor (FES). Several
methods already exist to transform the geometric shading factor into the effective shading factor
(Kawamura et al., 2003; Alonso-García et al., 2006; Karatepe et al., 2007; Karatepe et al., 2008;
Silvestre and Chouder, 2008; Ubisse and Sebitosi, 2009; Ishaque et al., 2011; Wang and Hsu,
2011; Rodrigo et al., 2013; Díaz-Dorado et al., 2014), many of which are based on I-V curves.
Even though most of these methods can be appropriately used here, this study focuses on the
use of an innovative GPU-based method for the derivation of the geometric shading factor. It
is coupled here with the particular method of Martínez-Moreno et al. (2010) to evaluate the
effective shading factor, without the use of I-V curves. This approach is selected because it is
both simple and accurate. It is implemented and adapted here as follows:
where FES is the effective shading factor, FGS is the geometric shading factor, NSB is the number
of active by-pass diodes, and NTB is the total number of diodes. This method considers that a
diode becomes active when the surface of at least one of the cells protected by that diode is
When the shadow evaluation is carried out with one element per PV cell, the resulting shading
level can only take two states: 100% if the element is diagnosed as shaded, or 0% if not. When
several elements per cell are used, it is possible to obtain an intermediate shading level for the
whole cell by combining the shading states of its elements. For example, if 3 out of the 4
elements that cover one PV cell are shaded, then the PV cell itself is evaluated as 75% shaded.
Fig. 7 illustrates the establishment of the PV cell shading levels from the shading state of the
elements that compose them. The top left image (a) shows a PV cell that is only composed of
one element, and for which only two shading levels (0% or 100%) are possible. The top right
image (b) shows a PV cell divided into four elements, allowing for five shading levels (0%,
25%, 50%, 75%, 100%). The bottom left image (c) shows a PV cell divided into nine elements,
resulting in 10 shading levels. The bottom right image (d) shows a PV cell divided into one
hundred elements, resulting in 101 shading levels. The parameter FGS is obtained from the
shading level of each cell.
Fig. 7: PV cell shading levels obtained from the shading state of its elements. The top left image (a) shows a PV
cell that is only composed of one element, and for which only two shading levels (0% or 100%) are possible. The
top right image (b) shows a PV cell divided into four elements, allowing for five shading levels (0%, 25%, 50%,
75%, 100%). The bottom left image (c) shows a PV cell divided into nine elements, resulting in 10 shading levels.
The bottom right image (d) shows a PV cell divided into one hundred elements, resulting in 101 shading levels.
The combination of the shading state of the PV cells with knowledge of their electrical
connection into the modules is used to determine which diodes are active. In practice, if the
simulation is carried out with 1, 4 or 9 elements per cell, the diode is considered active as soon
as at least one element is diagnosed as shaded. This corresponds to shading levels that
respectively amount to 100%, 25%, and 11%. Fig. 8 shows the sequence of operations that leads
to the identification of the active diodes: shadow evaluation, element by element (Fig. 8a),
shading level per PV cell (Fig. 8b), and diode activation status (orange = active; blue = inactive,
Fig. 8c). The parameters NSB and NTB can thus be derived, and Eq. (1) can be solved.
Fig. 8: Sequence of operations that leads to the identification of the active diodes: a) shadow evaluation element
by element, b) shading level per PV cell, c) diode activation status (orange = active; blue = inactive).
The shading losses evaluation can be carried out at any temporal resolution. For PV arrays that
are not subject to important or complex shading, a temporal resolution of one simulation per
hour is adequate. For more complex cases, a temporal resolution of one simulation every 10
minutes (or six simulations per hour) often leads to a good compromise between accuracy and
calculation time. For the most complex cases, a temporal resolution of one simulation each
minute might be necessary. All depends on the required accuracy and computing time
constraints. This is further illustrated in Section 3.3 when dealing with the case of a PV plant
using sun-tracking systems.
