Technical Review of Solar Distribution Calculation Methods: Enhancing Simulation Accuracy for High-Performance and Sustainable Buildings

Abstract

Solar energy utilization in buildings can significantly contribute to energy savings and enhance on-site energy production. However, excessive solar gains may lead to overheating, thereby increasing cooling demands. Accurate calculation of sunlit and shaded areas is essential for optimizing solar technologies and improving the precision of building energy simulations. This paper provides a review of the solar shading calculation methods used in building performance simulation (BPS) tools, focusing on the progression from basic trigonometric models to advanced techniques such as projection and clipping (PgC) and pixel counting (PxC). These advancements have improved the accuracy and efficiency of solar shading simulations, enhancing energy performance and occupant comfort. As building designs evolve and adaptive shading systems become more common, challenges remain in ensuring that these methods can handle complex geometries and dynamic solar exposure. The PxC method, leveraging modern GPUs and parallel computing, offers a solution by providing real-time high-resolution simulations, even for irregular, non-convex surfaces. This ability to handle continuous updates positions PxC as a key tool for next-generation building energy simulations, ensuring that shading systems can adjust to changing solar conditions. Future research could focus on integrating appropriate modeling approaches with AI technologies to enhance accuracy, reliability, and computational efficiency.

1. Introduction

Solar radiation is a key factor in determining the energy and thermal performance of buildings, influencing both energy efficiency and occupant comfort across various climates. In colder regions, solar radiation supports passive heating, reducing dependence on conventional heating systems [1]. Moreover, optimized shading devices can harness sunlight effectively, reducing the need for artificial lighting and achieving energy savings of up to 55% [2]. Solar technologies, such as photovoltaic (PV) and solar thermal systems, can significantly enhance energy performance by enabling buildings to generate renewable energy on-site—an indispensable step towards nearly zero-energy buildings. For instance, PV systems can reduce energy demand by up to 40%, while integrated solar and passive strategies can lead to substantial reductions in carbon emissions, reaching up to 70% in certain cases [3]. In high-rise buildings, PV-based strategies have shown potential in achieving net-zero energy performance, with energy use intensities as low as 17–24 kWh/m² per year [4].

In hot climates, excessive solar heat gain leads to challenges, potentially increasing cooling energy demands by up to 30%, especially in buildings with extensive glazing [5]. Effective shading solutions, such as overhangs and perforated panels, can mitigate this impact, reducing heat gain by up to 50% without compromising daylight [6,7]. Climate-adapted architectural strategies, focused on orientation and shading, have shown potential energy savings of up to 76% in tropical climates [8]. Moreover, solar radiation influences the thermal comfort of occupants [1,9]. Direct sunlight exposure, particularly in glazed spaces, can create localized overheating, with indoor temperatures rising above 26 °C in sunlit areas, thereby affecting thermal comfort [10]. This dynamic between solar radiation, shading, energy efficiency, and thermal comfort highlights its significance in sustainable building design.

Building performance simulation (BPS) tools play an important role in harnessing the benefits of solar radiation while mitigating its challenges. These tools enable precise evaluations of shading strategies—such as shading devices, self-shading, and surrounding obstructions—by using advanced computer simulations [7,11,12]. BPS offers a cost-effective alternative to field tests that are often costly and time-consuming. By enabling the preliminary assessment of multiple design options, BPS supports data-driven and energy-efficient decision-making [13,14,15,16]. Moreover, BPS simulations facilitate a detailed understanding of interactions between complex building components, helping designers to predict solar radiation impacts and optimize shading solutions for sustainability and comfort [13].

One of the key steps in assessing solar radiation’s impact on building energy performance is quantifying sunlight exposure across different building surfaces, distinguishing between sunlit and shaded areas. This solar shading analysis is primarily a geometric calculation, influenced by the building’s shape and the constantly changing position of the Sun throughout the year [17,18]. While the building’s geometry, shading devices, and nearby obstructions generally remain fixed, the solar position varies continuously with latitude, longitude, and time of day. Different BPS tools employ distinct techniques to analyze sunlight exposure on exterior building surfaces and the distribution of solar radiation inside, offering a detailed understanding of how each design feature affects the building’s energy efficiency.

The calculation of sunlit areas on exterior surfaces in BPS tools is typically conducted using trigonometric operations or projection-and-clipping (PgC) methods, each with distinct advantages and limitations. Trigonometric methods (TgM) are efficient for simple shading elements like overhangs and fins, offering quick calculations based on geometric principles [18,19]. However, P&C methods, which involve the projection and clipping of polygons, are better suited for modeling shadows cast by more complex shapes such as building façades, louvers, and nearby obstructions [20,21,22]. While these methods enable more detailed simulations, they are computationally demanding and may struggle with complex polygons [23,24].

Regarding the internal spaces of buildings, the distribution of direct solar radiation transmitted through windows is often simplified in most tools, with the assumption that all the solar beam radiation is uniformly distributed over the floor surface [25]. Although this assumption works for one-dimensional models, it leads to inaccuracies in more detailed simulations, particularly in spaces with significant glazing [26,27]. Accurate sunpatch distribution is crucial for predicting indoor temperatures and supporting thermal comfort studies as it helps to map mean radiant temperatures [28,29]. Rodler et al. [30] emphasized the importance of integrating 3D thermal models with precise sunpatch localization to optimize the energy performance of highly insulated buildings, which are particularly sensitive to internal heat gains.

As architectural design has evolved with parametric systems and digital fabrication has enabled more complex shading geometries, the need for precise sunlit area calculations has grown. Advanced shading systems and adaptive façades, such as perforated solar screens (PSSs), can significantly improve energy efficiency and occupant comfort [31]. Studies have shown that optimizing PSS variables, such as perforation percentage and shape, can increase daylighted areas by up to 50% and reduce energy demand by 55% [2]. This underscores the need for BPS tools to support intricate shading designs and dynamic façades in high-performance buildings.

However, the conventional methods face challenges when applied to complex architectural forms, such as concave or hollow geometries. To address this, the pixel counting (PxC) method has emerged, leveraging computer graphics technology to improve the accuracy of shading analysis. PxC calculates sunlit areas by rendering the building from the Sun’s perspective and counting the visible pixels on each surface. This method is particularly effective for complex shapes and is implemented in tools like EnergyPlus [32] and Domus [33], offering improved performance for modern architectural geometries [24,34,35].

Despite the advancements in computational models and the integration of sophisticated shading geometries in BPS tools, there remains a gap in comprehensive reviews specifically focused on sunlit area calculation methods within these tools. While the current literature extensively discusses shading devices and energy-saving strategies, it often overlooks the specific computational methods required to determine sunlit areas. For instance, Valladares-Rendón et al. [8] reviewed passive cooling strategies, including the use of shading devices, window-to-wall ratios (WWR), and building orientation, while Kirimtat et al. [7] analyzed shading simulations across various tools, and Al-Masrani et al. [36] explored the performance of shading systems in tropical buildings. More recently, Lionar et al. [6] provided an overview of self-shading façades, highlighting their energy-saving potential in warm climates. Complementary reviews have emphasized broader challenges in integrating performance metrics into shading device analyses. Avcı et al. [37] underscored the limited alignment between daylighting, visual comfort, and energy efficiency metrics, while Naik et al. [38] highlighted gaps in the evaluation of dynamic solar screens, particularly regarding their application across different geometries, climates, and architectural contexts. Additionally, Shum and Zhong [39] conducted a comprehensive review of smart solar shading systems, highlighting their potential for improving building energy performance in cold climates. The study emphasizes the suitability of automated shading systems with insulating materials, which can achieve significant energy savings during both heating and cooling seasons, while also identifying the need for season-specific control strategies. Despite these valuable contributions, none of these studies delve deeply into the specific methodologies required for accurate sunlit area calculations. This gap is significant as precise sunlit area assessments are fundamental for reliably simulating energy performance and thermal comfort in complex architectural designs.

