# Optimal tilt angles for bifacial photovoltaic plants across Europe based on cumulative sky and Typical Meteorological Year data

Apolline Ferry, Mattia Parenti, Martin Thebault, Christophe Ménézo, Marco Fossa

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Apolline Ferry, Mattia Parenti, Martin Thebault, Christophe Ménézo, Marco Fossa. Optimal tilt angles for bifacial photovoltaic plants across Europe based on cumulative sky and Typical Meteorological Year data. Solar Energy, 2025, 10.57745/8I4MZ. hal-05028604

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# Optimal tilt angles for bifacial photovoltaic plants across Europe based on cumulative sky and Typical Meteorological Year data

Apolline Ferry<sup>a,*</sup>, Mattia Parenti<sup>b</sup>, Martin Thebault<sup>a</sup>, Christophe Ménézo<sup>a</sup>, Marco Fossa<sup>b</sup>

## ARTICLE INFO

Dataset link: https://doi.org/10.57745/814MZ3

**Keywords:**

Bifacial PV plants

Solar radiation modeling

Optimal tilt angle

Europe

Topography

GIS

## ABSTRACT

The European targets for the Energy Transition by 2030 are strongly oriented toward the massive diffusion of PV plants, both at the building level and in large-scale utility systems on land. The expected PV capacity in Europe by 2030 is 900 GW, 4.3 times the 2022 level. Bifacial modules are currently the best solution for utility-scale PV systems, offering higher solar energy yields with minimal additional costs. This paper helps improve the efficiency of bifacial PV power plants at the European scale through the optimization of tilt angles in various configurations. Typical Meteorological Year data are retrieved from the PVGIS platform for a series of 2382 points uniformly distributed across Europe according to a regular 50 km × 50 km grid. A cumulative sky radiance is built for each location and an iterative search for the best tilt angle is performed to maximize the annual energy yield (on both front and back surfaces considering direct, diffuse and reflected energy) of south-facing modules, with both ground albedo and PV row spacing taken as parameters. Latitude is the primary influencing factor for optimal tilt angles, with an average value ranging from 26° for Greece to 36° for Finland. The results indicate that the Ground Coverage Ratio is also a significant parameter, while albedo has only a minor effect. The study of an Alpine region with a finer grid shows that mountainous terrain leads to reduced optimal tilt angles. A free web-based tool has been developed to provide optimal tilt angles for any location in Europe.

## 1. Introduction

In recent years, the worldwide transition to more sustainable energy models, driven by improvements in energy-efficient production systems and the broad integration of distributed renewable generation, has resulted in a significant increase in solar energy installations. This deployment, encouraged by specific policies implemented by numerous countries and organizations in response to the climate emergency, has gained speed due to a growing recognition of the necessity to diversify energy sources and reduce reliance on foreign energy supplies.

In this context, photovoltaic (PV) systems play a crucial role, with a total installed capacity reaching 1183 GW by the end of 2022. Notably, over half of this capacity has been added within the last four years, according to the 2023 IEA Report. Within the European Union, the PV installed capacity in 2023 amounts to 263 GW, considering 56 GW of new installations only in 2023 [1].

Recently, there has been a widespread installation of large-scale ground-mounted grid-connected photovoltaic (GCPV) systems using bifacial PV modules. These modules can absorb sunlight from both their front and rear sides [2], offering higher energy yields compared to

conventional monofacial modules [3]. It has been demonstrated that idealized standalone bifacial modules are able to produce up to 50% more energy than their monofacial counterparts in similar configurations [4]. However, in practice, this gain was never reached and typically ranges between 10% and 30%, depending on installation conditions and site characteristics [5-7], with the highest values obtained with highly reflective ground surfaces [3,8,9]. These higher gains enable increased energy density and reduced area-related cost [10], with only minimal technical modifications or additional costs when set up in optimized configurations.

The 2023 ITRPV report indicates that bifacial modules have become increasingly popular and are projected to secure a substantial portion of the PV market, potentially exceeding 70% of the market share of modules within the next decade [11]. While bifacial PV modules have been industrially produced since the 1980s [12,13], power plants using this technology are developed mainly since the 2010s, in various regions and climates [6,14,15]. This has led to a reduction in the system's Levelized Cost of Energy (LCOE) [16]: world-average LCOE of PV plants has witnessed an 89% reduction from 2010 to 2022,

<footnote><sup>a</sup> University Savoie Mont-Blanc, LOCIE UMR CNRS 5271, Le Bourget-du-Lac cedex, F- 73376, France</footnote>

<footnote><sup>b</sup> Dime Dept. of Mechanical, Energy, Management and Transportation Engineering, University of Genova, via Opera Pia 15a 16145, Genova, Italy</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: apolline.ferry@univ-smb.fr (A. Ferry).</footnote>
Nomenclature

Greek Symbols

<table><tr><td>χ</td><td>Angle between the direction of an element and the direction of the sun [rad]</td></tr><tr><td>γ</td><td>Azimuth angle [°]</td></tr><tr><td>Ω</td><td>Solid angle of a sky element [sr]</td></tr><tr><td>φ</td><td>Auxiliary angle for view factor calculation [°]</td></tr><tr><td>ρ</td><td>Albedo or reflectivity of the ground [–]</td></tr><tr><td>θ</td><td>Zenith angle [°]</td></tr><tr><td>θ<sub>w</sub></td><td>Incidence angle of beams on a surface [°]</td></tr></table>

Subscripts

<table><tr><td>el</td><td>Relative to a generic element of sky</td></tr><tr><td>gr</td><td>Relative to the ground</td></tr><tr><td>i</td><td>Relative to element i of sky</td></tr><tr><td>j</td><td>Relative to element j of ground</td></tr><tr><td>k</td><td>Relative to element k of module</td></tr><tr><td>mod</td><td>Relative to the module</td></tr><tr><td>opt_tilt</td><td>Relative to optimal tilt angles</td></tr><tr><td>tilt=lat</td><td>Relative to tilt angles equal to latitude angles</td></tr></table>