The energy losses evaluated for each time step of the simulation are integrated to quantify the
losses corresponding to longer time intervals, such as one day, one month, or one year. For
other applications, shorter temporal resolutions are also possible. In fact, this method is general
and, if necessary, can be used with extremely high temporal resolutions, such as one shading
simulation per second, while maintaining reasonable calculation times if the 3D scene is not
Under certain circumstances, there can be important differences between the geometric and the
effective shading factors, such as is illustrated in Fig. 9 for the day of the summer solstice and
with a shading evaluation carried out on 9 (3x3) elements per PV cell. During the morning, the
PV generator is affected by shadows cast by the chimney and the antenna. The shading from
the chimney affects exclusively the string of PV module corresponding to the two upper rows
and the shading from the antenna affects exclusively the string of PV modules corresponding
to the two lower rows. The chimney constitutes a broad and opaque object that casts an
important shadow, whereas the antenna is composed of many thin metallic parts that induce a
partial shading on a wide area. The temporal evolution of the geometric and effective shading
factors induced by this antenna on the lower PV modules string shows that at some moments
of the day, a relatively low geometric shading factor can be associated with a high effective
shading factor, such as at 7:45 AM solar time when the geometric shading factor is 8.2% and
the effective shading factor is 47.8%, i.e. almost 6 times higher. Whereas the evolution of the
geometric shading factor during the day is relatively smooth, the effective shading factor is
marked by several discrete peaks that are triggered by the nonlinear shading effects on the
diodes. Therefore, an error in the shading factor evaluation can have a larger impact on the
effective shading factor than on the geometric shading factor.
Fig. 9: Evolution of the geometric and effective shading factors induced by the antenna on the PV string
corresponding to the two lower rows of PV modules for the day of the summer solstice. A relatively low geometric
shading factor can be associated with a high effective shading factor.
The number of elements per PV cell on which the shading state is evaluated can bear an
important impact on the accuracy of the shading factors. Fig 10 presents a comparison between
the effective shading factors that are evaluated for the case of Fig. 9 (i.e., the shading from the
antenna on the lower PV string for the day of the summer solstice), and for four different
numbers of elements per PV cell: 1 (1x1), 4 (2x2), 9 (3x3), and 100 (10x10). For a low number
of elements (1 or 4), the evolution of the effective shading fraction shows several important
oscillations that are due to a lack of accuracy in the shading evaluation, in particular during the
morning because the shadows cast by the tiny parts of the antenna are not properly captured.
The curves are already much better with 9 elements per cell, and they get only marginally better
beyond this level, such as with 100 elements per cell. For most of the real problems, a spatial
resolution of 9 elements per PV cell corresponds to a very good compromise between accuracy
and use of computation resources (time and memory).
Fig. 10: Comparison between the effective shading factors that are evaluated for the shading caused by the antenna
on the lower PV string for the day of the summer solstice, for four different numbers of elements per PV cell: 1
(1x1), 4 (2x2), 9 (3x3), and 100 (10x10). Beyond 9 elements per cell, the improvement resulting from a higher
discretization is only marginally better.
Fig. 11 shows a sensitivity analysis between the error in the evaluation of the shading fraction
and the number of elements per PV cell, for the same example as in Fig. 9. This analysis
evaluates the calculation error in the geometric and the effective shading factors for different
numbers of elements per PV cell (1, 4, 9, 16, 25, 36, 49, 64, and 81), in comparison with the
high-resolution case of 100 elements, which is chosen as the baseline. Most of the error is
captured with 9 elements per PV cell, and only small improvements are shown for 16 and 25
elements. Beyond this level, no significant improvement is obtained from a more detailed
discretization, whereas the computation time and memory increase approximately linearly with
the number of elements considered.
Fig. 11: Error in the evaluation of the shading fraction induced by the antenna on the lower PV string during the
day of the summer solstice, as a function of the number of elements per PV cell (1, 4, 9, 26, 25, 36, 59, 64, and
81) and in comparison with the high-resolution case of 100 elements that is chosen as the baseline.
