1. Introduction
Despite a constant rise in concentration photovoltaic (CPV) cell efficiency, currently more than 45% [
1], CPV technologies struggle in the photovoltaic system market, as they require high precision tracking systems in order to take full advantage of direct solar radiation, and they also need appropriate cooling. Indeed, tandem solar cells endure localized high heat fluxes and high operation temperature gradients. Additionally, concentration optics deliver non-uniform irradiance profiles onto cell surfaces, due to their inherent ray tracing properties and manufacturing imperfections. Baig et al. [
2] and Franklin et al. [
3] found that non-uniform illumination induces efficiency losses.
Non-uniform illumination induces in-plane temperature gradients that follow complex time regimes according to ambient stimuli: slow day-night cycles, fast daytime cloud shading periodicity, and faster wind-induced tracking point perturbations. Wind influence on CPV systems performance has raised great interest in research: Chih-Kuang et al. [
4] quantified how much wind induces mechanical deformation in structures supporting Fresnel lenses. Chumakov et al. [
5] developed a methodology to evaluate CPV systems’ aerodynamic behavior, including wind tunnel experiments. Stafford et al. [
6] performed experimental observations to assess how meteorological data correlate with tracking errors. Another factor is soiling, studied by Sanchez et al. [
7] and Calvo-Parra et al. [
8], which reduces CPV system performance by scattering and shading effects.
Acceptance angle is another important parameter related to receiver flux density [
9]. This parameter depends on several factors such as the optical and mechanical precision in the construction of the CPV module and the quantity, type and quality of the optical elements for solar concentration. Small acceptance angles require greater precision in the tracking system, solar sensor, actuators, installation, and alignment [
10]. In some systems, a secondary optical element is employed to raise the concentration ratio, increase tolerance to tracking errors or achieve a uniform flux distribution in the receptor [
11,
12]; however, the manufacturing cost also increases. Hence, concentrators should be designed to use the minimum amount of materials and the simplest optical scheme [
11].
All these CPV-inherent factors trigger uneven and, to some extent, unpredictable thermal stress variations. They combine synergically to reduce concentration cell performance and life cycle; increasing the cost, thus restraining market penetration of CPV technology. Therefore, operation temperature predictions resolved in space and time are very valuable information and mandatory for CPV designers, issued from models that simulate and analyze the concentration, temperature and thermal stress level of the CPV cell. Hence, permitting scale concentrator apertures according to available solar radiation and CPV cell maximum operation temperatures, which range between 85–110 °C. By doing so, designers can increase the energy density and cell life-cycle, while decreasing cell cost.
Nowadays, given the complexity of CPV structures and irradiance heat fluxes spatial profiles, temperature fields are estimated by means of the finite element method (FEM) as reported in
Table 1. FEM studies are costly in computing time [
13] for any CPV optimization process, as they require a high number of simulations for parametric studies.
Most studies have dealt with steady-state analysis. This is not an intrinsic limitation of an FEM approach and, of course, transient simulations can also be performed, as shown by Renno et al. [
14], Li et al. [
15], Sweet et al. [
16], Theristis et al. [
22] and Oliverio et al. [
23]. FEM is fundamental regarding system design since it can deal with exact complex 3D structures and also with possible nonlinearities. However, this approach can be impractical if system real-time simulation in response to fast incident flux changes (cloud shading and wind perturbations), is needed. None of the listed studies report numbers concerning computing times.
A complementary alternative for thermal modeling under such specific circumstances, conditioned to maintain an acceptable level of simplification for the 3D system structure, would be to solve the heat equation analytically; along with its inherent advantages: direct parameter estimation and optimization, straightforward parameter sensitivity studies, and fewer computing resources needed than FEM.
The thermal quadrupoles method [
24] is suitable for heat transfer analysis of multilayered systems and has been extensively employed for space and time-resolved thermal simulations as varied as aircraft composite materials thermal characterization [
25], ultrahigh heat flux and temperature simulations in magnetically confined plasma facilities [
26], and thermal characterization of microchannel reactors in microfluidic systems [
27]. The method is based on space and time integral transformations, permitting work on the transformed space [
24,
26,
28,
29] within a linear relationship between temperature and fluxes at different system layer interfaces.
In this work, the first thermal quadrupoles implementation of a 3D analytic solution for transient heat transfer in CPV systems is reported. The thermal quadrupole model is associated with ray tracing analysis which provides custom irradiance maps as an input stimulus for the thermal model. Detailed sections are organized to highlight the contributions and advantages such a combination of tools brings to CPV systems engineering.
In
Section 2, Materials and Methods, the core of this work is described, starting with the procedure for irradiance profile estimation for a concentrator based on a Fresnel lens. Inspired by the specifications of a commercial Fresnel lens, simulations based on refractive surfaces generated by script are reported for the first time and preliminary comparisons between experimental and simulated irradiance spots are provided. Subsequently, a 3D analytic heat transfer model, based on the thermal quadrupole method, is introduced. A 2D time-resolved CPV temperature map can be computed from the combination of the absorbed flux map obtained from Tonatiuh and the thermal quadrupole model.
In
Section 3, simulation results are presented for a typical configuration of a CPV module, i.e., a Fresnel lens to concentrate the solar radiation onto a CPV cell coupled to a passive heat sink. A parametric study is presented, revealing the best set of parameters for design optimization in order to keep operation temperatures within safe conditions. Transient thermal variations induced by tracking point dynamic perturbations are reported for the first time.