Thus, there is a clear need for a comprehensive review that addresses the evolution, limitations, and future potential of sunlit area calculation techniques. As computational resources continue to advance, improvements in shading calculations and solar radiation distribution become particularly important for designing energy-efficient and nearly zero-energy buildings. This paper aims to address this gap by providing a review of the methods for calculating external and internal direct solar distribution, emphasizing their relevance in achieving accurate simulations for high-performance and sustainable building designs.

2. Materials and Methods

This study adopted a review approach to investigate methods for calculating sunlit and shaded areas in building energy simulation tools, conducted using primary scientific databases, namely Scopus and Web of Science, with a focus on publications from 1990 to 2024. This timeframe ensured the inclusion of both established methodologies and recent advancements in the field. A focused search strategy was implemented using specific keywords such as solar shading solar radiation, sunpatch, shading, and shadowing along with terms related to energy performance simulation and techniques for calculating sunlit areas, including trigonometric model, polygon clipping, pixel counting, shading calculation, shading modeling, and sunlit area calculation.

Articles were selected according to specific inclusion criteria: (a) a focus on methods for calculating external sunlit areas or internal solar distribution; (b) an assessment of accuracy, computational cost, and applicability; and (c) relevance to building energy performance and integration with simulation tools. Studies were excluded if they focused solely on natural lighting without addressing energy or thermal performance, or if they did not specifically address methods for calculating shading or sunlit areas.

The identified methods were categorized into two main groups: (a) external solar shading techniques, including trigonometric models, polygon clipping, and pixel counting, and (b) internal solar distribution methods, which include not only the same external sunlit area techniques but also homogeneous simplifications, as well as more complex ray projection and solar modeling approaches. These methods were then evaluated based on the following main criteria: accuracy, computational efficiency, and applicability. Accuracy was measured by the methods’ ability to precisely represent sunlit and shaded areas, while computational efficiency focused on processing time and performance, especially in complex geometries. Applicability was assessed in terms of how well the methods integrate with widely used energy simulation tools such as EnergyPlus, TRNSYS, ESP-r, and Domus. A comparative analysis was conducted to highlight the strengths and limitations of each method across these criteria. This analysis not only reveals gaps in the current literature but also identifies opportunities for future research and development, particularly in the context of sustainable building design.

3. Timeline of Solar Shading Methods in Building Performance Simulation

A timeline illustrating the key milestones for BPS tools is shown in Figure 1. Shading calculation methods have evolved in parallel with building performance simulation tools since the 1960s. In the early years, building models mainly focused on the sizing of heating, ventilation, and air conditioning (HVAC) systems using simple calculations for basic shading elements. During the 1970s and 1980s, as building energy simulation shifted towards passive design strategies, more accurate methods, such as polygon clipping, were developed to predict sunlit areas on surfaces. The 1990s brought significant advancements in personal computer hardware and software, which enabled the optimization of polygon clipping techniques and faster simulations of complex geometries. This period also saw the introduction of pixel-counting approaches, driven by the continuous growth in graphics hardware and libraries.

The next two sections explore two main categories: (a) external solar shading techniques, and (b) internal solar distribution methods. It is important to note that some techniques presented in the external sunlit area calculation section are also applied to internal sunlit area calculation. In other words, while the articles focus on different aspects of external shading and internal solar distribution, the same methodologies are employed in both contexts. Therefore, the external solar shading techniques are discussed first, followed by the internal solar distribution methods, without reiterating the details of the techniques.

4. Exterior Solar Shading Techniques

This section presents a detailed analysis of the existing studies on advancements in solar shading calculations, emphasizing the trade-offs between accuracy and computational efficiency, as well as the potential for real-time shading simulations.

Table 1. Studies related to exterior solar shading techniques.

Ref.	Year	Output Data	Method	Details	Tools	Comparison with Tools	Study Case	Context	Shadow Source	Studied Surfaces	Location	Simulation Timeframe	Max. Runtime
[40]	1994	GIC [%]	PxC	Uses orthographic projection from the Sun’s point of view; bitmap technique for pixel analysis; evaluates solar rights and shading requirements.	SHADING	Compared with traditional CAD methods	Modeled case	Building	Surrounding buildings, Trees and other vegetation	Façades and Openings	Israel	Specific Dates and Times	6 s per view
[46]	1996	GSC [%]	PxC	Projection of shadow-casting surfaces onto a receiving surface, combined with pixel counting to determine the proportion of the shaded area.	TRNSYS, Suncode	Not directly compared	Modeled case	Building	Overhangs	Façades and Openings	Germany	Monthly	Not specified
[47]	1997	GIC [%]	PxC	Uses orthographic projection from the Sun’s point of view; bitmap technique for pixel analysis; evaluates solar rights and shading requirements.	SHADING	Not directly compared	Real case	Urban	Surrounding buildings	Façades and Openings	Israel	Specific Dates and Times	Not specified
[17]	2000	Sunlit area [m²]	PgC	Handles complex surface geometries, including self-shading and external obstructions.	TRNSYS	Compared with Simulation and Standards	Modeled case	Building	Self-shading, Surrounding buildings	Façades and Openings	USA	Specific Dates and Times	Not specified
[48]	2006	Shaded area [m²]	Vectorial	Uses vector algebra for calculating shadow projections	Custom-developed model	Compared with Experimental	Real case	Generic	Overhangs	Generic surface	Thailand	Specific Times	Not specified
[19]	2011	Sunlit area [m²]	Vectorial	Uses vector algebra and homogeneous coordinate systems for shading	Custom-developed model	Not directly compared	Modeled case	Generic	Trees and other vegetation	Generic surface	Italy	Specific Times	Not specified
[34]	2012	PSSF [%]	PxC	A module calculates incident radiation using a pixel-based depth buffer and B-Spline interpolation.	Custom-developed model	Compared with Analytical	Real case	Building	Trees and other vegetation, Self-shading	Façades and Openings	USA	Annual	6.2 min
[49]	2012	PSSF [%]	PxC	Uses programmable shaders to handle complex transparent and perforated shading surfaces.	Custom-developed model	Compared with Simulation	Modeled case	Building	Trees and other vegetation	Façades and Openings	USA, Abu Dhabi	Specific Dates and Times	209 s
[50]	2012	Sunlit area [m²]	PgC	Improved method projects all polygons on a single plane orthogonal to the sunray; uses Vatti’s clipping algorithm for shadow intersection.	Custom-developed model	Compared with Standards and Experimental	Real case	Building	Surrounding buildings	Façades and Openings	Spain	Annual	8.3 min for a whole year, every Sun hour.
[51]	2013	PSSF [%]	PgC	Matrix-based coordinate transformations and polygon clipping.	Custom-developed model	Compared with OpenGL	Modeled case	Generic	Trees and other vegetation	Generic surface	Not specified	Not specified	264.7 s
[24]	2017	Sunlit fraction [%]	PxC	Experimental validation of two solar shading calculation techniques. PxC and PgC are compared based on their accuracy and computational time.	Domus, EnergyPlus, Shading II SketchUp plug-in	Compared with Simulation, Compared with Experimental	Modeled case	Building	Shading devices, surrounding buildings, trees	Façade surfaces	Brazil	Specific Times	Varies by software; Domus: <1 h, EnergyPlus: ~18 h
[35]	2017	PSSF [%]	PxC	Uses OpenGL to calculate shading with pixel counting. Transparency feature enables more accurate shading in vegetated or complex urban environments.	Modified version of EnergyPlus	Compared with Standards, Compared with Experimental	Modeled case	Urban	Surrounding buildings, Self-shading	Façades and Openings	USA	Annual	40–50% faster than EnergyPlus
[52]	2017	Shaded area [m²]	PgC	Projection of the geometry of shading elements onto a surface using 3D coordinates and resolves overlaps with the Planar-Polygon method and General Polygon Clipper (GPC).	Custom-developed model	Not directly compared	Modeled case	Generic	Kinetic façades	Generic surface	Repulic of Korea	Specific Dates and Times	Not specified
[53]	2019	Shaded area [m²]	TgM	Uses empirical equations based on trigonometric models for shading calculation	Custom-developed model	Compared with SketchUp	Real case	Building	Overhangs; Self-shading	Façades and Openings	China	Specific Times	Not specified
[54]	2019	Shaded area [m²]	PgC	Improved PgC method for solar radiation calculations, including direct and sky diffuse irradiance, and reflections on building surfaces.	Custom-developed model	Compared with Simulation	Modeled case	Urban	Surrounding buildings	Façades and Openings	Germany	Specific Dates and Times	Similar to Energyplus simulations
[55]	2022	Sunlit area [m²]	PxC	Comparison of four shadow modeling approaches: forward ray tracing, focused ray tracing, pixel-counting, and analytical areas. PV panels.	Custom-developed model	Compared with Simulation	Modeled case	Building	Overhangs	Façades and Openings	USA	Specific Dates and Times	Not specified
[56]	2023	Shading factor (SF) [%]	TgM	Mesh elements and solar exposure analysis of a building façade within linear and elliptical site contexts.	Custom-developed model	Compared with Experimental	Modeled case	Urban	Surrounding buildings	Façades and Openings	Canada	Annual	6.1 min per façade
[57]	2023	Shading factor (SF) [%]	TgM	Mesh elements and solar exposure analysis of a building façade within linear and elliptical site contexts.	Custom-developed model	Compared with Experimental	Modeled case	Urban	Surrounding buildings	Façades and Openings	Not specified	Annual	0.8 s per façade
[58]	2023	Shaded area [m²]	PgC	Uses a solid clipping technique that extrudes a 3D sunpath and calculates the intersections with building geometry to accurately compute shadow areas for solar regulation checks in BIM environments.	Custom-developed model	Compared with Simulation	Modeled case	Building	Overhangs	Façades and Openings	France	Specific Dates and Times	Not specified
[59]	2024	Shaded area [m²]	PgC	Combines polygon clipping, distance filtering, and the sunlight channel to calculate shadows in urban environments by eliminating irrelevant surfaces and focusing on the most relevant ones based on proximity, height, and the Sun’s position, significantly speeding up the process.	Custom-developed model	Compared with Simulation	Modeled case, Real case	Urban	Surrounding buildings	Façades and Roofs	China	Annual	10 times faster over the baseline runtime
[60]	2024	Shaded area [m²]	PgC	Uses multithreaded parallel processing to accelerate shadow calculations, combining sunlight channel and celestial sphere methods.	DeST	Compared with Simulation	Real case	Building, Urban	Surrounding buildings	Façades and Roofs	China	Annual	2.4× faster for complex buildings