The calculation of shading-induced energy losses for the specific PV installation of Fig. 2 is
illustrated here using hourly solar irradiation data provided for the weather station of
Denver/Centennial-Golden, in a Typical Meteorological Year (TMY3) format (Wilcox and
Marion, 2008). To increase the temporal resolution, 10-min irradiance data have been derived
from these hourly data by application of a simple interpolation procedure. Note that more
elaborate downsampling methods exist and could also be used if accuracy is deemed important
(e.g., Bright et al., 2015; Larrañeta et al. 2015; Zhang et al., 2018). The Direct Normal
Irradiance (Ebn) and Diffuse Horizontal Irradiance (Ed) data are taken as inputs to a simple
transposition model (Hay, 1993) to obtain the tilted components of the solar irradiance, and
ultimately the global tilted irradiance (Es). The selection of this particular model is based on
convenience and it is not critical for this exercise. Hence, any other accepted model with good
performance (Gueymard, 2009) would be suitable. Using the Hay transposition model for
convenience in what follows, the solar irradiance in the plane of the PV array is decomposed
into four components: Direct Tilted Irradiance (Ebs), Diffuse Tilted Circumsolar Irradiance
(Edsc), Diffuse Tilted Isotropic Irradiance (Edsi), and Reflected Tilted Irradiance (Ers). The
effective Global Tilted Irradiance (GTI, Egse) that is received by the modules is obtained by:
This equation explicitly considers that the effective shading factor is only applied to the Ebs and
Edsc components, and not to the Edsi and Ers, which constitutes a very good approximation in
most cases. In order to be strictly rigorous and take the losses associated with the other
components into account, it is possible to use the GPU to undertake other operations than the
rasterization (Everitt and Linkgard, 2002; Akerlund et al., 2007; Krivanek and Colbert, 2008;
Hasan et al., 2009), but these developments are out of the scope of this work. These additional
losses can also be evaluated by separate methods (Capdevila et al., 2013; Ivanova, 2013, 2014;
Rehman and Siddiqui, 2015, 2016).
Once FES and Egse have been obtained, the other energy losses (not caused by shading) can be
calculated using one out of the many existing PV simulation models or software. An effective
shading loss fraction is obtained at this stage. The examples developed below are handled with
SISIFO (Muñoz and Perpiñán, 2016; Carrillo et al., 2015), an open-source PV simulation code.
A flowchart summary of the overall shading methodology is presented in Fig. 12.
Fig. 12: Flow chart summary of the GPU-based shading methodology proposed here.
The 3D shading scenes and their corresponding energy losses are often represented by their
projection onto a 2D solar trajectory diagram, which is intuitive and explicit. This also allows
the use of input data that come from photogrammetry or “suneye” methods (Quaschning and
Hanitsch, 1998; Drif et al., 2008; Fan et al., 2012; Goss et al., 2014; Masa-Bote and Caamaño-
Martín, 2014). Alternatively, this kind of representation is very easy to obtain with a GPU-
based method and thus is used here to illustrate the results below.
Fig. 13 represents a conical projection of the shading scene on a sun trajectory diagram as seen
from the center of the PV generator. The sun positions over the course of the year are defined
by boxes of approximately 2.5 x 2.5º, as a consequence of the choice of a 10-min time interval,
given that the apparent sun’s movement along its trajectory is approximately 15º per hour. Fig.
13a shows a heatmap of the relative GTI values for an ideal shading-free scenario. It illustrates
that, for such a PV system on a roof facing south in the Northern Hemisphere, most of the
annual solar irradiation is received for sun positions that are relatively close to due South, which
is also where the shading has the higher potential for inducing energy losses. In turn, Figs. 13b
and 13c respectively show the geometric and effective shading factors over the year. Their
comparison illustrates that some objects can induce very little geometric shading, but can still
be responsible for a substantial amount of energy losses, such as the light pole or the tree in the
present case. In this situation, the effective shading factor is higher than the geometric shading
factor: even if the light pole is casting a small shadow, a large number of PV cells are affected
by it because their electrical connection is set in series. Fig. 13d shows the effective GTI shading
losses, normalized by the ideal shading-free GTI. It illustrates that most of the shading losses
occur when the sun is low over the horizon. Even though the relative shading losses are high
under such circumstances, the incident solar irradiance is low, so that the absolute shading
losses may not necessarily be significant.
Fig. 13: Conical projection of the shading scene on a sun trajectory diagram as seen from the center of the PV
generator. Fig. 13a shows the relative GTI values, normalized with respect to its annual maximum for a shading-
free scenario. Figs. 13b and 13c respectively show the geometric and effective shading factors over the year. Fig.
13d shows the effective GTI shading losses, normalized by the shading-free GTI.
