Development, Implementation, and Experimental Validation of a Novel Thermal–Optical–Electrical Model for Photovoltaic Glazing

Abstract

The use of semi-transparent photovoltaic (Solar PV) glass in buildings is an effective strategy for integrating renewable energy generation, solar control, and thermal comfort. However, conventional simulation models rely on global optical properties, neglecting spectral radiation and its propagation within the material. This limits the accurate assessment of thermal comfort, light distribution, and performance in complex systems such as multi-layer glazing. This study presents the development, implementation, and experimental validation of a numerical model that reproduces the thermal, electrical, and optical behaviour of semi-transparent Solar PV glass, explicitly incorporating radiative transfer. The model simultaneously solves the conduction, convection, and electrical generation equations together with the radiative transfer equation, solved via the finite volume method across two spectral bands. The refractive index and extinction coefficient, derived from manufacturer-provided optical data, were used as inputs. Experimental validation employed 10% semi-transparent a-Si glass, comparing surface temperatures and electrical power generation. The model achieved average relative errors of 3.8% for temperature and 3.3% for electrical power. Comparisons with representative literature models yielded errors between 6% and 21%. Additionally, the proposed model estimated a solar factor of 0.32, closely matching the manufacturer’s 0.29.

1. Introduction

Although the European Union (EU) energy policies have been making great efforts to improve the energy performance of buildings and promote the use of renewable energy for years, faster transition is necessary to achieve the targets of the 2050 net-zero emissions scenario [1]. According to the International Energy Agency [2], buildings account for approximately 36% of global final energy consumption. Within this consumption, heating, ventilation and air conditioning systems are typically responsible for around 50% of the energy used in buildings. With regard to lighting, it accounts for approximately 10% of the energy consumption in buildings.

Based on the distribution of residential buildings in OECD (Organisation for Economic Co-operation and Development) countries (2019), 40% are flats in multi-family buildings. Furthermore, almost 75% of European buildings—more than 220 million—remain energy inefficient, and most of them are expected to remain in use over the coming decades. In this context, there is a broad commitment to energy refurbishment and renovation of the building stock to reduce the total energy demands of buildings and, whenever possible, replace fossil energy sources with locally available renewable alternatives.

Traditionally, the integration of these energy sources into buildings has relied on systems based on rooftop-installed panels, both thermal and photovoltaic [3]. However, many buildings, particularly in urban environments, present difficulties for the viability of these renewable generation systems, including limited installation space, small rooftops or poor solar orientation.

This fact has driven the development of new systems in recent years [4], particularly in the field of solar energy harvesting, where innovative technologies such as photovoltaic glazing have emerged. Photovoltaic (Solar PV) glazing is suitable for limiting solar heat gain and daylight while generating clean electricity [5]. Solar PV glazing consists of two glass panes with an integrated Solar PV layer between them. According to Romaní et al. [6], these systems are classified as cell-coated glazing and homogeneous Solar PV glazing. Cell-coated systems use opaque silicon (Si) cells, such as monocrystalline silicon (c-Si) with an efficiency of 26.1% and multicrystalline silicon (m-Si) with an efficiency of 23.3%, resulting in non-uniform transparency. Homogeneous Solar PV glazing employs thin-film technologies such as amorphous silicon (a-Si) with an efficiency of 14.0% and an average visible transmittance (AVT) of 10–25%, cadmium telluride (CdTe) with 22.1% efficiency and an AVT of around 30%, and copper indium (gallium) selenide [Ci(G)S] with 23.4% efficiency and AVT values below 30%, thereby providing a more uniform transparency. Emerging technologies within this group, such as dye-sensitised solar cells (DSSCs), perovskite cells, organic cells, and inorganic cells, show potential with AVT in the range of 20% to 80%. For example, perovskite cells achieve 25.2% efficiency with AVT around 30%, and organic cells have up to 17.4% efficiency with AVT of 20–65% [7].

These Solar PV glazing systems can be integrated into buildings, replacing conventional glass and serving as semi-transparent façades in Building-Integrated Photovoltaics (BIPV) applications [8]. The development of these devices has attracted considerable attention in recent years, with a wide variety of designs and approaches being explored to maximise solar energy utilisation while fulfilling key architectural functions, such as daylight control and thermal comfort. Comprehensive reviews on this topic can be found in Chen et al. [9], Shi et al. [10] and Gosh et al. [11].

The reviewed studies range from vacuum Solar PV glazing applied in different climatic conditions to hybrid modules incorporating phase-change materials, systems with parametrically controlled blinds, bifacial thermochromic glass, and double-skin façades integrating photovoltaic blinds alongside air purification functions.

In terms of energy performance, experimental results show reductions in heating and cooling demand ranging from 30% to 58% depending on the climate and device configuration. Additionally, electricity generation increased by up to 25% compared with conventional systems, demonstrating a strong potential for energy savings and overall improvements in building efficiency. A detailed description of these experiments’ results is provided in Table 1.

Assessing the energy impact of these new developments requires analysing their performance across different climates and building typologies. Simulation is a fundamental tool in such scenarios. Given that Solar PV glazing is a key component in these systems, it is essential to have reliable and accurate models capable of predicting its thermal, optical, and electrical behaviour.

In scientific literature, some authors have opted for the development of one-dimensional thermal resistance–capacitance (RC) models. Alternatively, it is also common to find numerical models based on finite element or finite volume methods. Additionally, many energy studies assessing the impact of Solar PV glazing on buildings have been carried out using models available in major building energy simulation programmes (BEPS) such as EnergyPlus [12,13].

Early studies commonly employed simplified RC-based approaches, representing each glazing layer through equivalent thermal resistances and capacitances. Yao et al. [14] developed an integrated thermal–electrical RC model for semi-transparent crystalline silicon curtain walls, solving energy balance equations for each layer, including outer and inner glass panes, encapsulant, and PV regions, and coupling conductive, convective, and radiative heat transfer with a five-parameter equivalent electrical circuit. The model achieved a root mean square error (RMSE) of 1.10 W in electrical power generation.

Similarly, Wang et al. [15] proposed a quasi-steady RC model for a ventilated double-glazing system integrating Solar PV blinds and thermocatalytic materials, in which each layer was treated individually and linked through conductive, convective, and radiative exchanges. Their validation reported errors below 7.2% for electrical efficiency and 6.5% for thermal efficiency.