Symbols

<table><tr><td>a</td><td>Perez coefficient related to horizon brightening or darkening [–]</td></tr><tr><td>a<sub>r</sub></td><td>Angular losses coefficient [–]</td></tr><tr><td>b</td><td>Perez coefficient related to horizontal gradient of luminance [–]</td></tr><tr><td>c</td><td>Perez coefficient related to circumsolar intensity [–]</td></tr><tr><td>d</td><td>Perez coefficient related to circumsolar width [–]</td></tr><tr><td>DHI</td><td>Diffuse horizontal irradiance [W/m<sup>2</sup>]</td></tr><tr><td>DNI</td><td>Direct normal irradiance [W/m<sup>2</sup>]</td></tr><tr><td>E</td><td>Yearly irradiation on a surface [Wh/m<sup>2</sup>]</td></tr><tr><td>e</td><td>Perez coefficient related to backscattering light on earth surface [–]</td></tr><tr><td>Gain</td><td>Gain in irradiance by using optimal tilt angles rather than tilt angles equal to latitude</td></tr><tr><td>GCR</td><td>Ground Coverage Ratio [–]</td></tr><tr><td>G</td><td>Solar irradiance incident on a surface [W/m<sup>2</sup>]</td></tr><tr><td>GHI</td><td>Global horizontal irradiance [W/m<sup>2</sup>]</td></tr><tr><td>H</td><td>Height from ground to the center of PV modules [m]</td></tr><tr><td>IAM</td><td>Incidence angle modifier</td></tr><tr><td>L</td><td>Width of a PV row [m]</td></tr><tr><td>Lv</td><td>Radiance of a sky element [W/m<sup>2</sup> sr]</td></tr><tr><td>lv*</td><td>Relative radiance of a sky element [–]</td></tr><tr><td>N</td><td>Number sky patches [–]</td></tr><tr><td>obs</td><td>Obstruction factor [–]</td></tr><tr><td>P</td><td>Row pitch [m]</td></tr><tr><td>R<sup>2</sup></td><td>Coefficient of determination [–]</td></tr><tr><td>SVI</td><td>Sky View Index [–]</td></tr><tr><td>VF</td><td>View factor [–]</td></tr></table>

Acronyms

<table><tr><td>AWM</td><td>All-Weather-Model</td></tr></table>

<table><tr><td>CM SAT</td><td>Satellite Application Facility on Climate Monitoring</td></tr><tr><td>DTM</td><td>Digital terrain model</td></tr><tr><td>LCOE</td><td>Levelized cost of electricity</td></tr><tr><td>PV</td><td>Photovoltaic</td></tr><tr><td>RT</td><td>Ray tracing</td></tr><tr><td>TMY</td><td>Typical meteorological year</td></tr></table>

standing at 29% below the cost of the least expensive fossil fuel-fired alternative, as delineated in the latest report on power generation costs by IRENA [17]. Notably, Rodríguez-Gallegos et al. [16] demonstrated that bifacial PV modules achieved a lower LCOE than monofacial ones under the same configuration.

A common approach to estimate the optimal tilt angle is to set it equal or close to latitude [18] to maximize annual energy production, but this approach overlooks local factors such as cloud coverage patterns or topography [19]. In the existing literature, different works analyze the best tilt angle for monofacial modules, employing different optimization techniques [19], such as Genetic Algorithm [20], Simulated Annealing method [20], Particle-Swarm Optimization combined with non-linear time varying evolution [21], Artificial Neural Network [22] such as Radial Basis Function Neural Network [23], Decision Trees [22], Multi-Linear Regression [22–27] or correlations [26, 28], with experimental comparisons and validations [29]. Annual optimal tilt angles primarily depend on latitude, clearness index and azimuth [28], while seasonal optimal tilt angles are more influenced by the declination angle [26]. The optimal tilt angle depends on how cloud cover is modeled, especially in cloudy regions such Northern Europe, favoring the collection of diffuse rather than direct irradiance [19]. Thus, Smith et al. [30] considers the cloud distribution using a radiative heat transfer model to predict the optimal tilt angle, whereas Raptis et al. [31] consider real atmospheric conditions, including the effects of clouds and aerosols.

For bifacial PV systems, it is essential to accurately model the irradiance received by the PV modules, particularly the rear-side, and more specifically, the ground-reflected irradiance (eventually including reflections from neighboring arrays), while accounting for shading from the modules on the ground. Two fundamental methods are commonly used: view factors (VF) and ray tracing (RT).

RT is a rendering technique, designed specifically for 3D geometries, that uses stochastic rays to capture complex shading and reflections through interactions with surfaces. VF (or configurations factors) are simpler models that compute irradiance exchanges between surfaces while considering shading. Many studies on bifacial modules use either RT [3,32–36] or VF models [37–46], with some employing a combination of both. For example, MoBiDig simulation models use VF for the front-side irradiance in combination with RT for modeling the rear-side irradiance [7], while DUET shading algorithm pairs a 3D VF model with deterministic ray-object intersection [47,48]. Both methods have demonstrated good accuracy with strong agreement with field data [49,50]. However, RT models are computationally intensive, VF model has been therefore chosen here to keep a reasonable computation time.

In the optimization of monofacial PV systems, tilt angle is the primary design parameters, as it strongly influences the energy yield (along with azimuth angle, often set to south to maximize the energy yield). However, for bifacial PV systems, multiple installation parameters affect the optimization of the energy yield, and previous studies have examined the influence of various of these parameters on optimal tilt angle and maximum power output. Chudinzow et al. [38] evaluated the impact of several design factors on the bifacial energy gain, increasing the row-to-row distance enhances energy gain reducing ground shading. The gain also increases with the latitude, as the optimal tilt
angle rises with the lower sun altitude [51], allowing more ground surface to be seen by the system and increasing reflected irradiance on both sides. Increasing the ground clearance (up to a certain limit) increases the gain and the energy yield by enlarging the view field and increasing ground-reflected irradiance absorption [37,46]. The optimal tilt angle increases with lower ground clearance to capture more irradiance from both the ground and the sky [32]. Vertical east-west-facing configuration is then favorable for low ground clearance [37,46].

Tonita et al. [47] assessed the impact of ground coverage ratio (GCR) on the energy performance of bifacial modules, showing that high GCR increase shading losses. Thus, the optimal tilt angle decreases with high GCR to limit row-to-row shading, but remains higher than that for monofacial panels. Rodrigo et al. [42] further demonstrated that at high latitudes, a lower GCR is needed to maintain a given energy loss ratio due to the increased impact of shading from low sun elevations. In the context of fixed-tilt plants, it has been evidenced that self-shading can significantly impact energy production during a portion of the day, particularly in the winter, even at low latitudes in Europe [52]. Additionally, a lower GCR is needed for higher albedo and greater ground clearance for a constant energy loss ratio.

Several studies have shown that increasing ground albedo, using retro-reflective materials [3] or highly reflective surfaces [53], significantly improves bifacial gain and energy yield [37,46]. Optimal tilt angles are higher for high ground albedo to maximize reflections from the ground on both sides of the modules [32,51]. Alam et al. [53] also demonstrated that the rear-side irradiance correlates with the clearness index and increases with a higher diffuse fraction, which in turn enhances bifacial gain [37]. For cloudy conditions, a lower optimal tilt angle is required to expose the front side to a larger portion of the sky, but this angle is still lower than that of monofacial panels [32,51].

Large scale systems with multi-rows installations induce higher energy losses in bifacial systems than in monofacial systems due to ground shading that limits reflected irradiance [7]. In these configurations, optimal tilt angles are generally higher to reduce the shadowing area (up to 20° higher compared to single-module system) [33]. The effects of ground clearance and albedo are diminished in such cases due to the reduced impact of ground-reflected irradiance.