summarizes 21 studies on exterior sunlit area calculation methods, including the main details such as the methods used, output data, tools employed, and comparisons between different tools. Techniques like vectorial methods (VMs), trigonometric methods (TgMs), polygon clipping (PgC), and pixel counting (PxC) each offer distinct advantages in handling complex geometries, with varying levels of flexibility, computational efficiency, and accuracy.

4.1. Vectorial Methods

Vectorial methods (VMs) are widely used for calculating shaded areas in simpler architectural geometries, particularly when shading systems involve inclined planes, overhangs, or adjacent rectangular structures.

These methods use vector algebra to calculate shadows, with the Sun’s position and shading plane geometry represented as unit vectors. The calculation involves several steps: (a) the Sun’s position is provided by ${\widehat{v}}_s$ and the inclined plane by ${\widehat{v}}_y$ in a Cartesian coordinate system, where the Sun’s position is determined by its altitude $\alpha$ and azimuth ${\gamma }_s$, and the plane’s geometry by its tilt $\beta$ and azimuth $\gamma$. (b) The shadow of point P on the plane is calculated using:

$${\overrightarrow{y}}_s={\overrightarrow{y}}_p-{\displaystyle \frac{h}{cos\theta }}{\widehat{v}}_s,$$

(1)

where h is the height of the point and $cos\theta$ is the cosine of the angle between the sunray and the plane.

(c) The Cartesian system is transformed into a new one for more efficient shadow projection. A transformation matrix converts the vectors. (d) The shadow pattern is formed by connecting the calculated points, especially for simple shading devices like overhangs and fins. (e) The shaded area is calculated by determining the area of triangles formed by the shadow points and summing their areas using Equation (2):

$$\begin{array}{c}\hfill A_1={\displaystyle \frac{1}{2}}\left[\left|\begin{array}{cc}{y}_{21}& y_{31}\\ y_{22}& y_{32}\end{array}\right|+\left|\begin{array}{cc}{y}_{22}& y_{32}\\ y_{23}& y_{33}\end{array}\right|+\left|\begin{array}{cc}{y}_{23}& y_{33}\\ y_{21}& y_{31}\end{array}\right|\right].\end{array}$$

(2)

For irregular shading devices, additional points along the boundaries are traced, and the shaded area is computed by summing the areas of multiple triangles. Detailed derivation of the formulation can be found in [48].

Pongpattana’s model, which compares theoretical results with experimental data, has shown high accuracy in real conditions, with error margins of 0.25% for overhangs, 0.52% for fins, and 0.21% for eggcrate systems [48]. Cascone’s method has advanced VM by incorporating shading factors for direct, diffuse, and reflected radiation using homogeneous coordinates, offering a more comprehensive analysis of solar gains under varying sky conditions and including the effects of horizon profiles, obstructions, and vegetation [19]. While these methods are effective for assessing solar radiation in urban and natural environments, they remain computationally extensive, especially when modeling complex and dynamic geometries, such as irregular shading devices and vegetation.

4.2. Trigonometric Models

Trigonometric methods (TgMs) have long been valued for their computational efficiency, particularly in environments where geometries can be reduced to simple parameters, such as overhangs or self-shading elements [53]. TgMs’ ability to calculate shading within seconds, compared to several minutes required by more complex methods, makes them a practical choice for early-stage design simulations. This efficiency, combined with minimal hardware requirements and a low learning curve, makes TgMs accessible for professionals with limited technical expertise [56].

However, TgMs’ reliance on simplified assumptions limits their precision when dealing with irregular or complex geometries, such as those found in intricate urban environments or building façades. As urban geometries become increasingly complex—featuring non-convex, hollowed, or non-planar surfaces—TgMs’ simplifications may result in inaccuracies. Elmalky [56] addressed these limitations by introducing an enhanced formulation that accommodates diverse urban geometries and building inclinations, reducing computational time from 6.1 min to 0.8 s per façade while maintaining an average error margin of about 5% in urban grid scenarios [57]. Although these advancements improved its applicability, challenges remain, particularly when handling non-convex surfaces and dynamic environmental conditions, as well as integrating TgMs with BPS tools. The need for manual adjustments and programming further limits usability [57]. Integrating TgMs with more advanced methods, such as PgC and PxC, could enhance their adaptability and precision, especially for dynamic shading conditions and complex geometries.