Further simplifications were introduced by Yu et al. [16] and Ding et al. [17], who reduced the multilayer glazing to a one-dimensional RC network, assuming uniform temperature and solar absorption throughout the material. Instead of resolving layer-by-layer balances, they represented the entire glazing using one or a few thermal nodes and combined the boundary heat exchanges into global convective and radiative terms. Electrical generation was estimated via a temperature-corrected efficiency model, without explicit coupling to the electrical circuit. The optical domain was described using average absorption and emission coefficients, neglecting both wavelength dependence and incidence angle. These simplifications substantially reduced computational costs while maintaining acceptable accuracy, with coefficient-of-variation errors between 5% and 18%.

Among them, Fayaz et al. [18] used a 3D finite element model in COMSOL Multiphysics^® [19] to simulate hybrid PVT and PVT–PCM systems, representing the glass as a solid layer characterised by its thermal and optical properties. Validation under controlled conditions showed deviations below 1% in electrical efficiency. Huang et al. [20] also applied a 3D COMSOL to vacuum Solar PV glazing, incorporating both convective and radiative exchanges and obtaining errors below 5% for the overall heat transfer coefficient (U-value).

Other studies extended this approach to façade-scale applications. Youseff et al. [21] combined ANSYS Mechanical and ANSYS Fluent [22] thermal simulations with DIALux evo [23] optical modelling to evaluate transparent CdTe Solar PV walls, integrating measured solar radiation and ambient boundary conditions. Their results showed daily electrical generation of around 394 Wh/m² and electricity consumption reductions between 31% and 79% compared to conventional glazing. Du et al. [24] investigated transient heat transfer in glass–glass and glass–backsheet crystalline silicon modules using COMSOL, reporting temperature errors up to 9.7%. Likewise, Han et al. [25] employed finite element modelling in ANSYS to assess vacuum glazing, including conduction through support pillars and radiation between panes, achieving a 3.14% deviation in the simulated heat transfer coefficient.

Finally, some studies evaluating the energy impact of these Solar PV devices in buildings employed models already incorporated into major building energy simulation programmes. These models generally represented glazing using two main parameters: the overall heat transfer coefficient (U-value), which determined conductive heat transfer, and the Solar Heat Gain Coefficient (SHGC), which quantified the total solar radiation transmitted indoors per unit of radiation incident from the exterior [12]. For example, Wang et al. [26] assessed the energy performance of insulating glazing integrated with semi-transparent a-Si Solar PV modules. The objective was to develop a validated simulation model that incorporated measured electrical, optical, and thermal properties to simultaneously predict the electrical generation, thermal performance, and daylighting behaviour of the system. The WINDOW [27] and OPTICS [28] programmes were used to derive surface emissivity and thermal conductivity from the glass spectral data, which were subsequently exported to EnergyPlus. In parallel, the Sandia PV performance model [29] was employed to simulate electrical generation, accounting for the influence of the solar spectrum, angle of incidence, and operating temperature. Experimental validation was carried out by comparing measurements and simulations, yielding temperature deviations of up to 3.5 °C (MBE −1.2%, Cv(RMSE) 2.8%) and electrical power deviations of MBE 0.2% and Cv(RMSE) 22.8%.

A procedure similar to that described above was followed by Qiu et al. [30] for a vacuum Solar PV glass device. Their results showed that, in severe climates (e.g., Harbin), vacuum glazing could reduce energy consumption by up to 58%. Cooling energy savings ranged from 27% to 41%, depending on orientation; however, in moderate climates (e.g., Kunming), the system increased cooling demand, highlighting its limitations under different climatic conditions.

Although all the models reviewed were sufficiently accurate to simulate the overall energy performance of buildings, they lacked the level of detail required for lighting conditions, did not account for spectral discrimination of radiation, and did not consider the directionality of solar incidence factors that may be critical in certain applications. For instance, the louvre device developed by Alsukar et al. [31], described earlier, stands out for its comprehensive analysis of how solar radiation interacts with both the louvres and other system components. The interaction of these devices with the rest of the building is governed by how radiation is transmitted, reflected, or absorbed by the different elements. Consequently, any semi-transparent device integrated into a building requires a detailed radiative analysis, since its optical properties depend on both direction and wavelength. This radiative interaction between surfaces combines with other mechanisms, such as air movement between louvres, both forced and natural, which necessitate detailed calculations of the temperatures of the louvres and the surrounding medium. The study of such devices, therefore, involves the complex interplay of all these mechanisms. Moreover, as previously noted, spectral discrimination of radiation is crucial for analysing parameters related to both thermal and visual comfort. Thermal comfort is primarily influenced by infrared or long-wave radiation, whereas short-wave radiation plays a dominant role in visual comfort.

For this reason, the development of Solar PV glazing models that incorporate both thermal and optical dynamics is of particular interest. Although most of the models identified in the literature addressed thermal behaviour in detail, their treatment of optical behaviour was generally limited to the use of constant optical coefficients (absorptance, transmittance, and emissivity). In some cases, these coefficients were integrated over the short- and long-wave spectral bands; however, such integration was typically based on measurements taken in a single direction, usually at normal incidence.

The present work addresses these limitations by developing and experimentally validating a new numerical model for Solar PV glazing. The model accounts not only for the thermal behaviour of the device but also for the interaction of radiation within the semi-transparent glass, using wavelength-dependent refractive index and extinction coefficient data. Modelling radiative transfer through the medium with these properties enables the estimation of overall optical characteristics—such as absorptance, reflectance and transmittance—for both single- and multi-layer glazing configurations, while also capturing directional radiative mechanisms.

The model presented in this study is intended to serve as a foundation for future research involving more complex three-dimensional configurations, such as semi-transparent Solar PV louvre systems. In this context, the proposed formulation represents a significant advance over previous RC- or FVM-based approaches. In addition to experimental validation, a comparison with several widely used models from the reviewed literature has been carried out to assess the impact of incorporating internal radiative phenomena within Solar PV glazing.

Table 1. Overview of the analysis of various BIPV system configurations.

