The possibility of having adaptable curved solar modules, allows to think on having photovoltaic surfaces for localized energy production, in order not to depend exclusively on solar farms. These localized applications are precisely on roofs based on geodesic shapes [4], design of complex architectural components [5], amorphous facades in urban environments [6] including heat dissipation mechanisms [7]. Also, transportation applications [8] as vehicles in urban sector [9] in order to satisfy its own energy demand [1 0] are analyzed considering curvature of the roof surface [11]. In this way, photovoltaic energy presents a great advantage to distributed and democratized energy generation and use, because it is produced at the same place of consumption, so losses from the generation chain are minimized. This is the so-called “Prosumer” concept [12]. Most of the solar panels found commercially in the literature have completely flat geometries and consists of glass, encapsulant (normally EVA) and a reinforcing plate (Backsheet) usually made of aluminum [13].

However, for applications in transportation systems and buildings, it is possible to propose curved geometries that allow a better adaption to the required complex shapes which is further complicated by the configuration of conventional commercial solar panels due to the curved geometry the glass must take. This fact has severe implications in terms of conventional manufacturing. For this reason, it is necessary to explore other manufacturing techniques, in this case, the use of composite materials and its mechanical behavior will be analyzed.

A solar panel is an array of interconnected solar cells, most manufactured of silicon which converts the sun's energy into electricity. Solar cells are assembled in a sandwich arrangement above and below. The material on top of the cell must have good optical properties, this allows solar radiation can be absorbed as easily as possible, this material is called encapsulation. Similarly, the material at the bottom of the cell is called reinforcement, which must guarantee the structural strength. These layers can be seen in Figure 1.

Figure 1. Figure 1. Proposed Panel Configuration Source: created by the authors.

Photoelectric effect presented in solar cells transforms solar radiation into consumable electrical energy and heat, which has a significant negative implication on the overall efficiency of 75.58 % while reflection losses are approximately 6.98 % [14]. Energy losses due to heat are intrinsic to the photovoltaic module and must therefore be reduced during operation. This energy in form of heat is harnessed by hybrid systems such as thermal collectors [15], air cooling for water generation [16] and heat exchangers [8]. Also, from the optical modeling and the design of the encapsulation, it is possible to apply passive cooling techniques, based on the development of thin-film solar cells using solar spectrum splitting technique [17] in solar concentrators [18] which allows the elimination of high-energy solar photons in the infrared spectrum [19]. Materials in PV module must ensure the strength of solar cells and the manufacturing of the PV module with curved surfaces. From the point of view of mechanical modeling, there have been different studies focused mainly on the structural study of the complete solar panel under bending conditions in which the objective is to try to predict the propagation of cracks generated in the cell [20] and estimate the probability of failure as a function of mechanical fracture stresses [21].

Also, thermo-mechanical studies have been carried out in order to estimate the influence of temperature on the structural capacity of the PV module assembly [22]. Consequently, the overall behavior of the photovoltaic array is analyzed through multi-physics models, integrating the mechanical, thermal, optical, and electrical domains [23]. Finally, reinforcement studies were found using composite materials and core materials such as honeycomb to predict thermal fatigue [24]. However, there is no evidence of studies focused on the potential offered by composite materials for manufacturing solar panel reinforcements. The purpose of this paper is to analyze design strategies for curved photovoltaics modules according to the operating and environmental conditions, establishing a starting point to estimate the complete performance of the photovoltaic system. Methodology used in this work is discussed below.

Considering the high fragility of silicon solar cells and their low thickness in the order of micrometers, the encapsulated-reinforming arrangement must guarantee a high structural resistance for the operation loads and maintenance of the solar panel. Additionally, the manufacturing of photovoltaic surfaces with curvature causes the cell to take different shapes than the flat one, which causes stresses and deformations. These requirements and conditions will be described in the following subsections.

Manufacturing of solar panels using composite materials is based on the scheme shown in Figure 1. Reinforcement is made from the union of fabrics and a matrix material, while the encapsulation is a coating of a layer of translucent material that allows isolating the solar cells from the environment, adhering them to the reinforcement. For both parts it is a handmade process, so the mechanical properties obtained can have great variability between manufacturers, for this reason some properties are referenced from the literature and obtained from experimental tests.

To model the behavior of the composite material, it was necessary to use macromechanics models, which allow relating the mechanical properties of the fabrics with respect to the characteristics of the lamination process in reinforcement material [31]. For this purpose, a fully planar geometry reinforcement was assumed in order to obtain the deformations generated due to the operating conditions presented above. Also, a plane stress condition will be presented considering a reinforcement without curvature, in which the panel will only have interaction with the loads in the direction (1). This behavior is governed by (1).

(1)

Thus, a uniaxial loading state of 1000 N is applied for a flat glass fiber laminate consisting of 16 equally oriented layers, the area of fabrics will be 1 m×1 m. Furthermore, due to the behavior found between the Young’s modulus between direction and a transversely isotropic material can be assumed, which leaves (2).

(2)

where the compliance coefficients of the lamina as follows (3):

(3)

The S coefficients in (2) depend on Poisson’s and Young’s Modules of material. A symmetrical lamination is considered for the plies, so they have a specific thickness, specific material properties and a ply orientation, which for this case will be unidirectional through (1). The purpose is to find the deformation in direction and compare it with a finite element analysis to validate the results obtained in the simulation.

In this way, a numerical analysis in Matlab™ is implemented, which allows to relate the proposed equations by the macromechanical models with respect to the mechanical properties of the fabrics using conventional matrix operations. A deformation of ε₁= 0.047 mm is then obtained for stress condition assumed, this result will be compared with simulation in next section.

The purpose is to evaluate the analyzed condition in analytical model in order to validate the results, these are shown in Figure 6.