Table 1 summarizes the annual geometric and effective shading losses corresponding to the 3D
scenario of the house shown in Fig. 2, along with the relative contribution of each shading
object to the total. The geometric shading factor is 3.3%, and the effective shading factor is
14.5%, illustrating how much these two factors can differ from each other. This difference can
be even greater for some specific shading objects. This is, for example, the case of the antenna,
whose relative contribution to the annual geometric shading factor is only 12.2%, but whose
relative contribution to the annual effective shading factor is 56.5%. This happens because the
antenna casts a shadow whose surface is small, yet affects several modules during a large
fraction of the year.
Table 1: Annual geometric and effective shading losses corresponding to the 3D scenario of
the house shown in Fig. 2, along with the relative contribution of each shading object to the
Shading Factor (FGS)
Shading Factor (FES)
The main objective of this section is not to present the detailed results of the simulations, but
rather to illustrate the capabilities that the use of GPU acceleration can offer to evaluate shading
effects. This is done here through several case studies that are typical of the problems that are
encountered in the world of PV system engineering. Three application examples are proposed,
each being representative of a family of PV systems and their associated shading evaluation
problem: a BIPV system on a large and complex building, urban PV planning for optimal solar
resource utilization, and a PV plant equipped with tracking systems and installed on uneven
3.1 BIPV system on the Mineirao football stadium
The Mineirao football stadium (Fig. 14) is located in Belo Horizonte, the capital of the state of
Minas Gerais in the southeastern part of Brazil (19.8659° S, 43.9711° W). This stadium was
one of the sports facilities that hosted games for the football FIFA World Cup 2014. For that
occasion, a BIPV generator was installed on its roof, with a total capacity of 1.42 MWp. The
roof is made of 88 facets that cover a full circle of 360º around the stadium, with a constant tilt
angle of 8º and variable orientation. The BIPV generator is made of 88 sub-units. For each sub-
unit, the PV modules are installed in portrait mode along several files of 15 PV modules. The
PV generators located on the eastern and western slopes of the stadium are composed of five of
these files, whereas the generators on the northern and southern sides are composed of four files
only. For simplification and conciseness, the example discussed here uses a layout that is only
made of four files for each of the 88 generators, thus providing results that are relatively
intuitive and easy to visualize. Each PV module is composed of 60 PV cells so that the whole
PV system consists of a total of 316,800 PV cells. The stadium casts complex shadows on the
PV modules. Furthermore, Mineirao is located in the intertropical zone of the southern
hemisphere, which conveys relatively complex solar trajectories. Fig. 14 shows a Lidar-
generated representation of the stadium, which is not made publicly available by Google Earth.
Fig. 14: The Mineirao stadium as generated by Lidar and publicly available from Google Earth.
A team of researchers from Brazil has recently conducted an extensive study of the performance
of this PV system (Monteiro et al., 2017). In their work, they provided a detailed technical
description of the PV system, they reviewed the monitoring data from one complete year, and
compared them to PV energy simulations using PVsyst. In order to evaluate the shading losses
of the whole PV system with PVSyst, they needed to carry out one simulation per facet, i.e.
they had to repeat the simulation exercise 88 times. They also had to draw a simplified version
of the corresponding 3D shading scenarios themselves. That part of the shading evaluation is
reconducted here using the GPU-based approach described above. This can be done in one
single simulation, which represents a considerable time-saving improvement in terms of
engineering manpower because one simulation is needed instead of 88. The original 3D model
of the stadium has been downloaded from Google 3D Warehouse in a Collada format (.dae),
which is fully public and free. It was further simplified for the shading simulation tool, and was
then augmented by a depiction of the BIPV system, as shown in Fig. 15.
Fig. 15: 3D model of the Mineirao stadium as extracted from Google Earth and simplified, with the BIPV system
added in the shading simulation tool.
The hourly GHI time series are obtained in a TMY format from the PVGIS SARAH database
(Gracia Amillo et al., 2014) available from the Joint Research Centre (JRC,
http://re.jrc.ec.europa.eu/pvg_tools/en/tools.html), and then transformed into sub-hourly data
on the plane of array with SISIFO, which makes use of the Hay model for the horizontal-to-tilt
transposition. Fig. 16 shows the distribution of GTI over the stadium for a typical year. The
annual GTI is relatively homogeneous among the PV generators, mainly due to the low tilt of
the generators and the inter-tropical latitude of the location. Nevertheless, GTI is slightly higher
for the northern generators because these have a more favorable solar geometry in the southern
hemisphere. Overall, the annual GTI is the lowest for the southern generators, with intermediate
results for the eastern and western generators.