Author	Device Type	Thermal Model	Optical Model	Results
Qiu et al. [30]	Vacuum Solar PV glass	EnergyPlus + WINDOW	N/A	58% heating energy savings (Harbin–severe climates). Increased cooling demand in moderate climates (Kunming–moderate climates).
Wei et al. [32,33]	Hybrid Cd-Te-PCMG module (Solar PV with phase-change material).	Coupled optical–thermal–electrical model (parametric analysis of PCM thickness and PV coverage).	Global properties (no spectral discrimination).	Improvement of 0.53–0.99% in daily electrical generation.
Alsukkar et al. [31]	PV louvres integrated with Solar PV glass.	N/A	Parametrised optical model (DIALux evo) for light distribution and transmittance analysis.	Light uniformity of 0.70 with rear louvres. Optimal transmittance 50–70%.
Xu et al. [34]	Bifacial thermochromic Solar PV wall (BPVW-TC).	Thermal model with phase-change simulation (35.8 °C).	Global properties (no spectral discrimination).	Maximum efficiency 20.25% in summer. Thermal error 1.4 °C.
Yao et al. [14]	Semi-transparent Solar PV curtain walls.	RC model with thermal–electrical coupling (layered energy balance).	Average optical coefficients (no spectral discrimination).	RMSE 1.10 W in electrical generation.
Wang et al. [15]	Ventilated double glazing with Solar PV louvres.	Quasi-steady RC model with temperature correction.	Integrated absorption and emission coefficients (no spectral detail).	Maximum error 7.2% (electrical) and 6.5% (thermal).
Yu et al. [16]	Semi-transparent double-glazing Solar PV.	1D nodal model with average thermal coefficients.	Average optical coefficients (no spectral discrimination).	Cv(RMSE): 9.64% (PV temp.), 5.21% (rear glass temp.), 18.11% (electrical generation).
Ding et al. [17]	Semi-transparent Solar PV insulating glass.	RC model with uniform radiation absorption.	Constant optical coefficients (transmittance and absorptance).	Cv(RMSE): 8.56–17.28% for surface temperature and electrical generation.
Fayaz et al. [18]	Hybrid PVT and PVT-PCM systems.	3D model (COMSOL) with conduction equations and thermal source terms.	Integrated optical properties (surface emissivity and absorptance).	Differences < 1% in electrical efficiency (12.4% simulated vs. 12.28% experimental).
Huang et al. [20]	Vacuum Solar PV glass.	3D FEM model (COMSOL).	Surface emissivity and absorptance adjusted for long-/short-wave radiation.	Error < 5% in heat transfer coefficient (U-value).
Youseff et al. [21]	Transparent walls integrated with Solar PV.	3D model (ANSYS) for heat transfer and convection.	Lighting simulation (DIALux evo). Global optical properties in two spectral bands.	Daily generation ~394 Wh/m2. Electricity consumption reduced by 46.9–79.3%.
Du et al. [24]	Crystalline silicon Solar PV modules (GG and GB).	Multilayer heat transfer model (COMSOL) with glass, EVA, and silicon layers.	Integrated optical properties (absorptance and transmittance).	Maximum error 9.7% in cell temperature.
Han et al. [25]	Vacuum glass.	3D FEM model (ANSYS) for conduction through pillars and radiation between panes.	N/A	3.14% error in heat transfer coefficient.
Wang et al. [26]	Insulating glass with a-Si Solar PV.	EnergyPlus model with U-value and SHGC.	Spectral optical model (WINDOW/OPTICS) and Sandia PV simulation for electrical generation.	Temperature deviation up to 3.5 °C (Cv(RMSE) 2.8%). Electrical power error 22.8% due to spectral variability.

Note. N/A = Not applicable.

Table 1. Overview of the analysis of various BIPV system configurations.

Author	Device Type	Thermal Model	Optical Model	Results
Qiu et al. [30]	Vacuum Solar PV glass	EnergyPlus + WINDOW	N/A	58% heating energy savings (Harbin–severe climates). Increased cooling demand in moderate climates (Kunming–moderate climates).
Wei et al. [32,33]	Hybrid Cd-Te-PCMG module (Solar PV with phase-change material).	Coupled optical–thermal–electrical model (parametric analysis of PCM thickness and PV coverage).	Global properties (no spectral discrimination).	Improvement of 0.53–0.99% in daily electrical generation.
Alsukkar et al. [31]	PV louvres integrated with Solar PV glass.	N/A	Parametrised optical model (DIALux evo) for light distribution and transmittance analysis.	Light uniformity of 0.70 with rear louvres. Optimal transmittance 50–70%.
Xu et al. [34]	Bifacial thermochromic Solar PV wall (BPVW-TC).	Thermal model with phase-change simulation (35.8 °C).	Global properties (no spectral discrimination).	Maximum efficiency 20.25% in summer. Thermal error 1.4 °C.
Yao et al. [14]	Semi-transparent Solar PV curtain walls.	RC model with thermal–electrical coupling (layered energy balance).	Average optical coefficients (no spectral discrimination).	RMSE 1.10 W in electrical generation.
Wang et al. [15]	Ventilated double glazing with Solar PV louvres.	Quasi-steady RC model with temperature correction.	Integrated absorption and emission coefficients (no spectral detail).	Maximum error 7.2% (electrical) and 6.5% (thermal).
Yu et al. [16]	Semi-transparent double-glazing Solar PV.	1D nodal model with average thermal coefficients.	Average optical coefficients (no spectral discrimination).	Cv(RMSE): 9.64% (PV temp.), 5.21% (rear glass temp.), 18.11% (electrical generation).
Ding et al. [17]	Semi-transparent Solar PV insulating glass.	RC model with uniform radiation absorption.	Constant optical coefficients (transmittance and absorptance).	Cv(RMSE): 8.56–17.28% for surface temperature and electrical generation.
Fayaz et al. [18]	Hybrid PVT and PVT-PCM systems.	3D model (COMSOL) with conduction equations and thermal source terms.	Integrated optical properties (surface emissivity and absorptance).	Differences < 1% in electrical efficiency (12.4% simulated vs. 12.28% experimental).
Huang et al. [20]	Vacuum Solar PV glass.	3D FEM model (COMSOL).	Surface emissivity and absorptance adjusted for long-/short-wave radiation.	Error < 5% in heat transfer coefficient (U-value).
Youseff et al. [21]	Transparent walls integrated with Solar PV.	3D model (ANSYS) for heat transfer and convection.	Lighting simulation (DIALux evo). Global optical properties in two spectral bands.	Daily generation ~394 Wh/m2. Electricity consumption reduced by 46.9–79.3%.
Du et al. [24]	Crystalline silicon Solar PV modules (GG and GB).	Multilayer heat transfer model (COMSOL) with glass, EVA, and silicon layers.	Integrated optical properties (absorptance and transmittance).	Maximum error 9.7% in cell temperature.
Han et al. [25]	Vacuum glass.	3D FEM model (ANSYS) for conduction through pillars and radiation between panes.	N/A	3.14% error in heat transfer coefficient.
Wang et al. [26]	Insulating glass with a-Si Solar PV.	EnergyPlus model with U-value and SHGC.	Spectral optical model (WINDOW/OPTICS) and Sandia PV simulation for electrical generation.	Temperature deviation up to 3.5 °C (Cv(RMSE) 2.8%). Electrical power error 22.8% due to spectral variability.