Although numerous studies have examined the relationship between tilt angle, other design parameters and energy yield, most focus on specific locations [32,46] or a limited set of sites across a country or continent [38,53]. Other studies selected sites based on data availability, such as Barbosa et al. [51] who retained 218 locations from a 10° × 10° global grid using meteorological station data, or Baricchio et al. [45], who selected 102 locations based on the availability of Meteonorm data. Some other studies cover global scales, presenting worldwide maps but with low (unspecified) spatial resolution, aiming to provide general trends [39–41]. Finally, Rodríguez-Gallegos et al. [16] employed a 1° × 1° resolution to determine the LCOE of bifacial and monofacial technologies globally, but limited efforts were dedicated to accurate irradiance modeling, with no accounting for ground shading or rear-side irradiance inhomogeneity.

Furthermore, optimization of tilt angle as well as solar resource modeling in mountainous areas has been rarely addressed [54], even though mountains account for a significant proportion of both the world's land area (25%) and population (12%) [55]. Sharma et al. [56] and Awasthi et al. [57] derived correlations between optimal tilt angle and latitude or declination for locations situated in the Western Himalayan region for monofacial PV systems. Xu et al. [58] developed a methodology to obtain the optimal tilt angle for monofacial systems for high-altitude mountainous regions using a viewshed approach that considers the mountain terrain shading, based on Digital Elevation Model (DEM).

Despite the increasing consideration of various parameters in the optimization of bifacial system design, there is a notable lack of multi-scale studies, considering locations uniformly spread over a territory

with a detailed irradiance calculation. Furthermore, the mountainous areas have been largely overlooked in tilt angle optimization, particularly for bifacial PV systems.

The present study aims to create a European-scale map that presents the optimal tilt angle for south-facing bifacial plants, uniformly distributed on a 50 km × 50 km grid, as a function of the ground coverage ratio (related to the row-to-row spacing) and ground albedo. The optimization has been made by maximizing the total incident radiation on both the front and rear surfaces within the central section of the plant.

The main contribution of the present paper are thus :

* The calculation of the annual cumulative irradiance received by a south-facing bifacial system using a cumulative sky radiance approach and view factors that account for ground reflection and model obstructions from adjacent module rows.

* The optimization of the tilt angle for a south-facing bifacial PV system under various configurations, with the Ground Coverage Ratio and the average annual albedo as input parameters.

* The evaluation of the impact of mountainous terrain on optimal tilt angles with a detailed analysis focusing on the French Alps.

## 2. Methodology

### 2.1. Databases

Two types of data were needed for this study: geographical and meteorological databases. The selected databases had to provide spatial coverage encompassing the entire studied area, which is the European territory, with Ukraine being the easternmost country.

#### 2.1.1. Geographical databases

The geographical data required for this study are the geographic boundaries of European countries. The database used is the Global Administrative Areas (GADM), which contains vector descriptions of administrative boundaries [59]. The database was developed by the university of Berkeley (California). Several levels of sub-divisions are available, the country level was chosen for the study. To investigate the effects of the mountains on the optimal tilt angle, the delineation of mountains area in Europe was utilized. The vector description of the delineation was created by the European Environment Agency in 2008, based on a combination of thresholds of various criteria such as altitude, climate and topographical variables [60,61]. The boundaries of the French mountainous areas were defined using the French department-level from the GADM database.

#### 2.1.2. Meteorological databases

The meteorological database needed for this study is a database of Typical Meteorological Year (TMY) at an hourly time step. The web application PVGIS, developed by the European Commission's Research Center, is used as it contains several solar irradiation databases that have a wide coverage, including Europe [62]. These datasets are also accessible through the python library pvlib [63] with the iotools sub-package [64], that aims at retrieving solar irradiance datasets in a standardized way. The solar radiation database PVGIS-SARAH-2 is the default database for Europe. It is derived from the second version of the Surface Solar Radiation Data Set - Heliosat (SARAH-2) [65], produced by the Satellite Application on Climate Monitoring (CM SAF) collaboration and has been validated with ground measurement in Europe [66]. Satellite observations of the geostationary Meteosat satellite are used to retrieve the cloud index, which quantifies the reflective properties of the atmosphere [67]. This index is then utilized in a radiative transfer model to obtain solar surface irradiance [68]. However, high-latitude areas are not covered by the majority of satellite data, because the spatial coverage of geostationary satellite is limited to latitudes
<figure><img src="image_5.png" /><figcaption>Fig. 1. Workflow of the developed methodology to generate a European map of optimized tilt angle for single-axis ground-mounted bifacial photovoltaic systems.</figcaption></figure>

within the range ± 65° [66]. Then, the atmospheric reanalysis PVGIS-ERA5 database, from the European Centre for Medium-Range Weather Forecast, is used to fill out the missing data, in particular in polar areas [69]. Reanalysis datasets used a Numerical Weather Prediction model, combined with correction from ground meteorological stations and satellite data to extrapolate weather data to past years and also in space. The quality of reanalysis is however lower than satellite databases, due to a poor modeling of clouds, and with particular high uncertainties in mountainous and coastal areas, due to the low grid resolution [68].

The spatial resolutions of SARAH-2 and ERA5 dataset are 5 km and 25 km, respectively [64]. The irradiation data are further refined at a smaller spatial scale by PVGIS by incorporating local obstructions using a DTM with a resolution of approximately 90 m (3 arc-seconds). This enables the calculation of the actual horizon and determines periods when the sun is behind the mountains if any. These adjustments affect the partitioning of global irradiation into diffuse and direct components, as solar irradiation consists solely of diffuse radiation when the sun is blocked by mountains. Consequently, this modification can influence the optimal tilt angles by favoring the capture of sky diffuse and diffuse ground-reflected components.

## 2.2. General framework and preparation of meteorological data

In this section, the algorithm developed to determine the optimal tilt angle of a solar photovoltaic system for various locations across Europe is presented (see Fig. 1). The objective of the procedure is to find the optimal tilt angle for single-axis ground-mounted bifacial composed of three arrays at each defined location. It allows to produce a European map of optimized tilt angle. The method relies on open-access databases and partially on open-source tools, enabling reproducibility and comparison of the results. The algorithm is divided into three parts. The first part involves the preparation of geographical data and is done with QGIS. This GIS software, free and open-source contains a Python extension that facilitates acceleration and automation of processes. The boundaries of all European countries are obtained from the GADM database (see Section 2.1.1). In order to capture the variation of the optimal tilt angle across Europe, several thousand

points uniformly distributed across the continent are necessary. To achieve this, functions from the toscana Python-QGIS library [54] are used to create a grid (50 km × 50 km), and then obtaining the centroid of each tile and the corresponding coordinates. Only points located on land (2382 points) are retained to reduce the total number of points and the associated computation time. The second step is the obtention of the meteorological data. This part is implemented using a Python routine, specifically the pvlib library [63]. For each previously determined location, a TMY weather file with hourly temporal resolution is retrieved from PVGIS databases using the pvlib.iotools.get_pvgis_tmy function (see Section 2.1.2). The solar position, comprising solar elevation and azimuth, is then calculated for each hour of the year using pvlib.solarposition.get_solarposition. Only hours with a GHI above zero are retained for the rest of the process.

The last part consists of a MATLAB algorithm (see Section 2.3) designed to optimize the tilt angle of a south-facing bifacial PV plant, considering the yearly cumulative radiance distribution on the sky vault, calculated from the typical meteorological year (TMY) of the selected location in Europe.