Fig. 16: Distribution of the daily average GTI over the stadium for a typical year.
The shading evaluation is carried out on the basis of a typical year using 10-min time intervals,
resulting in 26,234 daylight situations. For each one of them, the shading evaluation is
conducted at the PV cell level, which means that 26,234 moments x 316,800 PV cells (≈9
billion) shading evaluations are conducted. Despite this large number of cases, the whole
shading evaluation process takes only several minutes on a standard personal computer. The
shading evaluation helps determine which by-pass diodes are activated at each instant, leading
to the calculation of the corresponding energy production of the whole BIPV installation. As
an example, Fig. 17 represents a heatmap of the daily-integrated shading losses on one part of
the BIPV installation during the summer solstice, i.e., December 21 at that location.
Fig. 17: Heatmap of the daily integrated effective shading losses fraction on one part of the BIPV installation
during the southern hemisphere’s summer solstice, December 21.
Fig. 18 shows the simulated effective fraction of shading losses for the whole stadium, at key
moments during the year: (a) Winter solstice (June 21), (b) Spring equinox (September 21), (c)
Summer solstice (December 21), and (d) Average year. To the best of these authors’ knowledge,
it is the first time that effective shading losses are calculated and represented in a single
simulation for a substantial number of tilted surfaces constituting a large PV system.
The annual evolution of the patterns of daily shading losses shows important seasonal changes.
For instance, the shading losses are globally higher during the winter season, when they reach
up to 13% for some parts of the stadium’s solar system. Conversely, they are lower during the
summer season — as low as 2% for some parts of the system. During the winter season, the
shading losses are higher on the eastern and western parts of the stadium because of the apparent
solar trajectory. The solar elevation is then lower and its azimuth is closer to the north, which
implies that shadows are then mostly cast in the north-south direction. Therefore, the major
blockers of solar radiation are the vertical walls located on the roofs of the stadium and that are
oriented along the east-west direction. This is particularly the case for the eastern and western
parts of the stadium. The situation is reversed at the equinox and summer solstice when the
solar trajectory is characterized by higher solar elevations, close to zenith at solar noon for many
days. Moreover, the solar azimuth is preferentially eastward during the early morning and
westward during the late afternoon. Consequently, the shadows are mostly cast in the east-west
direction, and the major solar radiation blockers are the vertical walls located on the roofs of
the stadium that are oriented along a north-south direction. Similar situations occur on the
northern and southern parts of the stadium.
The consideration of all shading losses over the whole year shows patterns that are intermediate
between the seasonal extremes, but closer to those occurring during the summer and equinox
periods. This is caused by the larger incident irradiance (GTI), inducing a higher weight with
respect to the annual total. Ultimately, the annual effective shading losses are evaluated to be
4.3% for the whole stadium. The lowest annual shading losses are observed on the sub-units
located on the eastern and western parts of the stadium (with a minimum at 2.2%) and the
highest shading losses are observed on the southern and northern facets (with a maximum at
Fig. 18: Effective shading losses fraction simulated on the whole stadium, at key moments of the year: (a) Winter
solstice (June 21); (b) Spring equinox (September 21); (c) Summer solstice (December 21); and (d) Whole year.
3.2. Urban PV planning in Boston
With the rapidly increasing installation of large amounts of PV systems in cities, optimal urban
planning has become of particular relevance and has been the object of intense research and
development in recent years (Mulcué-Nieto and Mora-López, 2015). GIS-based techniques
were first developed from image-based rendering approaches (Mardaljevic and Rylatt, 2003).