One of the most well-known trigonometric techniques, the ASHRAE method for overhangs and fins, employs trigonometric relationships to determine shading effects on window surfaces. As depicted in ASHRAE Handbook—Fundamentals [61], the shadow cast by an overhang is determined by the geometry of the overhang and the shadow-line angle, $\Omega$, which is determined by the solar altitude ($\beta$) and azimuth ($\gamma$) (for more details, see Figure 16 from [61]), by using Equation (3).

$$tan\Omega ={\displaystyle \frac{tan\beta }{cos\gamma }}.$$

(3)

The shadow width (SW) and shadow height (SH) produced by the vertical and horizontal projections can be calculated using the respective projections (PV and PH) and the angle $\Omega$:

$$SW=PV|tan\gamma |$$

(4)

$$SH=PHtan\Omega .$$

(5)

When the solar azimuth angle ($\gamma$) is between 90° and 270°, the fenestration is completely shaded. The sunlit (ASL) and shaded (ASH) areas of the fenestration vary throughout the day and can be calculated using the following Equations (6) and (7):

$$ASL=[W-(SW-RW\left)\right][H-(SH-RH\left)\right]$$

(6)

$$ASH=A-ASL.$$

(7)

where A is the total area of the fenestration product. For software-based calculations, McCluney (1990) describes an algorithm for determining the unshaded fraction of a window equipped with overhangs, awnings, or fins. The ASHRAE Handbook [61] provides a comprehensive set of equations to facilitate these calculations.

4.3. Discrete Element Analysis of Grids

The Discrete Element Analysis of Grids method is recognized for its simplicity and flexibility in managing various geometries, including both convex and concave shapes. This method involves discretizing surfaces into smaller grid elements, which enables efficient calculations of interactions and solar exposure. It was among the earliest techniques implemented in BPS tools such as NBLSD and NECAP, predecessors of DOE-2 and BLAST [50]. The method was initially introduced by Groth and Lokmanhekim [62,63,64] and is used in ESP-r [65].

In this method, shading elements are projected onto receiving surfaces along the Sun’s rays. The receiving surface is subdivided into a grid, with each grid cell represented by a central point (e.g., a 20 × 20 grid in ESP-r, as shown in Figure 2). Each grid cell is then classified based on whether it is sunlit or shaded according to the central point. By summing the contributions of all grid elements, the total sunlit and shaded areas are calculated [63,64].

Although the technique is highly versatile, its accuracy may be compromised when applied to coarse grids. As the mesh resolution increases to capture finer details, processing time tends to lengthen. Furthermore, the risk of inaccuracies rises, particularly when dealing with complex shapes or high-density grid configurations.

4.4. Projection and Clipping Techniques

Projection and clipping methods are widely used for assessing solar shading on building surfaces in BPS tools. Known for their high accuracy, these methods are particularly effective in calculating direct solar shading on planar surfaces. Unlike grid-based approaches, which rely on approximations, PgC ensures precise results by accurately representing the shadow polygon’s shape.

Despite their accuracy, PgC methods face challenges when applied to complex geometries, such as non-planar surfaces or polygons with holes. These difficulties are compounded by the increasing computational demands as the number of polygons grows, which can make simulations impractical due to excessive runtime, especially when dealing with large or irregular geometries. The computational intensity of PgC arises from the numerous geometric transformations and intersection calculations required, which can be prohibitive for complex shading scenarios [24,59]. Moreover, the successive computation of transformations and intersections can lead to a loss of precision in the results due to cumulative computational errors, primarily arising from round-off and truncation errors caused by floating-point arithmetic. PgC implementations often require advanced programming skills and specialized hardware, thus limiting their integration into BPS tools and hindering their widespread adoption in professional settings.

Several PgC algorithms have been developed to address these challenges and improve performance. Vatti [22] introduced a general clipping algorithm that processes surface and clipping polygon edges systematically, starting from the lowest edges. Greiner and Hormann [41] proposed a faster, more efficient algorithm for clipping arbitrary 2D polygons, particularly useful when polygons may self-intersect. Liu et al. [66] developed a memory-efficient algorithm for geoprocessing applications that rebuilds clipping polygons after each operation, streamlining the process. Maestre et al. [50] enhanced Vatti’s algorithm by combining it with plane projection for shadow calculations, improving efficiency and reducing computational time.

Recent advancements have further refined the PgC technique. Voivret [58] introduced a solid clipping technique that extrudes a 3D sunpath and calculates intersections with building geometry. This method, designed for Building Information Modeling (BIM) environments, offers a highly accurate way to compute shadow areas for solar regulation checks. Liu [59] integrated PgC with distance filtering and sunlight channel methods, significantly accelerating shadow calculations by focusing on the most relevant surfaces based on proximity, height, and Sun position. This approach has proven particularly effective in large-scale urban simulations. Zhou [60] demonstrated that combining PgC with multithreaded parallel processing can increase calculation speed by nearly 2.4 times for individual buildings and reduce computation time to just 5.4% of the original duration for city-scale simulations.

Despite the progress achieved, the incorporation of these PgC methods into BPS tools is still restricted, presenting challenges for their broader adoption in professional environments. Moreover, PgC’s computational demands continue to present a barrier, especially for complex shading scenarios requiring higher processing power. One potential solution is the use of advanced hardware, such as Graphics Processing Units (GPUs), which could significantly accelerate simulations and make PgC more feasible for large-scale urban modeling [59].

4.4.1. Steps Involved in Projection and Clipping

The process of projection and clipping typically involves four main steps: selection, exclusion, projection, and clipping [50]: (1) in the initial selection step, receiving polygons (RPs) and shading polygons (SPs) are identified (Figure 3a) [67,68]. (2) Next, shading calculations begin with a coordinate transformation, converting the 3D coordinates of all vertices to a 2D plane, which simplifies the projection of the SPs [17]. (3) The exclusion process involves removing portions of the SPs that are submerged within the RP by identifying and eliminating any submerged vertices. This step is performed prior to projection to prevent inaccuracies, such as false shadow projections (Figure 3b,c) [67]. (4) In the final steps, the projection and clipping processes are applied: SP vertices are projected onto the RP plane along the Sun’s rays, and clipping algorithms are then used to define the sunlit boundary [21,50,63].

4.4.2. Clipping Algorithms

Clipping algorithms implemented in BPS software are organized into two aspects: Boundary Evaluation and Polygon Clipping with Homogeneous Coordinates.

The Boundary Evaluation method, introduced by Weiler and Atherton [20], simplifies solar shading calculations by reducing the three-dimensional problem to two dimensions using a coordinate transformation. This process eliminates SPs that lie behind the RP, streamlining the calculation of solar shading.

The method determines the sunlit areas by identifying intersections between the edges of the RP and SP [17]. Two separate lists of intersection points are generated: one for the SP and another for the RP. A boundary traversal algorithm processes these intersections, alternating between the SP and RP boundaries, as shown in Figure 4. The process continues until all intersection points are processed, and the clipping is completed once the algorithm returns to its starting point [17].

The sunlit area is then calculated by

$$Area={\displaystyle \frac{1}{2}}\sum _{i=1}^n(x_i{y}_{i+1}-x_{i+1}{y}_i),$$

(8)

where $x_i$, $y_i$ are the coordinates of vertex i, with $x_1=x_n$ and $y_1=y_n$, and n represents the number of vertices. Since the receiving polygon is ordered counterclockwise and the shadow polygon clockwise, the sign of the area calculation indicates the orientation of the sunlit area: a negative value denotes counterclockwise ordering (sunlit area), while a positive value indicates clockwise ordering (shaded area) [63]. This technique ensures accurate results for non-convex polygons, supporting high-fidelity solar shading calculations [17,54].

Regarding methods related to Polygon Clipping with Homogeneous Coordinates, EnergyPlus provides a relevant example. The software integrates shadow algorithms derived from BLAST and TARP, employing coordinate transformation techniques similar to those proposed by Groth and Lokmanhekim [62] and Walton [69], as cited in [70]. These algorithms convert the coordinates of the RP, SP, and solar direction cosines into two-dimensional homogeneous coordinates, which are then used to define shadow overlaps.

EnergyPlus (v24.1) employs three clipping techniques:

• Convex Weiler–Atherton: This method is only accurate when both the casting and receiving surfaces are convex.
• Sutherland–Hodgman: This computationally simpler algorithm is used for non-convex surfaces. It clips each edge of the shading polygon against the edges of the receiving polygon, classifying the edges into four types based on their position relative to the clipping window.
• Slater–Barsky: Particularly efficient for rectangular surfaces, this algorithm uses parametric equations to calculate intersection points. It reduces computational effort by dividing the plane into regions, minimizing unnecessary calculations.