## 2.3. Optimization of tilt angle of PV bifacial modules

### 2.3.1. Sky vault discretization and cumulative sky calculation

The proposed model considers a 3D distribution on the sky vault of the diffuse component of irradiance, based on the Perez All-Weather-Model (AWM) [70]. The TMY files provide the mean hourly values of DNI and DHI along the entire year, which are used in Eqs. (1) and (2) to determine the correct irradiance distribution on the sky dome at each 1-h time step.

Given the TMY file for each considered location, the cumulative sky is calculated by integrating the hourly distributions of irradiance along the entire typical meteorological year [71]. In the example shown in Fig. 2, the solar analogs for each hour of the day are clearly visible, since the TMY files have a 1 h time discretization.

Moreover, the utilized database for TMY data takes into account the topology of the selected location, so the presence of mountains in the surroundings affects the distribution of radiation and allows to consider its effect on the best orientation of PV modules. In Fig. 2(a),
<figure><img src="image_6.png" /><figcaption>Fig. 2. Example of yearly cumulative sky distribution for the city of Genoa (Italy) (a) and for a mountainous location on French Alps (b).</figcaption></figure>

the cumulative sky distribution in Genoa (Italy) shows an unobstructed sky, while in Fig. 2(b) (location in French Alps) the presence of mountains obstructs the sun at certain positions (the “8 shape” of many analogues is interrupted).

To discretize the sky, as shown in Fig. 3, the sky vault is divided into rings with the same angular amplitude, and each ring is further subdivided into a number of elements (denoted by their azimuthal position) such that each sky patch subtends a similar solid angle, similar to the well-known approach of Tregenza [72]. Around 2300 patches are used in this study.

The Perez AWM model calculates, according to Eq. (1), a distribution of relative irradiance (setting relative zenith radiance equal to 1) depending on the position of the sun and the meteorological conditions at the given moment.

$$lv^* = f(\theta, \chi) = (1 + a \cdot \exp(b \cdot \cos(\theta))) \cdot (1 + c \cdot \exp(d \cdot \chi) + e \cdot \cos^2(\chi)), \quad (1)$$

where θ is the zenith angle of the considered point, χ is the angle between the point and the position of the sun, the coefficients a, b, c, d depend on the insolation conditions and affect the distribution: a is related to the horizon brightening or darkening, b to the horizontal gradient of luminance, c to the circumsolar intensity, d to the circumsolar width and e to the backscattering light on earth surface. Coefficients a, b, c, d depends on the position of the sun and the irradiance conditions, and are calculated according to Perez et al. [73].

Absolute values are then computed by ensuring that the integral of radiance by the entire sky on a horizontal surface equals the value of

<figure><img src="image_7.png" /><figcaption>Fig. 3. Example of discretization of the sky vault, every circle corresponds to the center of a sky element.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 4. Representation of the position of a sky element, in terms of cartesian coordinates (Xel, Yel, Zel) and polar angles (azimuth γ and zenith θ).</figcaption></figure>

diffuse irradiance (DHI) on the ground as in Eq. (2):

$$Lv_i = lv_i^* \cdot \frac{DHI}{\sum_{i=1}^{N} lv_i^* \cdot \cos(\theta_i) \cdot \Omega_i} \left[ \frac{W}{m^2 \text{ sr}} \right], \quad (2)$$

where Lv is the radiance of a sky element, $lv_i^*$ is the relative radiance, θ is the zenith angle of the sky element as defined in Fig. 4, Ω is the solid angle of the patch, N the number of sky patches.

Finally, the direct irradiance is considered to be irradiated from the element of sky closest to or containing the Sun's disk, so to the diffuse irradiance of the element is added the equivalent for the direct irradiance:

$$Lv_{i,Sun} = \frac{DNI}{\Omega_i} \left[ \frac{W}{m^2 \text{ sr}} \right]. \quad (3)$$

### 2.3.2. Calculation of yearly energy incident on modules and ground

The present model computes the distribution of incident radiation along the central section perpendicular to the modules and the ground of a bifacial system consisting of 3 rows of modules. The ground is discretized into M segments of constant length, and the approximation of infinite ground length is achieved by modeling the ground up to a sufficient distance from the modules (approximately 20 times the module height). The irradiance on each ground element is calculated using Eq. (3):

$$G_{gr,j} = \sum_{i=1}^{N} Lv_i \cdot \Omega_i \cdot \cos \theta_i \cdot \text{obs}_{i,j}, \quad (4)$$
<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Representation of angles involved in calculation of view factors between elements of ground and module.</figcaption></figure>

where, G is the yearly energy incident on the ground element j, Lv is the yearly radiation per steradian of the element i, obsᵢⱼ can take the values of 0 or 1 to account for shading of the modules between sky element i and ground element j.

While the sky vault is treated as a 3D surface, the PV module area (front and rear surfaces) is modeled as 2D (the central section of the PV rows is considered for the present analysis) and discretized into K elements of equal length, and the irradiance on each element is calculated using Eq. (5):

$$G_{\text{mod},k} = \sum_{i=1}^{N} L v_i \cdot \Omega_i \cdot V F_{i,k} \cdot \text{IAM}_{i,k} + \sum_{j=1}^{M} G_{gr,j} \cdot \rho_{gr} \cdot V F_{j,k} \cdot \text{IAM}_{j,k} \quad (5)$$

IAM is here the incidence angle modifier, calculated using Eq. (6), which considers the partial reflection of the incident radiation on modules due to the effects of the refraction on module glass cover.

$$\text{IAM} = \frac{1 - \exp(-\frac{\cos(\theta_w)}{a_r})}{1 - \exp(-\frac{1}{a_r})}, \quad (6)$$

where $\theta_w$ is the angle of incidence on the module element, $a_r = 1.57$ is the refractive index for Air/glass/Si configuration [74]. $VF_{i,k}$ are the view factors between sky element i and module element k, calculated with Eq. (7):

$$VF_{i,k} = \cos \varphi_{i,k} \cdot \text{obs}_{i,k}, \quad (7)$$

where obsᵢⱼ is equal to 1 or 0 if the element i is visible or not by the element k, $\varphi_{i,k}$ is the angle between the normal to element k and the line connecting element k and element i. $VF_{j,k}$ are the view factors between ground element j and module element k, calculated with Eq. (8):

$$VF_{j,k} = 0.5 \cdot (\cos \varphi_{j,k,1} - \varphi_{j,k,2}) \cdot \text{obs}_{j,k}, \quad (8)$$

where obsᵢⱼ is equal to 1 or 0 if the element j is visible or not by the element k, $\varphi_{j,k,1}$ and $\varphi_{j,k,2}$ are the angles between the direction parallel to the module and the line which connects the center of the module element k and the ends of the ground element j, as in Fig. 5.