More recently, the Lidar technology has constituted a major breakthrough and has resulted in
an ever-increasing amount of data, over an ever-increasing number of locations and with higher
accuracy. This development has opened the doors to new applications and techniques (Brito et
al., 2012; Strzalka et al., 2012; Lukač and Žalik, 2013; Redweik et al., 2013; Catita et al., 2014;
Santos et al., 2014; Freitas et al., 2015; Brito et al., 2017; Desthieux et al., 2018). The state-of-
the-art Lidar technologies can now provide 3D maps for whole cities, and evaluate their solar
energy potential based on the combination of solar irradiation maps and estimates of the fraction
of solar irradiation that is lost because of shading.
for solar-dependent urban planning. In 2007, Boston was designated by the U.S. Department of
Energy (DoE) as a Solar America City, whose funding helped the city form the Solar Boston
initiative (DoE, 2011). Its objective is the installation of solar technology on all feasible and
appropriate locations throughout Boston. To achieve its ambitious development plan, Solar
Boston has created a public online GIS-based map that helps building owners calculate their
rooftop solar potential (Boston Plan, 2018). The city also partnered with DoE’s National
Renewable Energy Laboratory (NREL) to complete a Lidar scan of the city, with the goal of
enhancing the Solar Boston map and improving its ability to calculate potential installation
sizes and outputs (Cerezo-Davila et al., 2016; Reinhart and Cerezo-Davila, 2016). The 3D
models of the city of Boston have therefore constituted an interesting basis for solar potential
studies, and have been used in several investigations focusing on this topic (e.g., Lorenz and
Döllner, 2010; Nguyen and Pearce, 2012). In particular, one study has presented the solar
potential for the case of one particular area of Boston called the Skyline with a great level of
detail, up to the solar irradiation losses caused by shading (Liang et al., 2014, 2015). The
remainder of this section revisits the latter solar potential study, taking the same urban area as
a reference, but including one additional important step: the solar potential is now estimated
with proper evaluation of the effective energy losses caused by shading, and also takes the
design of the PV systems themselves into account.
Fig. 19a shows the Skyline area as displayed by Google Earth. Fig. 19b shows the
corresponding 3D map as extracted from the online tool of Solar Boston, showcasing several
possible PV installations. These are shown for illustrative purposes only because they do not
exist yet, but their localization has been carefully selected for this exercise because of their
suitability. Some PV installations are mounted on building facades, whereas others are mounted
on rooftops. Some systems are even mounted on curved facades for architectural reasons.
Overall, there is a wide variety of system geometries. All the hypothetical PV installations
together amount to a total number of 29,872 PV modules, or 1,792,320 PV cells, representing
a total capacity close to 7.5 MWp.
Fig. 19: (a) Area of the Skyline urban district of Boston as displayed in Google Earth; (b) Corresponding 3D map
as extracted from the online tool of Solar Boston, showing hypothetical PV systems in dark blue color.
The irradiation values are obtained from the hourly TMY3 file for Boston, made available by
NREL (Wilcox and Marion, 2008). These values are then interpolated with SISIFO to obtain a
10-min time step, representing a total of 26,389 daytime moments for the whole year. A shading
evaluation is carried out here for each PV cell and for each 10-min moment. This represents a
total of ≈40 billion shading evaluations. Despite this massive amount of calculations for the
complete scenario, they were completed within only several minutes on a conventional desktop
computer. Fig. 20 illustrates some results that are obtained for the evaluation of the mean annual
shading time fraction on one particular zone of the Skyline area, whereas Fig. 21 illustrates the
results for another zone in terms of effective mean annual irradiance after detailed accounting
of all shading situations and their impact on the PV energy losses, as per Eq. (2).
Fig. 20: a) Mean annual shading time fraction on one particular zone of the Skyline area. b) Detailed view using
a different scale.
Fig. 21: a) Effective mean annual irradiance on one particular zone of the Skyline area. b) Detailed view with a
In order to evaluate the effective energy losses caused by shading, each PV generator is
associated with a particular electrical design, specifically considering how the PV modules
composing one single PV generator are connected in either series or parallel. This exercise is
undertaken for the particular area illustrated in Fig. 21b. The results of the shading simulation
are shown in Fig. 22 for the Autumn equinox (September 21). Two different kinds of electrical
connections are considered. In Fig. 22a, a configuration with micro-inverters is used so that
each module has its own inverter. In Fig. 22b, several modules are connected in series of 10 to
14 elements per string (depending on the arrangement of the PV modules). Both figures show
the daily-integrated effective shading loss fraction for each configuration and each string.