Note. N/A = Not applicable.

2. Materials and Methods

The methodology presented below comprises, first, the development of the proposed model, which is structured into two main parts: a detailed formulation of the governing equations, describing the physical mechanisms incorporated into the model together with the principal assumptions adopted; and the solution methodology, which in this study employs a numerical approach based on the finite volume method. Subsequently, the experimental procedures undertaken for model validation are described. In addition, a comparative analysis is carried out between the predictions of the proposed model and those obtained from several reference models selected from the most widely adopted in the reviewed scientific literature. This comparison, incorporating both experimental measurements and reference model outputs, focuses on surface and volumetric temperature distributions, as well as electrical power generation.

2.1. Proposed Model

2.1.1. Governing Equations

Figure 1 illustrates the different heat transfer mechanisms considered in the simplified theoretical model of the Solar PV glazing. As shown, the model accounts for convective exchange between both glass surfaces and the surrounding air, incident short-wave radiation (in terms of both magnitude and direction), long-wave radiative exchange with the environment, conductive heat transfer within the glass, internal radiative behaviour within the solid (including reflection, absorption, and transmission), and electrical power generation.

Although Solar PV glazing consists of several distinct layers (Figure 2), including conventional glass, EVA, and the Solar PV cell, the thermal and optical properties provided by the manufacturers correspond to the equivalent properties of the complete assembly. For this reason, in the present approach, the Solar PV glass is modelled as a single semi-transparent solid with equivalent homogeneous properties.

In particular, the energy balance equation within the solid is expressed as:

$$\rho c_p{\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }t}}=k{\nabla }^{2}T-\nabla ·q_{rad}-P_{elec}$$

(1)

where $\rho$, $c_p$, and $k$ are the density, specific heat at constant pressure, and average thermal conductivity of the Solar PV glazing, respectively; $P_{elec}$ is the electrical power generated by the Solar PV layer; ${\nabla }^{2}T$ is the thermal conduction term, proportional to the Laplacian of temperature; and $\nabla ·q_{rad}$ is the divergence of the radiative flux, obtained from the solution of the radiative transfer equation (RTE).

If the Solar PV glazing is assumed to be a medium with negligible scattering, the RTE simplifies to:

$$\widehat{s}·\nabla I_{\lambda }\left(\overrightarrow{r},\widehat{s}\right)=\alpha I_{b,\lambda }\left(T\right)-\alpha I_{\lambda }\left(\overrightarrow{r},\widehat{s}\right)$$

(2)

where $I_{\lambda }\left(\overrightarrow{r},\widehat{s}\right)$ is the radiative intensity as a function of wavelength λ, at position $\overrightarrow{r}$ and in direction $\widehat{s}$; $\alpha$ is the absorption coefficient, which coincides with the extinction coefficient (k) when scattering is negligible; and $I_{b,\lambda }\left(T\right)$ is the blackbody intensity at temperature T.

The coupling between Equations (1) and (2) is achieved through the divergence of the radiative flux, where $q_{rad}$ is given by:

$$q_{rad}={\int }_{4\pi }{\int }_0^{\infty }{I}_{\lambda }\left(\overrightarrow{r},\widehat{s}\right)\widehat{s}d\lambda d\mathit{\Omega }$$

(3)

where $\mathit{\Omega }$ is the solid angle. The angular distribution of $q_{rad}$ is determined at each interface in accordance with Snell’s law.

$$n_{1}sin{\theta }_1=n_{2}sin{\theta }_2$$

(4)

where $n_i$ is the refractive index of medium i, and ${\theta }_i$ is the angle between the direction of radiation propagation in that medium and the normal to the interface surface.

Typically, glass manufacturers provide the optical properties R (reflectance) and T (transmittance) for both the visible and infrared spectra. These are global properties that account for all rays resulting from multiple reflections and transmissions within the medium (see Figure 1). Each individual interaction depends on the surface reflection factor between the glass ($\rho$) and the surrounding medium, as well as the transmission factor within the glass ($\tau$). For a given wavelength $\lambda$ and a specific direction $\theta$, these factors are related to T and R as follows [35]:

$$R\left(\lambda ,\theta \right)={\rho }_{\lambda }+{\displaystyle \frac{{\rho }_{\lambda }{\left(1-{\rho }_{\lambda }\right)}^2{\tau }_{\lambda }}{1-{{\rho }_{\lambda }}^2{{\tau }_{\lambda }}^2}}$$

(5)

$$T\left(\lambda ,\theta \right)={\displaystyle \frac{{\left(1-{\rho }_{\lambda }\right)}^2{\tau }_{\lambda }}{1-{{\rho }_{\lambda }}^2{{\tau }_{\lambda }}^2}}$$

(6)

where $\rho$ is the reflection factor between the glass and the surrounding medium, and $\tau$ is the transmission factor of the glass for a given thickness.

Likewise, the refractive index n and the absorption coefficient α are determined as follows:

$$n\left(\lambda \right)={\displaystyle \frac{1+\sqrt{{\rho }_{\lambda }}}{1-\sqrt{{\rho }_{\lambda }}}}$$

(7)

$$\alpha \left(\lambda \right)=-{\displaystyle \frac{\ln \left({\tau }_{\lambda }\right)}{e}}$$

(8)

where $e$ denotes the thickness of the glass.