The effect of the obs factors is relevant especially for closely packed modules (i.e. for high GCR values) since it allows to take into account the shading from neighbor panels with respect to the sky dome and to the ground. Worth noticing once again that the cumulative sky distribution of radiance is obtained from both diffuse on the horizontal and beam hourly insolation, since the hourly beam component is referred to the sky patch closest to the current position of the sun

<table><caption>**Table 1**

Geometric dimensions and simulation parameters of the reference plant for validation.</caption><thead><tr><th></th><th>Configuration #1</th><th>Configuration #2</th></tr></thead><tbody><tr><td>H [m]</td><td>1.56</td><td>1.71</td></tr><tr><td>ρ<sub>gr</sub></td><td>0.3</td><td>0.4</td></tr><tr><td>L [m]</td><td colspan="2">1.61</td></tr><tr><td>GCR</td><td>0.35</td><td>0.5</td></tr><tr><td>β [°]</td><td>44</td><td>30</td></tr><tr><td>Plant orientation</td><td>South facing</td><td>45° South-West</td></tr></tbody></table>

<table><caption>**Table 2**

Parameters of the present simulations.</caption><tbody><tr><td>H [m]</td><td>1.56</td></tr><tr><td>L [m]</td><td>1.62</td></tr><tr><td>GCR</td><td>0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.7</td></tr><tr><td>Module azimuth</td><td>South facing</td></tr><tr><td>Albedo ρ</td><td>0.1, 0.2, 0.3, 0.4, 0.5</td></tr></tbody></table>

disk at each time step of the weather file. The present model seeks to combine elements of existing 3D ray tracing models, known for their high accuracy but long computation times, with 2D models that rely on View Factors. While the latter are computationally efficient, they lack the precision needed to meet the demands of optimal tracking in photovoltaic applications involving bifacial modules.

### 2.3.3. Validation of the present irradiance model

The present algorithm has been validated through the comparison with the results of Bifacial Radiance [75], developed by NREL and based of 3D ray tracing. The analysis has been carried out by comparing over one year the hourly and daily radiation incident on the front and back surface of a fixed PV array in two different configurations (orientation and geometry) as described in 1. The simulated system is located on the University of Genoa site (44.403°N, 8.960°E, altitude 66 m above sea level) and the weather data are extracted from the relative TMY file. Since the bifacial radiance documentation does not declare the implemented formula for the IAM, Eq. (6) has been implemented with a coefficient $a_r = 0.03$ to better align the results of the present algorithm with those of bifacial radiance.

The analysis reveals that the annual average discrepancy between the two data series is minimal, amounting to 0.45% for configuration #1 and 0.66% for configuration #2. More specifically, the average hourly deviation on the rear side alone is 2.36% and 3.33% for the two configurations, respectively. Furthermore, data from the two simulation tools demonstrate a strong linear relationship, with R² values exceeding 0.999 for the combined back+front side hourly data (Fig. 6(a)) and daily insolation of front+back side (Fig. 6(b)), here reported for configuration #1 because more similar to the systems analyzed within the present investigation.

The comparison between Figs. 6(a) and 6(b) indicates that integrating irradiance over the course of a day enhances the model's accuracy. Therefore, integrating over an entire year is anticipated to yield even greater accuracy.

### 2.3.4. Tilt angle optimization

The determination of the best tilt angle has been made for each tile of the 50 km × 50 km grid on Europe, by maximizing the total energy yield on front and back surface of the modules, calculated using the algorithm described in Section 2.3.2. A three rows system has been analyzed, the parameters of the simulation are described in Table 2, while the geometry is shown in Fig. 7.

The optimization is made with a 1° resolution for the tilt angle, since the energy yield has negligible variations for smaller variations of tilt angle. The optimal tilt angle is determined by calculating the annual energy yield at each 1° step and selecting the angle corresponding to the maximum yield. To analyze the effect of the presence of multiple rows on best tilt angle, different values of Ground Coverage Ratio (GCR)
<figure><img src="image_10.png" /><figcaption>Fig. 6. (a) Hourly values of front+back side irradiance over the entire year and (b) daily values of front+back side insolation, results from the present model and from reference algorithm for configuration #1.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 7. 2D schematic of the bifacial PV system used in the algorithm, showing the dimensions – PV row width (L), ground clearance (H), tilt angle (θ) and row pitch (P) – considered for optimizing the irradiance capturing.</figcaption></figure>

has been considered. GCR is a non-dimensional number defined by Eq. (9) as follows:

$$GCR = \frac{L}{P} \qquad (9)$$

Finally, multiple values of albedo have been considered. The albedo of the ground depends strongly on the location, and it varies during the year, because of the evolution of the vegetation and the possible presence of snow. As a consequence, the albedo is imposed as a parameter in the range of 0.1 to 0.5 to cover most of the cases, and the results of

the present simulation can be used to cover the needs of the design of new bifacial PV plants.

## 3. Results

### 3.1. Presentation of the studied area

In the present study, the methodology described in the previous section is employed to determine the optimal tilt angle for bifacial solar systems consisting of three arrays across an entire continent. The study focuses on the European continent, including all countries whose territories are considered as part of Europe, excluding Russia, Kazakhstan or Turkey, and geographically bounded by the Bosphorus, Dardanelles and Gibraltar straits. Cyprus is also included as part of the European union, despite being considered as part of the Asian continent in some definitions. For spatial consistency, several British territories were also included: five British crown dependencies and overseas territories (Akrotiri and Dhekelia, Gibraltar, the Bailiwick of Guernsey, the Bailiwick of Jersey and the Isle of Man), as well as the Faroe Islands, an autonomous territory of Denmark. However, other territories that are geographically distant from mainland Europe, such as Åland, Svalbard or Greenland were excluded from the analysis.

Europe is predominantly composed of lowlands areas, with elevations generally below 180 m. The European Plain, extending from the French-Spanish border to Northern Europe and the Ural Mountains in Russia, is the largest uninterrupted plain on Earth. Southern Europe, in contrast, consists mainly of islands and peninsulas, and is predominantly composed of mountains and plateaus. These two regions are separated by major mountain chains, including the Pyrenees, the Alps and Carpathian Mountains. The climate in Europe is mainly temperate, with hot summers and cold winters, and is warmer than in other regions of the world at the same latitude due to the Gulf Stream. The large coastal areas of the continent experience an oceanic climate, characterized by low annual temperature variations but high precipitation levels. In contrast, Southern Europe has a Mediterranean climate, with mild wet winters and hot dry summers. The boundaries between these different climatic zones are created by mountain ranges, which exhibit an alpine climate with low temperatures and high precipitation.

In this study, a 50 km × 50 km grid was utilized to determine the optimal tilt angle, with the coordinates of the centroid of each grid tile used for the calculations, and also to represent the results. 0.5 is a common GCR value for existing bifacial PV plants with fixed tilt angle [47], while an albedo value of 0.2 is typical for land surfaces [76,77]. These values will be used initially to present the results.

Finally, a focus has been made on a highly heterogeneous part of Europe, the Alps, for which a resolution of 50 km × 50 km is insufficient in view of the variations of terrain conditions (e.g. valleys, ski resorts, mountain peaks). Using the same methodology, a 5 km × 5 km grid has been used in order to achieve higher precision in an area characterized by high topographical and climatic heterogeneity.