After determining the vertices, they are sorted in a clockwise orientation to facilitate area calculation. The area of the receiving surface is considered positive, while shadow overlaps are treated as negative, and overlaps between two shadows are considered positive [70].

4.5. Pixel Counting Technique

The pixel counting (PxC) technique represents a significant advancement in calculating sunlit surface areas by exploiting modern graphics hardware capabilities. Initially proposed by Yezioro and Shaviv [40], this method uses orthogonal projections from the Sun’s perspective at specific times and positions, with the Sun’s viewpoint treated as the origin (see Figure 5). The sunlit area is determined by counting pixels of the resulting image based on their color using a bitmap technique.

4.5.1. Original Implementation of PxC

In its original implementation, two orthogonal projections are generated: one for the analyzed surface (rendered in a specific color in front buffer) and another for the entire scene, including obstructions (back buffer). By comparing the pixel colors between the two buffers, the method identifies the unshaded pixels of the analyzed surface, which correspond to the sunlit area ($A_{sunlit}$). The sunlit fraction is then calculated as the ratio of sunlit area to total surface area, expressed as $A_{sunlit}/A_t$, where $A_t$ is the total area of the surface.

This approach was later integrated into tools such as the “Shading” plug-in for SketchUp, facilitating shadow analysis in architectural design [71]. However, a notable limitation is its lack of integration with BPS software.

4.5.2. Enhancements for Computational Efficiency

Jones et al. [23] proposed three enhancements to the PxC method [40] to improve computational efficiency: (1) enlarging the orthogonal view to maximize pixel resolution and reduce pixelization effects (Figure 6a), (2) using the depth buffer (Z-buffer) for sunlit area calculations, and (3) utilizing the functionalities provided by the OpenGL graphics library to streamline the computational process. These modifications were further refined in subsequent work by Jones et al. [34].

The orthogonal view was optimized by aligning the frustum’s clipping planes with the Sun’s rays, creating a rectangular volume defined by six planes (four parallel to the Sun’s rays and two orthogonal planes). This adjustment ensures accurate projections that tightly enclose the receiving polygon (Figure 6b). The sunlit fraction is then calculated in a single rendering pass using the Z-buffer, which tracks pixel depth, with closer pixels overwriting those further away. The sunlit fraction is then calculated in a single rendering pass using the Z-buffer, which tracks pixel depth, with closer pixels overwriting those further away [72].

With these modifications, the calculation of the sunlit fraction begins by rendering the entire building and all obstructions, excluding the receiving surface. This step ensures that all potential shading effects on the receiving surface are accurately accounted for. As part of the third enhancement, OpenGL’s hardware-accelerated Occlusion Query is utilized, a feature available in cross-platform graphics libraries designed to efficiently count visible pixels. Once the receiving surface is rendered, the occlusion query calculates the exact number of unoccluded pixels, denoted as N, representing the portion of the surface directly illuminated by the Sun.

The area of each pixel in the model space ($A_p$) is determined by the dimensions of the orthogonal projection and the resolution of the depth buffer. This is completed by calculating the relationship between the width and height of the orthogonal projection (defined by the right (R), left (L), top (T), and bottom (B) extents) and the resolution of the depth buffer, which is provided by the number of pixels in width (w) and height (h), as presented in Equation (9).

$$A_p={\displaystyle \frac{R-L}{w}}·{\displaystyle \frac{T-B}{h}}.$$

(9)

This relationship yields the area of one pixel in the image plane within the model space, helping to understand how the buffer resolution and projection geometry influence the accuracy of the 3D representation. Consequently, the total sunlit area can be obtained by multiplying $A_p$ by N. In this way, analytical polygon Boolean operations are replaced as shown in Equation (10).

$${\displaystyle \frac{A_s}{A_t}}cos\theta \approx N·A_p·{\displaystyle \frac{1}{A_t}},$$

(10)

where N is the number of visible pixels on surface and $A_p$, provided by Jones et al. [34].

Although polygon clipping methods are effective in precisely defining the boundaries of shadow areas on building surfaces, PxC approaches, while faster, suffer from pixelation effects, leading to approximations in shadow delineation. However, with advancements in graphics card technologies and higher screen resolutions, the error caused by pixelation becomes negligible. Rocha et al. [24] evaluated various aspects of PxC algorithms implemented in Domus, including screen resolutions (128 × 128, 256 × 256, 512 × 512, and 1024 × 768), using a complex shading case study. They found that, starting from a resolution of 256 × 256 pixels, the results were nearly identical, with only a negligible absolute difference. In a similar context, Jones et al. (2011) demonstrated that a resolution of 512 × 512 pixels provides results with precision to three decimal digits, ensuring high accuracy when simulating the sunlit fraction on building surfaces [23].

Regarding computational cost and efficiency, studies have shown that PxC is the least time-consuming technique for solar shading predictions [24,34,35]. This method is particularly advantageous for simulating complex geometries without significantly increasing processing time thanks to its ability to leverage the parallel computing power of GPUs, for instance, via the OpenGL library. However, although the implementation of pixel counting is simpler compared to methods that require complex computational geometry algorithms, such as polygon intersection calculations, it is still dependent on graphics hardware, which may not always be available or compatible with certain GPUs, posing a barrier to its use [51].

PxC was initially slow to be adopted in BPS tools, as highlighted by Yezioro [40] in the SHADING tool. Shaviv [47] pointed out its limitations, including its reliance on static conditions and inability to assess dynamic shading systems. Despite this, efforts have been made to integrate PxC with dynamic simulation tools. Niewienda and Heidt [46] developed the SOMBRERO tool, which uses PxC to compute the geometric shading coefficient (GSC) for determining shaded areas over time and location. This tool was incorporated into TRNSYS and SUNCODE for accurate passive solar heating and cooling predictions. Moreover, PxC has been integrated into the Domus software (v1.7) [33], which uses methods from Yezioro and Shaviv [40] and Jones et al. [23] to calculate sunlit fractions and direct solar energy on surfaces. EnergyPlus also offers PxC integration, allowing users to choose between the CPU-based polygon clipping method, GPU-based PxC, or external shading data.

Other recent advancements in hardware-accelerated shading algorithms have been useful to improve the performance of PxC techniques. Hoover and Dogan [35] introduced some modifications to enhance efficiency, including a probabilistic transparency model, where pixels are pseudo-randomly discarded during rendering based on opacity. Additionally, the rendering process was optimized by rendering the entire scene at once rather than surface by surface. A stencil buffer was added to handle intersecting surfaces, ensuring that each pixel was counted only once. Arias [55] explored various shadow modeling techniques, including PxC, for photovoltaic (PV) panels, finding PxC to be more accurate and easier to be implemented compared to other methods.

4.6. Key Features and Classifications

Regarding the exterior shading calculation methods previously discussed, Figure 7 summarizes the main features of each technique. The simplest method is the ASHRAEapproach, which employs trigonometric relationships to calculate shading on windows. The other methods are categorized into two groups: polygon clipping and pixel counting.

Each of these methods has its strengths and limitations, and the choice of method should consider the complexity of the geometry, the required level of precision, and the computational resources available; see

Table 2. Comparison of advantages and limitations of shading calculation methods.