Fig. 22: Daily-integrated effective shading loss fraction caused by different electrical configurations per string on
September 21. a) Micro-inverter installation (one inverter per module). b) Conventional parallel-series
arrangement (10 to 14 modules in series, depending on the geometrical arrangement of modules).
3.3 PV plant with 2-axis trackers
PV plants with sun-tracking systems may be designed with a wide diversity of tracking and
back-tracking strategies and installation conditions, which in turn translate into complex
shading simulations (García et al., 2008; Díaz-Dorado et al., 2011, 2017; Perpiñán 2012; Araki,
2014). In 2005, for instance, a PV plant project was planned to be built at Villajero, located
close to Madrid, Spain. The proposed plant had eighteen 2-axis trackers for flat-plate PV
modules, with a total capacity of 216 kWp, and was to be installed on irregular terrain.
Independently of the project, a high-voltage power line, including electric pylons, was projected
to cross the field. This situation represented a complex exercise from the point of view of
shading simulations and of the subsequent optimization of the layout. These specific studies
were undertaken using the tools that were then available (Leloux, 2005). In that process, a
topographical field study was carried out and its results were stored in a CAD format. The 3D
shading scenario was studied with the help of a 3D rendering method based on conventional
CAD software, and the PV shading losses were roughly estimated from what was visualized.
This was combined with PV simulations using an in-house software tool. More than a decade
later, this project is revisited here using the GPU-based method for a more precise shading
The topographic data and the PV system layout are directly imported from the CAD file. A 3D
mesh of the field is generated with the solar trackers on top of it, keeping the initial layout
unaltered, as shown in Fig. 23.
Fig. 23: 3D mesh of the field with the 18 solar trackers of the PV project.
The texture of the actual terrain is downloaded from Google Maps. A 3D rendering of the
electrical pylons is obtained from the 3D Warehouse online object library. Fig. 24 shows the
corresponding 3D scene of the PV plant at an arbitrary moment.
Fig. 24: 3D scene of the PV plant at an arbitrary moment. a) Overview of the PV plant. b) Partial view showing
the specific shading experienced by one PV tracker.
Fig. 25 shows the particular situation on September 21 (fall equinox) at 7:00 AM, and provides
an idea of the typical mutual shading that takes place when the sun is low. Terrain irregularities
result in asymmetrical shading profiles on a daily or seasonal basis.
Fig. 25: Mutual shading between trackers on September 21 in the early morning (7:00 AM solar time).
To evaluate the shading losses, hourly GHI time series are obtained in a TMY3 format from the
PVGIS CMSAF database (Urraca et al., 2017, 2018), and then transformed into sub-hourly data
on the plane of array of the trackers with SISIFO. The shading from the electric pylon is the
most demanding aspect of this evaluation (Fig. 24b). Fig. 26 shows the fraction of the daily
integrated solar irradiation that is lost by one particular tracker on September 21, evaluated
using three different time steps: a) 1-hour, b) 10-min, and c) 1-min. The shape of the shadow
that is cast by the pylon on the tracker evolves swiftly within a time interval that is much shorter
than one hour. Therefore, the shading evaluation using an hourly time step is very inaccurate
because the determination of the shading state of each PV cell during the whole hour interval
is derived from one single particular moment of this hour. A 10-min time step provides a
significant improvement. Nevertheless, a 1-min time step is necessary to obtain a smooth
simulation and accurate results.
Strictly speaking, the shading evaluations should also take into account the effect of the
penumbra, which appears when only a portion of the solar disk is obscured by the occluding
body. This arises when the shadows are cast by elements whose angular size as seen from the
shaded PV cell is smaller than the apparent solar disk. This situation can appear when the
distance between the PV cell and the shading object is relatively large compared to the size of
this object or parts of this object. This typically happens in the presence of objects such as
electric pylons, antennas, fences, or vegetation. For example, consider an electric pylon whose
typical width of the metallic sections of its lattice is ≈10 cm. Given the angular diameter of the
sun (≈0.5º), the penumbra phenomenon appears when the electric pylon is located at more than
20 m from the PV generator, which is a very frequent situation. No shading simulation tool for
PV is currently capable of considering the penumbra. Consequently, there is no satisfactory
solution to the evaluation of the shading losses for PV projects that include the presence of such
objects. The proper consideration of the penumbra with a GPU-based method is actually
feasible but is out of the scope of this study because it cannot be carried out with a simple
rasterization process. Nevertheless, a good approximation can already be obtained by using a
time step that is short enough so that the evolution of the shading status of a PV cell from umbra
to penumbra happens in a time interval that is longer than the time step chosen for the
simulation. In most cases, a time step of 1-min is necessary to achieve this condition.