It should be noted that glass manufacturers typically provide R and T data for normal incidence on the glass surface ($\theta =0$). However, reformulating these properties in terms of n and α, and consequently integrating the RTE into the energy balance within the solid, enables the modelling of radiation rays in any direction.

Both glass surfaces are coupled to the surrounding medium through the following expressions:

$$q_{rad}^{lw}=\epsilon \sigma \left(T_{si}^4-{T_{rd}}^4\right)$$

(9)

$$q_{conv}=h\left(T_{si}-T_{air}\right)$$

(10)

where $q_{rad}^{lw}$ is the heat flux due to long-wave radiation, $q_{conv}$ is the convective heat flux, $\epsilon$ is the surface emissivity in the infrared, $\sigma$ is the Stefan–Boltzmann constant, $h$ is the convective heat transfer coefficient, $T_{si}$ is the surface temperature of each face, and $T_{rd}$ is the exterior surface temperature, which, depending on the glass configuration, can correspond to the surface temperature of distant objects (approximated as the air temperature) or the sky temperature ($T_{sky}$) approximated by the following expression proposed by Duffie et al. [36]:

$$T_{sky}=T_{air}{\left[0.711+0.0056T_{dp}+0.000073T_{dp}^2\right]}^{1/4}$$

(11)

where $T_{sky}$ and $T_{air}$ are in Kelvin, and $T_{dp}$ (dew point temperature) is in degrees Celsius.

The convective heat transfer coefficients were modelled separately for each surface. For the upper surface, directly exposed to solar radiation, the Dittus-Boelter correlation was applied [37], which is specifically intended for forced convection under turbulent flow conditions over a surface. Given the exposure of the upper surface to wind, forced convection dominates in this case. For the lower surface of the Solar PV glass, the selected correlation corresponds to natural convection over the lower surface of a hot plate in a horizontal configuration [37].

Electrical power generation is estimated based on the variation in the cell efficiency, corrected for temperature [38]:

$${\eta }_{ \beta }={\eta }_{ref}\left(1-{\beta }_0\left(T_g-T_{ref}\right)\right)$$

(12)

where ${\eta }_{\beta }$ is the cell efficiency, defined as the ratio between $P_{elec}$ and $I$ (horizontal solar irradiance), ${\eta }_{ref}$ is the reference efficiency at the reference temperature $T_{ref}$, $T_g$ is the Solar PV glass temperature and ${\beta }_0$ is the cell temperature coefficient. It should be noted that ${\eta }_{\beta }$ represents the electrical conversion efficiency of the solar PV cell itself, rather than the overall conversion efficiency from incident solar radiation to electricity [39].

2.1.2. Numerical Resolution

The energy equation within the solid was solved using the finite volume method, employing the Least Squares Cell-Based approach for gradient evaluation, and a first-order implicit scheme for temporal discretization.

Electrical power was calculated at each time step based on the total incident solar radiation and Equation (12), and is included as a negative volumetric source term, as shown in Equation (1).

The radiative source term (Equation (3)) was solved using the Discrete Ordinates (DO) model [40], which solves the radiative transfer equation for a finite set of discrete directions within the angular domain. An angular discretization of 8 × 8 divisions in the polar and azimuthal planes was applied, along with a spatial resolution of 3 × 3.

A spectral discretisation was performed in two bands: short-wave (0–2.5 μm) and long-wave (2.5–5000 μm). Specifically, the average transmittance (T) and reflectance (R) values reported by the manufacturer for these two spectral ranges (see

Table 2. Thermal and optical properties of Onyx a-Si Dark 10% PV glass [41].

Model	Onyx a-Si Dark 10% PV Glass (Onyx Solar Energy S.L)
Dimensions [mm]	1245 × 635 × 7.15
Light transmission short wavelength [%]	10.8
Light reflection short wavelength [%]	8.3
Light transmission long wavelength [%]	0
Light reflection long wavelength [%]	16
Nominal peak Power [Wp]	29
Open-circuit voltage [V]	47
Short-circuit current [A]	1.11

), together with the corresponding refractive index ($n$) and absorption coefficient ($\alpha$) calculated for each band from the global T and R data of the glass using Equations (5)–(8), are presented in

Table 3. Spectral-band properties of the Solar PV glass.

Band	T [%]	R [%]	n [-]	$\mathsf{\alpha }$ [m−1]	Ɛ [-]
0–2.5 μm	10.8	8.3	1.75	289.51	-
2.5–5000 μm	1	15	2.25	599.11	0.84

. The long-wave emissivity was determined in accordance with Kirchhoff’s law.

A minimum global transmittance (T) was imposed for the long-wave band to ensure numerical stability.

A mesh independence analysis was performed (

Table 4. Mesh independence analysis.

	Number of Elements	Upper Surface Temperature [°C]	Lower Surface Temperature [°C]	Upper Surface Relative Change [%]	Lower Surface Relative Change [%]
Mesh 1	175,960	28.56	28.47	-	-
Mesh 2	792,456	28.50	28.32	0.21	0.53
Mesh 3	1,755,450	28.54	28.36	0.14	0.14

), using the average surface temperatures as the control variable. Based on the results, the intermediate mesh was selected for the simulations, as finer meshes did not significantly improve accuracy and increased computational cost (temperature variations were less than 1% between the intermediate and fine meshes).

The final mesh was structured, with an approximate element size of 3 × ${10}^{-3}$ m. The domain dimensions were 1245 × 635 mm with a thickness of 7.15 mm, corresponding to Onyx a-Si Dark 10% PV glass (Table 4).

To ensure an accurate representation of the variation in incident radiation throughout the simulation period, the solar position parameters, including solar altitude, $\beta$, azimuth angle $A_Z$, solar declination $\delta$, were determined using the following equations:

$$\sin \beta =\cos \mathrm{L}\cos \delta \cos \mathrm{H}+\sin \mathrm{L}\sin \delta$$

(13)

$$\cos Az={\displaystyle \frac{\sin \beta \sin \mathrm{L}-\sin \delta }{\cos \beta \cos \mathrm{L}}}$$

(14)

$$\mathsf{\delta } [°]=23.45\sin \left[{\displaystyle \frac{360·\left(284+\mathrm{d}\right)}{365}}\right]$$

(15)

where L is the local latitude, d is the number of days since the beginning of the year, and H is the hour angle, which converts the local solar time (LST) into the number of degrees the sun moves across the sky.

$$\mathrm{H} [°]=15(\mathrm{LST}\mathrm{ }-\mathrm{ }12)$$

(16)

The system model was implemented in Ansys FLUENT 23^®. The simulation time step was set to 5 min and a convergence criterion of 10⁻⁴ was applied to residuals. The solid was initialised homogenously at the initial temperature of the glass at the start of the experimental test.