### 3.2. Influence of latitude

Using the methodology developed in this study, the optimal tilt angle was calculated for each 50 km × 50 km tile across the entire studied area. The use of GIS databases enables the visualization of the results through geographical representations and facilitating the identification of regional patterns and the assessment of the impact of latitude and longitude. The European map of optimal tilt angles (Fig. 8) shows that optimal tilt angles are higher in the northern countries compared to southern Europe. A variation of up to 20° is observed for a GCR of 0.5 and an albedo value of 0.2, highlighting the influence of latitude on the optimal tilt angles. In high-latitude regions, the significant variations in the sun's path over the year along with the lower solar altitudes make higher tilt angles more suitable for maximizing energy collection throughout the entire year. This supports
<figure><img src="image_12.png" /><figcaption>Fig. 8. Map of Europe showing the optimal tilt angle calculated using the previously developed methodology for GCR = 0.5 and albedo = 0.2. The optimal tilt was determined for each 50 km × 50 km grid tile.</figcaption></figure>

initial conclusions found for bifacial systems [51]. The same trend is observed for monofacial PV systems but optimal tilt angles are lower in bifacial systems to minimize the ground shading, with difference up to 10° in high-latitude areas.

In contrast, variations across a constant latitude are less visible than those across longitude for the entire Europe. The standard deviation of tilt angles per latitude is generally between 1.5° and 2° in central Europe (with higher values in the southern and northern regions of Europe due to reduced land area). This limited variation is due to similar sun paths at the same latitude. However, greater variations at constant latitudes can be observed in areas corresponding to mountainous regions, particularly in the Alps, where the standard deviation increases up to 2.8°, as discussed in more detail in Section 3.3.

Some correlation can be found between the optimal tilt angle and latitude as apparent in Fig. 9, where linear fitting lines are superimposed to calculated points and correlation coefficients R² are shown. From Fig. 9 inspection it can be noticed that the correlation coefficient R² is ranging from 0.4 to 0.82, with the higher values corresponding to the lower ground coverage ratios. Latitude cannot be thus the only independent variable for best tilt of solar surfaces, as already stressed in a recent paper [28]. The variation in optimal tilt angles with latitude is not only related to the different solar trajectories in the sky (Sun elevation), but also to the local weather conditions during the year and presence of terrain.

Additionally, an anomalous result gap is observed around a latitude of 65°, characterized by particularly high tilt angles in polar areas (Figs. 8 and 9). This discrepancy corresponds to the limit of coverage of the PVGIS-SARAH database and the transition to the use of PVGIS-ERA5 database in polar areas. It has been showed that ERA5 provides higher irradiance values than SARAH in high-latitude areas (where two datasets overlaps) [68]. Consequently, the optimization algorithm is sensitive to the database used and to the irradiance levels. Although a higher order polynomial could better fit the present datasets in Fig. 9, the linear approximation was retained as the discrepancy appears to result from the change of database above 60°. To minimize discontinuities in the results, it is therefore crucial to use a consistent database whenever possible.

The distribution of optimal tilt angles by country is illustrated in Fig. 10 using boxplots, which confirm that northern countries (Finland, Sweden, Norway and Island) have higher average tilt angles compared

<figure><img src="image_13.png" /><figcaption>Fig. 9. Optimal tilt angle as a function of latitude for different GCR values and albedo = 0.2. A linear regression, including the regression equation and R² coefficient, has been added for each dataset corresponding to a specific GCR. For the sake of clarity, results for only four GCR values are presented – other values exhibiting similar trends.</figcaption></figure>

to other European countries. As expected, the greater dispersion of values in these countries reflects their significant north-south geographic spread, which results in important differences in the solar trajectory and weather conditions between northern and southern regions. Low tilt angle values correspond to outliers in several countries, including Spain, France, Austria, Switzerland, and Italy. These points are located in mountainous regions, such as Pyrenees and the Alps. The next section will therefore examine in greater detail the influence of terrain on optimal tilt angles.

### 3.3. Variations of optimal tilt angles in mountainous territories

As mentioned above, lower tilt angles are observed in mountainous areas (Fig. 8), such as the Alps or the Pyrenees, compared to the value in the surrounding areas. To gain more insights into these variations, the same methodology was applied to the French part of the Alps using a higher grid resolution (5 km × 5 km). This is presented in Fig. 11, in which the altitude in the French Alps is plotted (Fig. 11(b)) with the optimal tilt angle (Fig. 11(a)).

In the Alps, the variation of the optimal tilt angle is significant, with a decrease in the optimal angle observed in the areas with the highest topography and highest altitude (Figs. 11 and 11(b)). The highest grid resolution allows for a more detailed observation of these variations: a difference up to 8° in optimal tilt angle was observed between the mean value of the 5 km × 5 km grid and the value obtained with the 50 km × 50 km grid. A standard deviation of up to 3.17° in optimal tilt angle is also observed for the 5 km × 5 km value within each grid tile of the 50 km × 50 km grid. One indicator, developed by Ferry et al. [54] in a previous study was used here to analyze the results.

The Sky View Index (SVI) represents the proportion of visible sky relative to flat areas. A low SVI indicates that the location is affected by obstructions at the horizon (e.g. mountains) that limit the visible sky. It is defined as:

$$SVI = \frac{\text{Visible sky area from a location}}{\text{Unobstructed visible sky area}} \qquad (10)$$

As demonstrated by Ferry et al. [54], SVI decreases in mountainous areas, as it does for points of the 5 km × 5 km grid that fall at the bottom of the valleys surrounded by their mountains, which due to topographical features that obstruct a portion of the sky. On the other hand, points at lower altitude and located in almost flat areas are characterized, as expected, by a higher SVI. The variation of optimal tilt angles with SVI is shown in Figs. 11(b) and 11(c) and in Fig. 12.
<figure><img src="image_14.png" /><figcaption>Fig. 10. Boxplot of the optimal tilt angle by European country (in alphabetic order) for GCR = 0.5 and albedo = 0.2.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 11. Map of the French part of the Alps showing the (a) optimal tilt angle for a GCR value of 0.5 and an albedo value of 0.2, (b) the altitude and (c) the SVI. The optimal tilt angle was determined for each 5 km × 5 km grid tile.</figcaption></figure>

Optimal tilt angle decreases as the SVI decreases, because far masks caused by neighboring mountains, block the sun, particularly when it is a low altitude, such as during the early morning or late afternoon. Therefore, the optimal tilt angle decreases to achieve the optimal orientation when the sun is high in the sky.

### 3.4. Influence of the GCR and of albedo

The Ground Coverage Ratio (GCR) plays a major role in optimizing the tilt angle of solar arrays. Indeed, a high GCR corresponds to arrays positioned close together, while a low GCR indicates that the arrays are more widely spaced, thus functioning more like a single array that is not affected by shadow from neighboring arrays. European distributions of optimal tilt angles, for different GCR values – 0.05, 0.5 and 0.7 – are shown in Figs. 13(a), 13(b), 13(c), respectively. It can be seen that the optimal tilt angle decreases as GCR increases, since a lower tilt angle reduces the influence from close neighboring arrays, as it minimizes shading effects and increases the visible sky for the arrays.