Figure 6. Figure 6. Deformation analytical model Source: created by the authors.

Finally, it is observed that the results obtained in the simulation correspond to those estimated in the analytical model, with respect to the average deformation of 0.047782 mm. With this value, there is an error of 1.7 %. This fact allows to propose more realistic load conditions for solar panel reinforcement and curved geometries, which will be discussed in next section.

Taking into account the validation of the simulation conditions for the flat panel, a more realistic scenario can then be proposed, including the curvature effects estimated above.

A load condition will then be assumed for direction 1, 2 and 3 of 1000 N. The approximate thickness for the analysis is 6 mm for each of the materials, in this way 16 layers are used for Kevlar and Fiberglass, while 20 layers are used for Carbon Fiber. For this purpose, an analysis in ANSYS®, specifically in composites ACP module is implemented, which allows establishing in detail different characteristics of the process, such as ply thickness and type of material. Effects on the deformation of Carbon Fiber, Glassfiber and Kevlar are analyzed and shown in Figure 7.

Figure 7. Figure 7. Total Deformation in Carbon Fiber, Glass Fiber and Kevlar, respectively Source: created by the authors.

It is observed that the distribution of deformations is very similar for the materials analyzed, presenting the same patterns, according to the load conditions assumed, size finite element used was 10 mm. Likewise, Kevlar has less deformation with a value of 1.20 mm, followed by Carbon Fiber with 1.5 mm and finally, Glass fiber with 1.73 mm, precisely in Directional Deformation Z. In order to be more specific, the directional deformations are analyzed in the established coordinate system, which are shown in Table 3.

Table 3. Table 3. Final Deformations and mass

Properties	Kevlar	Glass Fiber	Carbon Fiber
δx [mm]	0.059	0.045	0.032
δy [mm]	0.021	0.025	0.022
δz [mm]	1.201	1.737	1.005
m[kg]	9.910	12.010	8.530

Source: created by the authors.

It can be seen that for plane state, the magnitude of the deformation does not change substantially, while the load in the direction 3 does have an important effect of the total deformation. In addition, it is necessary to analyze the stresses in each layer, as shown in Figure 8 and Figure 9.

Figure 8. Figure 8. Shear Stresses Per Ply Source: created by the authors.

Figure 9. Figure 9. Maximum Principal Stresses Per Ply Source: created by the authors.

A very similar behavior is observed between Carbon Fiber and Glass fiber. The layers in contact with the cell receive a lower stress that increases according to the layer order. In contrast, Kevlar offers the lowest stresses in the intermediate layers of the reinforcement, while the outermost layers reach the maximum values. Likewise, shear stresses generated by the joint between the fabrics, which is an important factor in the failure of solar panels due to delamination which causes 10 % of the failures of solar panels [32]. This behavior is shown in Figure 8. Thus, it is observed that the shear stresses between layers have very low values.

Taking into account the minimum radius of curvature allowed in the cell, an analysis of the stresses and deformations obtained for different radius of curvature is performed, starting at a value of 6.5 m found previously, this behavior is shown in Figure 10 and Figure 1 1.

Figure 10. Figure 10. Deformation vs. Radius of Curvature Source: created by the authors.

Figure 11. Figure 11. Maximum Stress vs. Radius of Curvature Source: created by the authors.

It should be noted that the panel tends to become flatter at higher values of radius of curvature. Thus, as the panel becomes flatter, the stress values also increase. Similarly, a deformation analysis is carried out for each of the materials, which is shown in Figure 10.

A big difference can be seen in the deformation behavior of the material, where Kevlar achieves the highest values, while Glassfiber and Carbon Fiber behave in a very similar way.

Unlike the analysis presented for the reinforcement from orthotropic materials, the structural study performed for the encapsulation can be done with shell elements, due to their isotropic nature. Thus, a series of simulations were performed varying the thickness of the encapsulation, assuming that the solar cell is symmetrically immersed in the material for a radius of curvature of 6.51 m with thicknesses from 1 mm to 3 mm. Table 4 shows the results obtained for the total deformation and von Mises stress for the previously selected encapsulation materials.

Table 4. Table 4. Stresses and Deformations in Epoxy Resin

Material	t [mm]	σVM [MPa]	d [mm]
Epoxy Resin	1	65.75		1.60
	2	53.44		0.91
	3	49.75		0.70

Source: created by the authors.

Von Mises stress decreases as the thickness of the encapsulated material increases. The 2 mm thickness is desirable, as it is only 4 MPa different from the 3 mm thickness. The stress and deformation distribution for the 2 mm thickness is shown in Figure 12.

Figure 12. Figure 12. Von Mises Stresses and Deformations Distribution for Epoxy Resin Encapsulation Source: created by the authors.

Considering supports at both ends of the panel and a load of uniformly distributed in the center, a distribution of deformations is obtained in which the maximum value of 0.91 mm is found at the point of load application. However, for the case of the stresses, it is found that the maximum magnitude of 53.44 MPa is close to the panel supports, while at the close point of load application, the stresses decrease considerably. This fact may affect the reliability of the solar panel discussed above. Therefore, for solar panel applications with curvature, it is recommended to use monocrystalline cells, since they present a fracture stress lower than 80 MPa, maintaining the assumed failure probability lower 5 % for perpendicular loads [21].

The accuracy and confidence of the reinforcement results is based on the convergence graph, for nodes number in comparison with Total Deformation, this is shown in Figure 13.

Figure 13. Figure 13. Mesh Convergence Analysis Source: created by the authors.

It is observed that from 3000 nodes, the values for deformation are stabilized in 1.17 mm, which allows to establish the characteristics of the mesh in the FEM study to obtain adequate values with the lowest computational cost.