2.2. Experiment

For the experimental test, a semi-transparent amorphous silicon PV glass panel with 10% transparency was used, which was installed horizontally on the roof of UCA-SEA Innovation Centre, Algeciras, Spain. Its main characteristics are detailed in Table 4.

As shown in Figure 2, the photovoltaic glass comprises several layers, including the outer glass panes, the encapsulant, and an internal matrix of opaque a-Si PV cell wiring. The spatial distribution and density of this wiring directly influence the effective optical properties of the glazing, particularly its visible transmittance. For this experiment, low-transparency glazing, characterised by a high PV-wire density, was intentionally chosen to maximise solar absorption within the material and thereby evaluate the performance of the proposed model under strong thermo–optical coupling conditions.

The density, thermal conductivity, and specific heat were approximated based on typical glass properties and the U-value reported by the manufacturer (

Table 5. Estimated properties of Onyx a-Si Dark 10% PV glass.

Property	Value
Density $\left[{\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^3}}\right]$	2500
Thermal conductivity $\left[{\displaystyle \frac{\mathrm{W}}{\mathrm{mK}}}\right]$	0.8
Specific heat $\left[{\displaystyle \frac{\mathrm{J}}{\mathrm{kg} \mathrm{K}}}\right]$	500

).

The test was conducted in the city of Algeciras (south of Spain), on 21 May 2025, between 11:20 and 16:30, ensuring that no shadows affected the device. The experimental setup is shown in Figure 3. During the test, data were collected every five minutes, including ambient temperature, panel surface temperatures, incident radiation on the horizontal plane, and wind values. Additionally, the output voltage was recorded to calculate the generated electrical power.

To monitor the temperature on both surfaces of the Solar PV glass, K-type thermocouples connected to a Testo 176T4 data logger were employed on the upper and lower faces of the glass. The thermocouples exhibited an uncertainty of ±2.2 °C or ±0.75% of the reading, whereas the data logger had an accuracy of ±0.3 °C within the range −100 to +70 °C and ±0.5% at higher ranges. The average glass temperature was then calculated as the mean of the temperatures measured on the upper and lower surfaces. These sensors enabled precise measurements of the thermal distribution over time. Simultaneously, the ambient temperature was obtained from a meteorological station located next to the glass, allowing correlation between atmospheric conditions and the thermal response of the system.

The incident solar radiation on the Solar PV glass surface was recorded using a pyranometer (Delta T SPN1), ensuring accurate measurement of total horizontal radiation. The instrument had an overall accuracy of ±5% for daily integrals, ±5% ± 10 W/m² for hourly averages, and ±8% ± 10 W/m² for individual readings.

To determine the convective heat transfer coefficient, wind speed measurements were carried out using an anemometer. These measurements enabled the assessment of the impact of forced convection on the thermal dissipation of the Solar PV glass.

To evaluate the electrical power output of the Solar PV glass, a 46 Ω load resistor was installed to allow the panel to operate within its optimal range during the experiment period. The output voltage was measured using a multimeter (PeakTech 3440, Ahrensburg, Germany) with an accuracy of ±0.05%, enabling the calculation of the generated power, $P_{elec}$, under different environmental conditions.

The adjustment of ${\beta }_0$ was performed by fitting Equation (12) to a sample of the experimental data using the least squares method. The validation of ${\beta }_0$ was carried out using the full set of experimental data, including measurements from additional days beyond those employed in the study of Section 3. The measured versus estimated efficiency is shown in Figure 4, and the RMSE was found to be 0.09%.

2.3. Reference Models

2.3.1. RC-Based Models

Two numerical RC (Resistance–capacitance) models were developed to reproduce the coupled thermal–electrical behaviour of the semi-transparent a-Si Solar PV glass under identical boundary conditions. These models are similar to those employed in references [16,17].

The first RC model (Reference Model 1) simplifies the Solar PV glass into a single thermal node representing its average volumetric temperature, as shown in Figure 5a. At each time step, the energy balance accounts for absorbed solar radiation, thermal losses through convection and radiation at both surfaces, and electrical generation losses. The temperature-dependent electrical efficiency is estimated by Equation (12), and the resulting electrical power is calculated as follows:

$$P_{elect}=-{\displaystyle \frac{{\eta }_{\beta }·I}{e}}$$

(17)

where $P_{elect}$ denotes the electrical power output, I [W/m²] is the total solar irradiance incident on the panel, and e [m] is the panel thickness.

The heat fluxes at the top and bottom surfaces, including radiative ($q_{rad}$ [W]) and convective ($q_{conv}$ [W]) contributions, are defined in Equations (9) and (10). The complete energy balance of the single-node model is expressed in Equation (18), using an implicit formulation:

$$\begin{gathered} -q_{rad,s}(t_{n+1})-q_{conv,s}(t_{n+1})+\alpha ·I(t_{n+1})·A-{\eta }_{\beta }·I(t_{n+1})·A-q_{rad,i}(t_{n+1}) \\ -q_{conv,i}(t_{n+1})-\rho ·C_p·V·{\displaystyle \frac{T_g\left(t_{n+1}\right)-T_g\left(t_n\right)}{∆t}}=0 \end{gathered}$$

(18)

where $q_{rad,s}$ [W] and $q_{rad,i}$ [W] represent radiative heat exchanges at the upper and lower surfaces, respectively, $q_{conv,s}$ [W] and $q_{conv,i}$ [W] are the convective heat exchanges at the upper and lower surfaces, $\alpha$ is the absorption of the semi-transparent Solar PV glass; $\rho$ [kg/m³] the density of the glass, $C_p$ [J/kgK] the specific heat capacity of the glass, V [m³] the volume of the glass, and $T_g$ [°C] the average volumetric temperature of the Solar PV glass.