The optimal tilt angle also varies with latitude, as described in Section 3.2, and this variation is more pronounced when the GCR is low. The slope of the linear regression is much lower for high GCR, as shown in Fig. 9, with angles ranging from 25° to 65° for a GCR value of 0.05 and only from 15° to 42° for a GCR value of 0.5. The optimal tilt angle is close to latitude for low GCR, whereas the influence of latitude is less significant for high GCR: the average difference between optimal tilt and latitude is only 7° for a GCR value of 0.05 but increases to

<figure><img src="image_16.png" /><figcaption>Fig. 12. Optimal tilt angle as a function of SVI for various GCR values and an albedo value of 0.2 for the French part of the Alps. A linear regression, including the regression equation and R² coefficient, has been added for each dataset corresponding to a specific GCR. The fit of the linear regression improves as the GCR decreases.</figcaption></figure>

21° for a GCR value of 0.5. The sensitivity of the optimal tilt angle to latitude is therefore lower at high GCR, as the presence of multiple rows induces lower tilt angles to mitigate the impact of shading.
<figure><img src="image_17.png" /><figcaption>Fig. 13. Map of Europe (50 km × 50 km grid) showing the optimal tilt angle methodology for an albedo = 0.2 and a GCR = 0.05 (a), 0.5 (b) and 0.7 (c).</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 14. Map of Europe (50 km × 50 km grid) showing the difference in optimal tilt angles (in degrees) between albedo = 0.1 and albedo = 0.5, for GCR = 0.05 (a) and GCR = 0.5 (b).</figcaption></figure>

The influence of albedo on the optimal tilt angle is much less significant than the influence of GCR, with an average variation of only 3° and -2° between albedo values of 0.1 and 0.5 for GCR values of 0.05 and 0.5, respectively. However, the direction of the variation in the optimal tilt angle depends on the GCR, as it is observed that the optimal tilt angle increases with increasing albedo for a GCR value of 0.05 but decreases for a GCR value of 0.5 (Fig. 14), with the transition occurring at a GCR value of approximately 0.3.

This trend arises from the fact that higher albedo results in a greater fraction of energy reflected by the ground. The optimal tilt angle, as albedo increases, will therefore be the one that assures the collection of the reflected component. For low GCRs, the reflected component is better captured with higher tilt angles, while at high GCRs, lower tilt angles are more optimal due to the obstruction caused by adjacent rows.

In Fig. 15 is reported the yearly cumulative radiation (front + back) on 1 m² of PV module. As expected, the energy yield is increasing with increasing albedo and with decreasing GCR and latitude.

### 3.5. Energy gains from optimal tilt angles

The algorithm developed in this study allows for the optimization of the tilt angle of bifacial PV systems by maximizing the irradiation

captured by the entire system, including both the front and back faces of the central rows. This enables the calculation of the gain in irradiation collected by the arrays when using the computed optimal tilt angle compared to the irradiation received when the tilt angle is set to the latitude, which is the usual choice for tilt angle for monofacial modules [78]. The gain (in percentage) can be defined as:

$$Gain = \frac{(E_{opt\_tilt} - E_{tilt=lat})}{E_{tilt=lat}} \cdot 100, \qquad (11)$$

where E is the yearly incident energy on the front and back side of modules.

The gain achieved by using the optimal tilt angle, rather than a tilt angle equal to the latitude, varies depending on the region, the GCR and the albedo (Fig. 16). The average European gain for a GCR value of 0.5 and an albedo value of 0.2 is 5.2% (0.56% for a GCR value of 0.05 and an albedo value of 0.2, and 5% for a GCR value of 0.5 and an albedo value of 0.5). The highest gains are observed for a GCR of 0.7 and an albedo of 0.1 (average gain of 9.1%), particularly in northern Europe and the Alps, whereas the lowest gains are observed in southern Europe.

However, as shown in Figs. 16 and 17(a) a discontinuity is observed around a latitude of 65° due to the change in databases, as
<figure><img src="image_19.png" /><figcaption>**Fig. 15.** Map of Europe (50 km × 50 km grid) showing the yearly insolation received by both the surfaces of the module with the optimal tilt (front and back irradiation) (kWh/m²/year) for GCR = 0.05 and albedo = 0.1 (a), and albedo = 0.5 (b), and for GCR = 0.5 and albedo = 0.1 (c) and albedo = 0.5 (d).</figcaption></figure>

already stressed in previous paragraphs. In northern Europe, the ERA-5 database provides higher irradiation values compared to the SARAH database, resulting in a lower potential gain from using the optimal tilt angle. The linear correlation between gain and latitude becomes stronger when points with a latitude above 65° are excluded, with an R² value reaching 0.88 for a GCR of 0.5 and an albedo of 0.2. The gain is also higher in mountainous regions, where it increases with lower SVI values (Fig. 17(b)), corresponding to areas with more complex topography and steep-sided valleys. The presence of terrain relief also results in a larger difference between the optimal tilt angle and the latitude.

Therefore, these results are particularly interesting for plants with high GCR, at high latitudes and in mountainous regions, where the optimal tilt angle deviates more significantly from the latitude value. In these regions, determining the optimal tilt angle is crucial due to the lower levels of irradiation.

### 3.6. Discussions and limitations

In this study, the optimal tilt angle was determined for each 50 km × 50 km grid tile as a compromise between spatial resolution and computation time. As mentioned in Section 3.2, significant variations in optimal tilt angle were observed between neighboring cells due to changes in mountain shading or local climate. To achieve higher precision, it would be necessary to use a higher-resolution grid (e.g., 5 km × 5 km), as was done for the local study in the Alps to account for specific local features, even if spatial heterogeneity is much lower across most of Europe compared to mountainous regions like the Alps.

Moreover, the spatial resolution of the SARAH-2 meteorological data is only 5 km (and 25 km for ERA-5). Thus, if a grid size smaller than the resolution of the meteorological database is selected, it may be beneficial to consider weather data from neighboring grid tiles using a
<figure><img src="image_20.png" /><figcaption>**Fig. 16.** Map of Europe showing the gain (%) between irradiation received by the module with the optimal tilt (front and back irradiation) and irradiation received at a tilt equal to the latitude for a GCR value of 0.5 and an albedo value of 0.2.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Fig. 17.** Gain between yearly insolation received by the module with the optimal tilt (front and back irradiation) and irradiation received at a tilt equal to the latitude as a function of (a) latitude at the European scale and (b) SVI for the French part of the Alps for various GCR values and an albedo value of 0.2. A linear regression, including the regression equation and R² coefficient, has been added for each dataset corresponding to a specific GCR.</figcaption></figure>

spatial average, as was done by Ferry et al. [54], to reduce the spatial
disparity induced by the tiling of weather data.