Fig. 26: Fraction of the daily-integrated solar irradiation that is shaded on one tracker at the equinox (September
21), evaluated using three time steps: a) 1-hour, b) 10-min, and c) 1-min.
Once the shading patterns have been obtained, the corresponding energy losses are evaluated
using the method described above. In the case of this PV plant, 60 PV modules are mounted on
each tracker in landscape mode along 10 rows and 6 columns. The PV modules are electrically
connected between them into 6 strings of 10 modules, in line with the DC voltage range of the
inverters, and they are mounted on the trackers as shown in Fig. 27a. Fig. 27b shows the
effective shading loss fraction corresponding to the same tracker and the same day as for Fig.
26. The impact of the series connection is clearly observed. The benefit of such a study is that
it can help reorganize the electric wiring and optimize the rows/columns arrangement to
decrease losses on an annual basis.
Fig. 27: a) Connection of the PV modules in strings for one PV tracker. b) Daily integrated effective shading losses
fraction for one PV tracker on September 21.
This work has shown a method that makes use of the rasterization features of modern GPUs for
the simulation of complex 3D shading scenes applied to PV systems. The proposed method can
accommodate a very high spatial resolution that reaches well beyond the PV cell level while
maintaining short calculation times.
The method was applied to the shading evaluation pertaining to three very distinct types of PV
application: a BIPV installation on a football stadium, the urban PV planning in a large city,
and a PV plant with trackers on an irregular field crossed by electric pylons. This contribution
aims to constitute a further step towards the extended use of GPU-based methods in PV shading
simulation software. In its present form, the proposed method only evaluates the shading
dynamic that affects the direct component of solar radiation. This approach paves the way for
future GPU-based methods that will also evaluate the shading effects of the other radiation
components (sky diffuse and ground reflected), using more detailed procedures that are
currently under development.
Apart from the evaluation of shading losses, GPU-based methods are also well-suited — in
combination with a bidirectional reflectance distribution function (BRDF) method (Bartell
1981) — to solve other problems related to solar energy, such as (i) the calculation of the
reflected irradiance that reaches the backside of bifacial PV modules (Guerrero-Lemus et al.,
2016), (ii) the glaring effect of PV modules (Chiabrando et al., 2009), or (iii) the optical losses
in concentrating photovoltaics (CPV) systems (Sala and Antón, 2012).
The present tool is implemented using WebGL and can be directly run locally on the user's
client-side from any Web browser supporting HTML5, without requiring the installation of any
additional specialized software. Since the tool can be directly operated online, the use of 3D
objects libraries available online is possible. This approach also offers more interactive
capabilities, thus constituting a suitable pathway to integrated science, technology, engineering,
and mathematics (STEM) education (Xie et al., 2018; Gonzalez et al., 2019). This also opens
the door to more advanced built-in learning tools for the users and to the implementation of
gamification techniques, which introduce game-design elements and game principles in non-
game contexts (Marsh and Thoo, 2016; Marsh and Stravoravdis, 2017). Gamification
techniques are intended to improve the users’ experience and motivation and to make their
journey easier towards a stimulating learning experience and mastery, in this case in the use of
the software itself and also in the underlying topics, such as solar energy engineering.
The City of Boston and its Solar Boston initiative constitute a bold step towards a higher
penetration of the PV generation in the urban environment. The free and open access that is
kindly provided to the 3D maps of the city has been of great value to previous works and ours,
and it will likely continue in the future. Similarly, free access to high-quality solar irradiation
data provided by NREL and JRC constitutes an exceptional source of data. This example will
hopefully be followed by other institutions worldwide. This publication is partially based upon
work from COST Action CA16235 PEARL PV (https://www.pearlpv-cost.eu), supported by
COST (European Cooperation in Science and Technology). COST is a funding agency for
research and innovation networks. COST Actions help connect research initiatives across
Europe and enable scientists to grow their ideas by sharing them with their peers. This boosts
their research, career, and innovation.
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