Method	Advantages and Limitations
Vectorial Methods (VMs)	Advantages: Effective for simple geometries (e.g., overhangs, inclined planes). High precision in shadow projections.
	Limitations: Computationally expensive for complex geometries. Requires considerable hardware resources.
Trigonometric Models (TgMs)	Advantages: Extremely efficient in computation time. Ideal for early-stage design simulations. Low learning curve.
	Limitations: Relies on simplified assumptions. Accuracy diminishes with irregular geometries. Integration challenges with BPS tools.
Discrete Element Analysis of Grids	Advantages: Highly flexible, simple. Handles convex and concave shapes. Easy to implement.
	Limitations: Accuracy compromised with coarse grids. Long processing time with high-density configurations.
Projection and Clipping Techniques (PgCs)	Advantages: High precision for planar surfaces. Handles a wide range of geometries, especially non-convex surfaces.
	Limitations: Computationally intensive. Difficulty handling irregular shapes. Requires specialized hardware and programming.
Pixel Counting (PxC)	Advantages: Very efficient in computation time. Leverages GPU power for large models. High accuracy with better hardware.
	Limitations: Dependent on compatible graphics hardware. Pixelation effects cause small inaccuracies.

.

5. Internal Solar Distribution Methods

Accurate calculation of internal sunpatches – areas directly illuminated by solar radiation within building interiors – is essential for predicting thermal comfort, energy efficiency, and daylight performance.

Table 3. Studies related to internal solar distribution methods.

Ref.	Year	Location	Output Data	Study	Sunlit Patch	Tools	Comparison with Tools	Study Case
[42]	1990	-	Sunpatch area, solar radiation impact on walls, floor, ceiling	Vectorial analysis, analytical methods for projection	Vectorial analysis, analytical methods for projection	Custom-developed model	Comparison with conventional methods	Rectangular single-zone room
[26]	1997	Denmark	Solar heat gain, surface temperature, heating/cooling loads	Comparison of four simulation programs calculating solar radiation based on the building’s geometry, considering direct and diffuse radiation.	Different sunpatch models are used in the tested software, respectively: grid elements, volume, area factor, decided by the user.	DEROB-LTH, SUNREP (TRNSYS), FRES, tsbi3	Compared with Simulation	Glazed space with adjacent room
[43]	1999	France	Sunpatch area, solar radiation flux, surface temperature, thermal comfort	3D dynamic simulation with sunpatch modeling and heat transfer calculations, computing solar radiation distribution, including direct and diffuse radiation, with surface interactions (absorption, reflection).	Projection and polygon clipping	TRNSYS, custom heat transfer and sunpatch models	Compared with Experimental	Room with multiple obstacles (furniture and occupant), highly glazed wall configuration
[44]	2001	-	Sunpatch area, surface temperature	Multizone unsteady thermal simulation with 3D modeling for solar radiation and thermal exchanges, using volumetric modeling with thermal zones, solar radiation simulation, and multi-reflection calculations.	Solar ray projection: solar rays are traced along a square grid to calculate sunlit areas by determining intersections with building surfaces. The grid size (dx) controls the accuracy of the normal solar flux, accounting for opaque, transparent, or masked surfaces.	SIMULA_3D software	Not directly compared	Glazed space with adjacent room
[73]	2002	France	Sunpatch area	Delaunay triangulation for mesh generation on walls, ceiling, and floor, used to model solar radiation over time.	Uses a triangular mesh, with visibility calculated through ray tracing by casting rays from each triangle’s center towards the sun, checking for obstructions, and considering the angle of incidence to model solar radiation distribution.	Computational tools for triangulation, ray tracing for solar load modeling	Not directly compared	Rectangular single-zone room
[74]	2003	Denmark	Sunpatch area	3D model creation using BSim2002 for thermal, moisture, and daylight simulation, with solar analysis using XSun and OpenGL for virtual tours.	Projection and polygon clipping by Xsun	BSim2002 (SimView, XSun, SimLight), Radiance, DirectX for visualization	Not directly compared	Real Building
[75]	2009	Greece	Solar radiation impact on walls, floor, ceiling, thermal loads, surface temperatures	Numerical methodology based on view factor theory, linked to TRNSYS, for solar radiation distribution on internal surfaces, considering multiple openings and orientations.	Solar radiation distribution through view factor methodology, considering geographical location and opening orientation.	TRNSYS for thermal simulation, Fortran for view factor-based distribution	Comparison with TRNSYS absorptance-weighted area ratio method for solar distribution	Single-zone and dual-zone building models with varying window placements
[29,30]	2013 2016	France	Solar heat gain, temperature distributions, heating and cooling demand, energy consumption	3D transient heat transfer model with sunpatch projection and time-stepping solver, incorporating heat transfer and sunpatch projection onto internal surfaces with an adaptive time step.	The sunpatch is projected on walls using geometrical tests, and thermal properties are calculated for each control volume.	Custom-developed model, HEAT3 meshing for 3D calculations	Comparison with 1D models (M1D, M1D, sp) that simplify solar radiation distribution and neglect 3D heat transfer	Rectangular single-zone room with high insulation
[76]	2014	Tunisia	Sunpatch area, temperature distribution, thermal comfort	Numerical simulation coupling sunspot model with ZAER for solar radiation distribution and air temperature, calculating sunspot distribution based on Sun position and room geometry.	Divides walls into facets, projects solar rays from their centroids towards the window to determine irradiation, computes projection coordinates using vector equations, and calculates the sunspot area by summing the irradiated facet areas.	ZAER software for thermal comfort and energy simulation, Fortran for sunspot model	Comparison with other simulation models (e.g., TRNSYS, EnergyPlus)	Two parallelepiped rooms
[45]	2015	Greece	Sunpatch area, solar heat gain	Pattern-based simulation of sunlit areas for solar heat gain, optimizing glazing size and placement based on solar angle and room geometry.	Using a four-character sunlit pattern, projecting solar rays through four vertices of an orthogonal glazing onto interior surfaces (floor, left wall, back wall) to calculate direct solar radiation impact, forming footprints labeled 1-B, 2-L, 3-F, and 4-L.	Custom-developed model	Not directly compared	Single-zone building model with varying glazing sizes and placements, and wall orientations
[77]	2016	Algeria	Surface temperature, surface temperature	Numerical simulation using TRNSYS for room and floor heating, and FLUENT for sunpatch modeling and solar radiation simulation.	Moving sunpatch simulated using Ray Tracing model as proposed in FLUENT.	TRNSYS 16 and FLUENT 6.3 software, Ray Tracing model for sunpatch calculation	Comparison with models that neglect sunpatch effects	Single room with south-facing window, multiple orientations (South, West)
[78]	2019	France	Sunpatch area, surface temperature	Pixel counting technique for sunpatch calculation using 2D orthogonal projections from the Sun’s perspective.	Pixel counting with occlusion query and Z-buffer techniques	Domus, Shading II SketchUp plugin, EnergyPlus	Compared with simulation	Rectangular single-zone room with various shading elements (e.g., cobogó), window configurations
[79]	2021	France	Indoor temperature, radiant floor temperature, temperature gradients in the slab	Numerical simulation using a validated 2D finite-difference model (FDM) with dynamic sunpatch heat load (moving patch) for transient conditions.	Simulated sunpatch exposure with heating film, dynamic sunpatch movement (3 × 20 cm every 1.5 h).	Matlab for parametric studies and model simulations	Comparison with other studies	Full-scale test cell with radiant floor heating system, varying sunpatch intensity, different air ventilation rates
[9]	2023	France	Floor surface temperature, air temperature, thermal comfort, thermal manikin data	Experimental study with dynamic sunpatch simulation using heating films on the floor and a thermal manikin for comfort evaluation.	Moving sunpatch simulated by heating films in different positions (1 m × 0.5 m per patch).	SketchUp for sunpath calculation, ANSYS Fluent® for solar intensity	Not directly compared	Case with small window (5.8% WWR), case with large window (11.6% WWR)

summarizes 15 studies on interior sunlit area calculation methods, highlighting the evolution of techniques like projection, ray tracing, and view factor methods.