The second RC model (Reference Model 2), similar to those used by other authors ([14,15]) is illustrated in Figure 5b. This model discretizes the Solar PV panel into three thermal nodes: the top surface, the glass core and the bottom surface. This configuration allows for a more realistic representation of conduction through the glass thickness while capturing independent convective and radiative exchanges at each surface. Conduction between the surface nodes $\left(q_{cond}\right[W\left]\right)$ and the core are described by Equation (19), and the energy balance at the core node is formulated in Equation (18). The electrical power generation is calculated using temperature-dependent efficiency, now based on the instantaneous temperature of the central node.

$$q_{cond}={\displaystyle \frac{{A·(T}_{Si}-T_g)}{e/k}}$$

(19)

where k [W/m·K] is the thermal conductivity of the material.

2.3.2. Thermal FVM-Based Model

As a complementary approach to the proposed model, a simplified thermal model was also implemented, following modelling strategies commonly found in the literature ([21,24]). This model (Reference Model 3) aimed to evaluate the impact of neglecting radiative heat transfer and spectral optical properties on the thermal and electrical behaviour of the Solar PV system.

The model geometry, illustrated in Figure 6, was divided into three solid domains, each assigned the same thermal properties as those used in the proposed model. However, no internal radiation model was included. The top surface of the upper solid was subjected to the ambient temperature and a convective boundary condition, using the external heat transfer coefficient obtained from the detailed model simulation. Likewise, the bottom surface of the lower solid was exposed to ambient air, applying the corresponding heat transfer coefficient.

The intermediate solid, representing the Solar PV layer, incorporated a volumetric source term to simulate electrical power generation. This source was defined using the same temperature-dependent efficiency model adopted in the proposed model, based on the temperature coefficient ${\beta }_0$. Additionally, the total horizontal irradiance was applied as a heat flux boundary condition on the top surface of the upper solid, thereby approximating the solar input as a uniform surface source. This simplification neglects spectral absorption, transmission or internal radiative exchanges within the glass.

3. Results

3.1. Experimental Results

The thermal behaviour of the semi-transparent Solar PV module was closely linked to the level of incident solar irradiance recorded during the experimental period. The maximum total irradiance measured on the horizontal plane was 1009.4 W/m², while the minimum was 725.9 W/m². This global irradiance consisted of both direct and diffuse components, with direct irradiance ranging from 639.2 W/m² to 905.1 W/m², and diffuse irradiance varying between 76.4 W/m² and 116.5 W/m². Variations in irradiance were influenced by atmospheric conditions, including solar altitude, wind speed and transient cloud cover.

The highest volumetric temperature recorded during the experiment reached 54.1 °C, with the upper surface peaking at 56.3 °C and the lower surface at 52.1 °C (Figure 7). Conversely, the lowest volumetric average temperature measured was 42.7 °C, corresponding to surface temperatures of 44.1 °C (upper) and 40.5 °C (lower). The results indicate a consistent thermal gradient across the panel thickness, resulting from direct solar exposure on the upper surface and natural convection effects on the lower surface. Ambient temperature during the experimental period ranged from 21.3 °C to 25.8 °C.

Regarding the measured electrical performance, the efficiency ranged between 2.3% and 3.7%. A gradual decrease of approximately 0.5% was observed over the experiment. This reduction coincided with the heating of the glass and the increase in solar irradiance during the same period (Figure 8).

3.2. Proposed Model Validation

To assess the reliability of the numerical model, a comparative validation was carried out between the simulation results and the experimental data. The validation considered both the thermal behaviour of the semi-transparent a-Si Solar PV glass and the electrical energy generated under real operating conditions.

For thermal performance, three temperature indicators were evaluated: the volumetric average temperature of the Solar PV glass, the top surface temperature, and the bottom surface temperature. Figure 9, Figure 10 and Figure 11 show the evolution of these temperatures throughout the monitored period. Each temperature profile is presented in a separate graph, comparing the experimental data and the simulation results. Additionally, upper and lower error bars were included to account for sensor uncertainties. Curves corresponding to the ±10% limits have also been plotted (dashed lines).

The average relative error in the volumetric temperature was 3.8%, with a RMSE of 2.4 °C. The top surface exhibited a higher deviation, with a relative error of 6.2% and an RMSE of 3.9 °C. The bottom surface showed a slightly lower deviation, with a relative error of 3.5% and an RMSE of 2.0 °C. The temperature ranges predicted by the simulation were consistent with experimental trends: the top surface reached a maximum of 53.4 °C, the bottom surface 53.0 °C, and the average volumetric temperature 53.3 °C. The minimum temperatures recorded during the simulation were 44.0 °C at the top surface, 41.4 °C at the bottom, and 43.6 °C within the glass volume.

The evaluation of the electrical performance is presented in Figure 12, showing the evolution of the electrical power output for two cases: the experimental measurements, obtained from actual irradiance and measured panel efficiency, and the simulated electrical power predicted by the proposed model. To account for uncertainties in the experimental data, each measurement is represented with upper and lower bounds corresponding to the estimated measurement error. The two series representing the ±10% limits have also been plotted.

The relative error between the simulated and the experimental values was 3.3%, with a RMSE of 1.2 W/m². In comparison, the deviation between the theoretical estimates and the experimental results was 1.7%, with an RMSE of 0.7 W/m², indicating good agreement and supporting the reliability of the β-based correction method implemented in the simulation. The simulated electrical power output ranged from 26.8 W/m² to 34.1 W/m², reflecting the expected variations in response to changes in solar irradiance and operating temperature.

3.3. Reference Models Comparison

The models, referred to as Reference Model 1 and Reference Model 2 in the methodology, differ in their level of detail and formulation of the internal energy balance. As described in the methodology, Reference Model 1 employs a simplified thermal node representing the glass volume and estimates electrical power output using a temperature-dependent efficiency model. In this case, the predicted volumetric glass temperature ranged from 43.6 °C to 65.4 °C (Figure 13), while the calculated electrical power generation (Figure 14) varied from 24.9 W/m² to 27.3 W/m². When compared with the experimental data, the mean relative error was 21.4% for temperature and 14.9% for electrical power (

Table 6. Comparison of RMSE and Relative Error between proposed and reference models for temperature, electrical generation and solar factor.