The present simulations were conducted for a bifaciality coefficient of 0.95. High bifaciality coefficients lead to higher tilt angles, particularly in high-latitude regions to increase the proportion of reflected irradiance on the rear side. However, the impact of this coefficient on optimal tilt angles is limited, with an average change of less than 1° for a bifaciality coefficient of 0.65 (for a GCR value of 0.5 and an albedo value of 0.2.), as back-side irradiance generally accounts for less than 10% of the total irradiance.

Additionally, accounting for albedo variation throughout the year could improve accuracy of irradiation calculation. Mountainous regions, in particular, experience significant seasonal changes in albedo, as the presence of snow during certain periods of the year notably affects albedo values (with an albedo around 0.8 for snow) and, consequently, optimal tilt angles. Since an annual cumulative sky irradiance from TMY data is used in this study, a single albedo value is required in the calculation of the annual irradiation distribution on the modules. However, the study has demonstrated that the influence of albedo on the optimal tilt angle is limited. Moreover, the ground reflectivity has been imposed as parameter of the simulations, so that the users can set the most appropriate value for each specific case.

Finally, another possible limitation is the lack of a detailed electrical
model accounting for the effect of self-shading on mismatch due to
irradiance inhomogeneity in PV cells. However, this effect is mitigated
in modern modules, which are equipped with multiple bypass diodes
that reduce mismatch losses.

4. Online tool to visualize the optimal tilt angle

In order to facilitate a broader accessibility to the present results and support the development of PV plants and optimization of electricity production, an online, free-access tool has been developed. This tool provides access to the results of this study and enables users to easily visualize the data.

The tool is a Jupyter Notebook hosted on Binder, that should simply
be run to access the visualizations and the interactive map. It is
available with the whole result dataset in an online deposit [79].

The maps developed in this study (at European scale and also for
the French Alpine region) could be displayed, including the optimal
tilt angles, the solar irradiation collected by the arrays, and the gain
achieved from using the optimal tilt for the various GCR and albedo
values (see Section 2.3.4). Additionally, the tool allows users to deter-
mine the optimal tilt angle for any location in Europe by either clicking
on a map or entering specific latitude and longitude coordinates. The
optimal tilt angle of the desired location is then interpolated using
Inverse Distance Weighting (IDW), based on the values from the four
nearest points as done by Urraca et al. [68]. An example of the web
interface is shown in Fig. 18.

This tool could be highly valuable for local authorities, developers,
and decision-makers in assessing the feasibility of installing bifacial
solar systems in their regions, while estimating the potential energy
yield. It is particularly relevant for mountainous areas, where bifacial
PV plants can offer significant advantages due to the high albedo and
the availability of large space for installation. Bifacial PV plants can
be a more effective solution than roof-mounted systems, which face
challenges in winter due to snow accumulation. Snow removal is more
difficult for roof-mounted installations, and snow guards can prevent
snow melting, thus reducing the potential energy generated. In these
regions, the energy gains from using the optimal tilt angle, rather than a
tilt equal to the latitude, are also important as explained in Section 3.5.
<figure><img src="image_22.png" /><figcaption>Fig. 18. Illustration of the webtool (Jupyter Notebook).</figcaption></figure>

## 5. Conclusions

The present study has enabled the creation of a European-scale map illustrating the influence of different factors on the optimal collection of irradiance by fixed-tilt bifacial PV plants. This map, made accessible through the development of a web tool, is available to a wide range of energy stakeholders and can be used for the design and sizing of bifacial PV plants. A key innovation, compared to existing tools, is the consideration of self-shading within the arrays using view factors, as well as far shading effects from mountainous terrain through the consideration of a 3D cumulative sky distribution, calculated as sum of hourly sky radiance distributions determined using the Perez AWM model. Moreover, the blending between the implementation of a 3D sky and 2D distribution of irradiance on modules and ground offers a great compromise on computational time and accuracy of the method, as demonstrated by the comparison with the Bifacial Radiance ray tracing software. The effect of refraction on the module surfaces was accounted for by employing an Incidence Angle Modifier (IAM) model for the air-glass-silicon configuration. The simulation was conducted for 2382 locations across Europe, considering south-facing photovoltaic (PV) systems, ground coverage ratios (GCRs) in the range of 0.05 to 0.7, and albedo values between 0.1 and 0.5.

The results indicate that the primary factors influencing the optimal tilt angle are latitude and GCR. High variations are observed between the different European countries, with for example an average value of 36° for Finland, 29° for France, 27° for Italy and 26° for Greece, for a GCR of 0.5 and an albedo of 0.2. Higher GCR values reduce the dependency of the optimal tilt angle on latitude due to the increased shading effect from adjacent PV rows. Ground albedo has a limited impact on the optimal orientation, while local climatic conditions introduce a variability of approximately 5° in the optimal tilt angle for locations at the same latitude. As demonstrated by a more refined analysis of the French Alps, the correlation between tilt angle and SVI is significant. Specifically, higher SVI values lead to a reduction in the optimal tilt angle, facilitating improved collection of solar radiation from the unobstructed sky.

The present study demonstrates that using optimal tilt angles, compared to the reference case where tilt angles are set equal to latitude, can improve annual energy yield by up to 15% in high-latitude regions

and up to 10% in mountainous areas. This gain is especially pronounced at higher GCR values, increasing from a European average of 0.56% for a GCR of 0.05 to 5.2% for a GCR of 0.5, which is a typical value for utility-scale PV plants.

Future works could focus on determining optimal azimuth for bifacial PV systems, particularly in mountainous regions. While bifacial systems in the Northern Hemisphere are typically facing south, local topography could significantly influence their optimal azimuth. Additionally, a cell-level electrical model could be implemented to assess PV potential on a European scale, considering the potential impact of rear-side irradiance inhomogeneity and mutual shading on front side, which could induce mismatch effects. Finally, an evaluation of the irradiance captured by bifacial PV plants and the associated PV potential could be conducted by considering the seasonal variation of albedo, particularly in mountainous areas where snow cover can significantly enhance the performance of bifacial power plants

## CRediT authorship contribution statement

**Apolline Ferry:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Resources, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Mattia Parenti:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Resources, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Martin Thebault:** Writing – review & editing, Supervision, Methodology, Conceptualization. **Christophe Ménézo:** Writing – review & editing, Supervision, Project administration, Methodology, Funding acquisition, Conceptualization. **Marco Fossa:** Writing – review & editing, Writing – original draft, Supervision, Project administration, Methodology, Funding acquisition, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

This research is a part of the Italian National Program PON "Ricerca e Innovazione" 2014-2020, Action IV.4 "Dottorati e contratti di ricerca su tematiche dell'innovazione" and Action IV.5 "Dottorati su tematiche green". This work has been supported by European University UNITA, the Grand [La]Bo project, CITEE French Swiss Chair and by the French National Research Agency, through the Investments for Future Program (ref. ANR-18-EURE-0016 - Solar Academy). The research unit LOCIE is a member of the INES Solar Academy Research Center.

## Data availability

Our processed data, maps and interactive webtool are accessible publicly on an online deposit [79] https://doi.org/10.57745/8I4MZ3.

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