In general, BPS tools have relied on simplified assumptions, such as modeling solar radiation as uniformly distributed across floors, with approximately 60% of the radiation directed to the ground surface [63]. While this approach offers computational efficiency, it fails to capture the complex and dynamic nature of solar radiation patterns, which are particularly important in buildings designed to optimize passive solar gains. This oversimplification overlooks the varying distribution of solar radiation, which significantly influences heating loads and thermal comfort, especially in spaces with extensive glazing. Studies have shown that neglecting the dynamics of sunpatches can result in heating power prediction errors of up to 15% [28], underscoring the limitations of these assumptions in accurately modeling solar effects.

Over time, methods for calculating sunlit areas have evolved from basic geometric projections to more advanced models based on view factors, reflecting a shift towards greater precision and analytical depth. Although these advances have not driven major changes in simulation software, they have led to incremental improvements, each contributing valuable insights into solar dynamics and their impact on building performance. This cumulative progress illustrates the collaborative nature of scientific research, enhancing our understanding of solar radiation in buildings.

One early method, introduced by Messadi [42], used vectorial analysis to determine the angular boundaries of sunlit areas in simple rectangular geometries. While groundbreaking at the time, this technique is limited to single-window rectilinear configurations. Expanding on this, Kontoleon [45] developed a “sunlit pattern” method that uses geometric projections of the vertices defining the glazing area onto internal surfaces. This method dynamically identifies sunlit areas based on the Sun’s position, projecting footprints onto interior walls and floors for each vertex. While accurate, this method is also constrained by its applicability to rectangular geometries.

Polygon clipping and ray tracing techniques have become widely used to evaluate interior direct solar radiation. Trombe et al. [43] applied PgC methods to map solar radiation in complex enclosures, incorporating factors such as furniture and occupants. By adapting algorithms from image processing, this approach demonstrated temperature differences of up to 10 °C between sunlit and shaded surfaces, highlighting its potential for thermal comfort analysis. Similarly, XSun [74] employed a double-projection method in BPS tools, which accounted for both wall thickness and solar angles. This method resulted in highly precise internal sunlit polygons, enabling improved visualization and energy analysis. Boukhris et al. [76] introduced a facet-based ray tracing method that reduced the computation time by 30% compared to traditional polygon clipping techniques while maintaining similar levels of accuracy.

Another important approach was used by Groleau and Marenne [44], who employed a mesh-based technique in Simula-3D, subdividing surfaces into triangular or quadrangular cells. Solar flux was calculated for each cell based on its visibility to the Sun, providing a precise determination of sunlit areas. The accuracy of this model depends on cell size: smaller cells improve precision but significantly increase computational costs.

Despite these advances, the computational cost of precise sunpatch calculations remains a significant challenge. Techniques such as PxC have emerged as efficient alternatives. Rocha et al. [78,80] demonstrated that pixel-based methods could simulate non-convex geometries with high computational efficiency, achieving accuracies within 5% of experimental measurements. The study revealed that the Domus PxC algorithm outperformed others, with a maximum error of only 0.2 m² in sunlit area calculations compared to the experimental results. In contrast, EnergyPlus, which uses the PgC method for sunlit pattern calculations, yielded similar results, with discrepancies under 5%. The surface temperature differences between sunlit and shaded areas were found to reach up to 4 °C, emphasizing the necessity for precise sunpatch modeling in thermal comfort assessments. Furthermore, Rocha et al. [78] demonstrated the efficiency of the PxC technique in Domus, showing that the method could produce stable results for complex geometries, such as T-shaped rooms with perforated shading elements (cobogós), in under an hour. This performance was notably superior to traditional polygon clipping methods, such as the Weiler–Atherton algorithm used in EnergyPlus, which struggles with non-convex geometries.

These advanced techniques have increasingly been integrated with thermal and energy models, significantly improving predictive accuracy. Rodler [30] coupled geometric sunpatch modeling with a three-dimensional conduction model, improving both spatial and temporal predictions. Li [9] further validated the thermal impact of dynamic sunpatches on radiant floors using experimental data, showing that sunlit zones experienced temperature increases of up to 8 °C compared to shaded areas.

In conclusion, while the established methods provided a foundation for sunlit area calculations, the integration of more advanced techniques, coupled with thermal and energy modeling, has resulted in significant improvements in both accuracy and computational efficiency. These developments offer more reliable predictions for building energy performance and occupant comfort, ensuring a better understanding of solar dynamics in buildings.

6. Solar Shading Calculation Features Across BPS Tools

Building on the previous sections, which outlined the main techniques for solar shading calculations, this section examines how these methods are implemented in specific BPS tools, focusing on the approaches used for external solar shading and internal solar distribution, with a review of their application in tools such as EnergyPlus, TRNSYS, ESP-r, and Domus, as listed in the BEST (Building Energy Software Tools) directory maintained by IBPSA-US [81].

Table 4. Features of solar shading and distribution in different BPS software.

Software	External Shading	Internal Distribution	Pre-Processing	Simulation Frequency	Default Time Step (min)	Simulation Method
EnergyPlus	Polygon Clipping; Pixel Counting	Floor incidence (homogeneous); Polygon Clipping	No	1 day every 20 days	15	Finite Differences
TRNSYS	ASHRAE; Polygon Clipping	Absorptance-weighted ratios; Polygon Clipping	Yes	1 day every 30 days	60	Finite Differences
ESP-r	Analysis of grids	Analysis of grids	Yes	1 day every 30 days	60	Finite Differences and Finite Elements
Domus	Pixel Counting	Pixel Counting	Yes	1 day every 7 days	20	Finite Differences

provides a comparative summary of the techniques and features used for solar shading calculations across different BPS tools, highlighting their methods for external shading, internal solar distribution, pre-processing requirements, simulation frequency, and default time steps.

6.1. EnergyPlus

EnergyPlus is a comprehensive open-source BPS tool developed from the most popular features of BLAST and DOE-2 [82], designed to model energy consumption and water use in buildings with advanced capabilities for thermal, HVAC, and fenestration simulations.

For solar shading calculations, EnergyPlus offers two primary methods for solar shading calculations: the projection with polygon clipping and the pixel counting technique (PxC), as proposed by Jones et al. [23]. The polygon clipping method includes algorithms such as Convex Weiler–Atherton, Sutherland–Hodgman, and Slater–Barsky (specific to rectangular surfaces). The PxC method, with a default resolution of 512 pixels, is integrated into EnergyPlus, allowing users to choose between the CPU-based polygon clipping, GPU-based PxC, or external shading data.

If PxC is selected but GPU hardware is unavailable, EnergyPlus (v. 22.1) defaults to polygon clipping and issues a warning. While polygon clipping generally provides shorter runtimes for fewer than 200 shading surfaces, PxC becomes more efficient as the number of surfaces increases. In systems with multiple GPUs, users may need to adjust settings to ensure optimal performance.

For internal solar distribution, EnergyPlus employs two approaches. The first, known as the Homogeneous Distribution on the Floor method, assumes that the solar beam is evenly spread across the floor surface [25]. In this method, the incident solar beam is absorbed proportionally to the floor’s solar absorptance. Any solar radiation reflected from the floor is then redistributed uniformly across all interior surfaces and combined with transmitted diffuse radiation. This approach offers a practical and computationally efficient means of modeling, although it may not accurately capture the complexities of solar distribution in complex spaces. The second is a Projection-Based Method, which calculates the transmission of sunrays through exterior windows and their distribution onto all interior surfaces, including the floor, walls, and windows. Any overlapping areas caused by multiple windows are addressed using the same polygon clipping techniques utilized for external shading [70].

In EnergyPlus (v22.1), the calculation of overlap areas is performed using shading routines, where the zone’s exterior window acts as the “sending” surface (radiation source) and the internal faces serve as “receiving surfaces”. This approach yields precise results but requires that zone surfaces be fully enclosed and strictly convex. For direct solar radiation entering a zone through an interior window, EnergyPlus does not track the specific surface impacted by the solar beam. Instead, the radiation is treated as diffuse and is uniformly distributed across all interior surfaces within the zone [70].