RMSE (Relative Error, %)	Reference Model 1	Reference Model 2	Reference Model 3	Proposed Model
Temperature (°C)	11.2 (21.4)	5.1 (9.8)	3.5 (5.7)	2.4 (3.8)
Electrical generation (W/m2)	4.8 (14.9)	2.0 (5.7)	0.8 (2.4)	1.2 (3.3)
Solar Factor (%)	-	-	-	10.3 (3.0)

). The corresponding RMSE values were 11.2 °C and 4.8 W/m², respectively.

In contrast, Reference Model 2 adopts a more refined approach by introducing a third thermal node and explicitly accounting for conduction between the upper surface, the glass core and the bottom surface. This allowed for a more accurate representation of internal gradients and transient behaviour. For this model, the predicted glass temperature ranged between 37.8 °C and 49.4 °C (Figure 13), while the electrical power output varied from 26.8 W/m² to 35.9 W/m² (Figure 14). Reference Model 2 demonstrated markedly improved accuracy, with a mean relative error of 9.8% for temperature and 5.7% for electrical power. RMSE values were 5.1 °C and 2.0 W/m², respectively.

The performance of the simplified FVM model (Reference Model 3) was evaluated against the experimental data. Regarding thermal behaviour (Figure 13), this model exhibited an average relative error of 5.7% for the volumetric glass temperature, with a corresponding RMSE of 3.5 °C. For electrical power generation (Figure 14), the simplified model showed an average relative error of 2.4% and an RMSE of 0.8 W/m².

Concerning optical properties, the Solar Factor (g) was calculated as the ratio of the total heat transmitted through the upper surface (considering the spectral and directional characteristics within the radiation domain) to the direct incident radiation on the same surface, resulting in a value of 0.32. This is in close agreement with the manufacturer’s reported value of 0.29. It is noteworthy that the manufacturer determined this value according to the UNE-EN 410:2011 standard [42], which defines g based on the global optical properties (transmittance, reflectance, and absorptance) of the complete glass at normal incidence.

The proposed model achieves average relative errors of 3.8% for temperature and 3.3% for electrical generation, outperforming the simplified RC and FVM models (which reported thermal errors of 6.5–17% and electrical errors of 1–18%, see Table 1). It also compares favourably with other detailed 3D numerical approaches that neglect internal radiative phenomena. The models reported in the literature (Yao [14], Wang [15], Yu [16], Ding [17]) provide reasonable approximations for global energy simulations; however, they lack both spectral and directional resolution, which limits their applicability to detailed thermal and visual comfort analysis. Even the 3D numerical models (Fayaz [18], Huang [20]) treat radiation only superficially, without explicitly solving the RTE.

In contrast, the proposed model explicitly solves the RTE across two spectral bands. Consequently, the estimation of optical properties using this approach becomes particularly relevant when the glass forms part of a complex multi-pane system. By coupling one or more glass layers, each characterised by its refractive index and absorption coefficient, the model enables the calculation not only of the Solar Factor (g), but also of the band-integrated reflectance (R) and transmittance (T) for any direction, even in cases where these values are not previously known.

4. Conclusions

This study addresses a recurring limitation in current photovoltaic glazing models: these models often rely on global optical properties and simplified representations of radiative exchange, which limit the accurate representation of spectral absorption, directional effects, and internal radiation propagation in semi-transparent systems. This methodological gap is especially critical in advanced devices, particularly in multilayer configurations or in applications where thermal and visual comfort strongly depend on the spectral response of the glazing. The proposed methodology is particularly valuable for analysing complex glazing systems in which the global optical properties of the assembly are not known a priori, such as multilayer configurations or assemblies with varying absorption and reflection characteristics. It is especially relevant when the interaction between the PV-cell wiring and the glass needs to be represented to support geometric and configuration optimisation. Under such conditions, the model enables the accurate modelling of optical and thermal phenomena that conventional formulations are unable to capture.

From an implementation perspective, the model introduces several methodological enhancements. First, it employs the band-integrated reflectance and transmittance values provided by the manufacturer to derive, through optical inversion, the refractive index and absorption coefficient for each spectral band. This approach allows the reconstruction of the directional and spectral behaviour of the glazing without requiring detailed wavelength-dependent optical data. Second, the formulation simultaneously couples thermal mechanisms (conduction and convection), radiative processes (via the radiative transfer equation solved using the finite volume method), and electrical generation (through the temperature-dependent efficiency model). This fully coupled framework provides spatial temperature distributions, short-wave and long-wave radiative interactions, and electrical output under realistic operating conditions.

The comparison with the three reference models developed in this study highlights the advantages of the proposed formulation. Among the RC approaches, Reference Model 2 provided the most representative performance, with temperature and electrical generation errors of approximately 9.8% and 5.7%, respectively. Reference Model 3, based on a simplified FVM approach, yielded improved predictions; however, its thermal and electrical errors remained higher than those obtained with the detailed model presented here. This demonstrates that explicitly resolving internal radiative transfer and spectral optical properties leads to an improvement in predictive accuracy.

In conclusion, the model presented here not only outperformed the reference models and the approaches reported in the literature in predicting both temperature and electrical generation but also established the foundation for its application to complex configurations, such as multilayer glazing and different PV-wire arrangements, enabling the prediction of global thermal and optical properties that are not known a priori.

Certain limitations should, however, be acknowledged. In the present study, the Solar PV glass was modelled as a homogeneous medium, neglecting the individual layers of glass, encapsulant, and photovoltaic film; this aspect will be addressed in future work.

5. Future Work

Future work will address these limitations. Specifically, the model will be extended to explicitly represent multilayer glazing systems, in which each layer is characterised by its own thermal and optical properties. In parallel, future implementations will incorporate photovoltaic wire geometries within the numerical domain, allowing the model to reproduce the effects of wire distribution and shading on optical and electrical performance. These developments will support the design and optimisation of advanced solar PV glazing technologies, including adaptive and architecturally integrated configurations for next-generation Building-Integrated Photovoltaics (BIPV).

Additionally, although the selected glazing provided a suitable test case for assessing the model under pronounced radiative and thermal interactions, further validation using PV glasses with different transparency levels and wire densities will be necessary. Such configurations would introduce optical behaviours that differ substantially from those of the nearly opaque device analysed here, thereby enabling a more comprehensive verification of the model’s ability to predict global optical and thermal properties in semi-transparent photovoltaic systems. Consequently, the experimental campaign will be extended to include glazing types with varying degrees of visible transmittance and internal PV-wire arrangements.