In addition to the solar shading algorithms, the method used to calculate solar angles (altitude and azimuth) significantly impacts the accuracy of sunlit surface fraction results. In EnergyPlus, these calculations rely on the solar declination angle and the equation of time, derived from Astronomical Algorithms by Meeus [83], as cited in [70].

EnergyPlus calculates shading and insolation dynamically during the simulation process rather than as a pre-processing step. By default, calculations are performed for one representative day every 20 days, using a 15 min time step as default. This approach enables precise and adaptable shading assessments, directly integrated with the simulation workflow.

6.2. TRNSYS

TRNSYS [84] (TRaNsient SYstem Simulation Program) is a modular simulation environment designed for transient systems, with a focus on thermal and electrical energy applications. Its standard library contains approximately 150 components, including HVAC systems, multizone buildings, wind turbines, and electrolyzers. Users can customize these components or develop new models to meet specific simulation needs. The platform also features a graphical interface that facilitates system assembly and simulation [85].

TRNSYS employs different modules to handle direct solar radiation: Type 34, Type 56, and TRNSHD. Type 34 evaluates beam, diffuse, and ground-reflected radiation using an ASHRAE algorithm ( [86]), to determine the external sunlit fraction. Type 56 focuses on internal insolation distribution using absorptance-weighted area distribution [17,87], which enables the proportion of solar radiation reaching each interior surface to be determined based on the material properties and surface areas of the elements within the space. TRNSHD, a stand-alone tool specifically developed for TRNSYS simulations, employs a polygon clipping method known as the Boundary Evaluation method to calculate both external shading and internal solar distribution [20,50].

In the TRNSHD method, internal surfaces are treated as shading elements, obstructing solar access to receiving surfaces. The remaining sunlit portions of these surfaces, determined through external shading calculations, define the “receiving polygon”. The clipping process is applied iteratively: each wall and window in the zone is clipped relative to the sunlit area from external shading. This iterative process progressively determines the new sunlit area of the receiving window, accounting for overlaps with internal surfaces. The sunlit fraction ($S{F}_i$) of internal surfaces is then calculated as

$$S{F}_i={\displaystyle \frac{A_{s,total}-A_{S,after}}{A_{s,total}}},$$

(11)

where $A_{s,\mathrm{after}}$ represents the sunlit area of the receiving window after the clipping process, and $A_{s,\mathrm{total}}$ is the initial sunlit area derived from external shading calculations. However, TRNSHD is currently restricted to convex zones as it does not allow internal surfaces to cast shadows on one another [17,63].

For Sun position calculations, TRNSYS uses Type 16, a solar radiation processor that determines solar azimuth and altitude angles through trigonometric relationships [88,89,90].

To optimize computational efficiency, solar shading calculations are performed as a pre-processing step. Shading patterns are computed and stored hourly for one representative day per month, reducing the computational load during simulation.

6.3. ESP-r

ESP-r [91] is a multi-domain simulation tool developed over three decades, designed to model building performance across thermal, airflow, HVAC, and electrical systems. Its modular approach dynamically integrates solvers based on model complexity, enabling simulations to range from individual rooms to entire neighborhoods that are tailored to specific project needs [65].

For solar shading calculations, ESP-r employs two distinct methods based on the complexity of the geometry. For simple geometries, such as overhangs and fins, it uses an ASHRAE procedure similar to TRNSYS. For more complex geometries, it applies a Discrete Element Analysis of Grids to calculate sunlit and shaded areas ([62] as cited in [50]). The grid-based approach is also used for internal solar distribution calculations [92].

In ESP-r, for instance, a grid is applied to the internal surfaces of a zone, with unshaded grid points on transparent surfaces serving as radiation sources at a given time step. Solar rays are projected into the space, enabling the software to calculate the percentage of sunlit points. While the accuracy of this method is influenced by the grid resolution, ESP-r effectively accommodates rooms of varying complexity by accounting for all internal surfaces [92].

Like TRNSYS, solar shading calculations in ESP-r are performed as a pre-processing step to reduce computational demand during simulations. By default, the percentage of sunlit points is calculated and stored hourly for one representative day per month. Solar angles (altitude and azimuth) are determined using an algorithm described by Clarke [93].

6.4. Domus

Domus [94] is a whole-building simulation tool designed for analyzing thermal comfort and energy performance. It incorporates models for coupled heat and moisture transfer, accounting for both vapor diffusion and capillary migration [33,95].

For solar shading calculations, Domus employs orthogonal projection and PxC techniques. Users can select between two PxC approaches: a two-phase PxC method, based on the Shading II plug-in [40], or a one-phase PxC method, derived from the procedure proposed by Jones et al. [34]. Both methods are implemented using OpenGL functionalities, enabling efficient handling of complex geometries.

The PxC process is conducted as a pre-processing step, where sunlit patterns are calculated every 7 days with a 20 min time step. Solar angles (altitude and azimuth) are determined using the algorithm developed by Reda and Andreas [96]. As already discussed, this approach ensures precise solar shading assessments while optimizing computational efficiency.

7. Conclusions

The accurate calculation of solar distribution and sunlit areas is critical for optimizing the energy performance and thermal comfort in buildings, particularly in the context of the increasing emphasis on sustainable design strategies. This study provides a comprehensive review of the methods employed in the calculation of sunlit areas and shading for building energy performance simulations, ranging from traditional techniques to advanced methods such as trigonometric, projection and clipping (PgC), and pixel counting (PxC). Each of these methods offers varying levels of accuracy and computational efficiency, with trigonometric and vectorial methods still prevalent due to their simplicity and low computational cost. However, more advanced techniques like PgC and PxC show considerable promise for simulations involving complex geometries, particularly in non-convex and dynamic environments.

A key finding of this review is the growing importance of leveraging graphics processing capabilities, such as GPUs, to enhance the efficiency of these simulations. Building performance simulation tools, including EnergyPlus, TRNSYS, ESP-r, and Domus, have progressively incorporated these methodologies, yielding more accurate and computationally efficient results, especially for complex urban scenarios. Nevertheless, challenges remain in ensuring the precision of sunlit area calculations, particularly when modeling irregular surfaces or implementing dynamic shading systems that require continuous updates during simulations.

Although the existing methods are adequate for a broad range of applications, future developments could benefit from integrating more sophisticated modeling approaches, as mentioned in the Future Research section.

Future Research

Some topics of future research that could be addressed include the integration of more advanced computational techniques, such as artificial intelligence (AI), into the calculation of sunlit areas and shading. Machine learning algorithms, in particular, could enable adaptive modeling that responds dynamically to varying conditions, such as the interaction of shading systems with environmental factors over the course of the day or throughout different seasons. This would enhance the versatility of simulations across a broader range of climates and architectural contexts.

Additionally, the combination of methods like PgC and PxC with 3D surface modeling and real-time visualization technologies presents an exciting opportunity for improving simulation accuracy. These advancements would facilitate more detailed evaluations of shading impacts on both thermal comfort and energy performance. Further exploration of GPU-based solutions is also crucial for reducing computation time and enabling the simulation of large datasets with greater efficiency.

However, while advancements in simulation techniques offer new levels of accuracy and efficiency, comparing quantitative results, such as temperature, thermal load, and heating and cooling demands, requires further research. We suggest that future studies address this by conducting simulations in a controlled and consistent manner, using the same input parameters across different methods. This would enable a more precise and objective analysis of how these techniques influence energy performance parameters.

Another promising direction for future work involves the development of comprehensive global databases that consolidate data on adaptive shading strategies, building materials, and regional climatic conditions. These databases could be integrated into simulation tools to offer personalized design recommendations, supporting the adoption of efficient shading technologies and contributing to the enhancement of energy performance in buildings worldwide.

Finally, the validation of these emerging simulation techniques through experimental studies is relevant. Ensuring their reliability in real-world applications will be critical to the broader adoption of advanced computational methods for the design of high-performance sustainable buildings.