The contribution of this work is twofold: on the one hand, time-resolved surface temperature profiles in response to time-varying 2D irradiance patterns have been analytically estimated, providing a powerful tool for CPV system designers and paving the way for future thermal stress assessment, which in return would allow more accurate life-time and thermal breakdown predictions. The model covers full time regimes: transient and steady state, as shown by simulations performed with an engineering design scope. Theoretical temperature gradients induced by fast tracking point perturbations are presented for the first time and discussed. Simulation execution times are as short as 9 s for a full month of CPV system operation computed with irradiance stimulus inspired by real data.
On the other hand, Fresnel lens ray tracing implementation by means of the Tonatiuh open source software is reported for the first time. This is relevant as Fresnel lenses are extensively employed in CPV systems. A preliminary comparison between a real lens spot and the spot generated by script is discussed, indicating directions for more in-depth studies. However, this work’s focus is on the thermal quadrupole model, its capabilities, and the advantages it brings in combination with the Tonatiuh open source ray tracing software.
4. Discussion
Script-based Fresnel lenses simulations for concentration photovoltaics were proposed. They allow the use of Tonatiuh ray tracing open source software to represent refraction objects, and obtain synthetic receiver flux maps for any worldwide date, hour and latitude. Characteristics of a commercial Fresnel lens have been introduced to the Tonatiuh script, and experimental and simulated flux distributions were compared. It should be pinpointed that commercial lens is employed in order to inspire the script-based Fresnel lens model.
For design purposes, the script-based lens was employed, as its design was inspired in the real lens, and it provides a synthetic irradiance spot similar to the real one. Additionally, it provided the maximum theoretical flux, and consequently, the highest foreseen operation temperatures.
Tonatiuh’s set of synthetic flux profiles under perfect tracking conditions throughout daytime and seasons were then combined with measured solar irradiance series, which include variations induced by cloud shading. Such a synthetic flux voxel was then introduced into the thermal quadrupole model. Indeed, an analytic model for CPV cell temperature maps estimations, based on the thermal quadrupoles method has been proposed. It accurately predicts time-lapse CPV cell surface temperature maps under any worldwide irradiance conditions, for any set of experimental or simulated irradiance fluxes. Such a thermal model, coupled with Tonatiuh ray tracing analysis, is a powerful combination of tools for CPV systems design.
Indeed, a parametric analysis to keep the system’s upper operation temperature below the maximum limit according to CPV cell specifications was developed. Computing time can be as low as a few seconds, even for a temperature map calculation spanning over a month, provided there is a known spatial function for flux. The proposed method simplifies the design of solar concentration systems and would allow studying the performance and the life cycle of concentration photovoltaic cells, through an accurate computation of thermal stress along years of operation.
The scope of the proposed methodology does not claim to replace FEM analysis. Instead, we think it provides a complementary tool that can be employed at the first stages of CPV system design to provide FEM models (more accurate for complex 3D structures representation) with narrowed and fine-tuned ranges of relevant design parameters and time regimes, minimizing computer resources usage. Furthermore, this method opens up interesting opportunities for improvement of CPV systems reliability studies.
As shown in
Table 1, all prior works concerning thermal response of CPV systems consist of 3D models for average temperatures or numerical simulation by Finite Element. To our knowledge, this is the first time analytic computation of CPV cell surface temperature resolved in space and time has been proposed. The same is true for the association of the thermal quadrupole method and ray tracing analysis.
5. Conclusions and Perspectives
The thermal quadrupole model, in association with ray tracing analysis software Tonatiuh, applied to transient temperature maps, resolved in time and space was reported for the first time. Such an approach, by its analytic essence, is suitable for CPV engineering, including parametric studies, design optimization, and reliability studies. Additionally, the so-performed studies are executed in less computing time, as short as 9 s for a full month of simulated operation, with a generic laptop computer.
Nowadays these tasks are accomplished with FEM tools, which are undeniably useful and mandatory when systems structures are geometrically complex and show non-linearities. Accordingly, the claim is not replacing FEM, but rather complement it, by providing them with narrowed ranges for design parameter optimization, when working with CPV designs whose complexity cannot be properly represented by thermal quadrupole models. We would like to draw reader’s attention to another field that could benefit from this research: CPV cell reliability.
Potential for Reliability Studies
CPV cell thermal breakdown has been extensively studied and is the recognized main cause of cell failures and lifespan shortening [
46,
47,
48,
49,
50,
51,
52,
53,
54]. To our knowledge all reported works employ simplified steady state models and operation condition statistics for the assessment of cyclic degradation. With FEM simulations, it is hard to foresee the implementation of transient simulations over months and years of operation with real ambient and irradiance data. Nevertheless, with the approach proposed in this work, computation times are very short and hence, it is reasonable to contemplate thermal stress simulation with real irradiance data over full years of operation. Furthermore, including higher time resolution computations which permit assessing the influence of fast angular perturbation of tracking point (wind, vibrations) is also conceivable.
The consequences on lifespan shortening induced by these temperature variations, represented with better accuracy by this work method, are still to be studied. The same is true for thermal breakdown forecasting based on full knowledge of CPV thermal response over years with high time resolution.
Finally, concerning the implementation of Fresnel lenses by refractive surfaces generated by script with Tonatiuh, which is also a novelty presented in this work: Proper validation of the script-based model by comparing real and scrip lenses, including their geometry, is worth its own research and is beyond the scope of this work, focused on the association between the thermal quadrupoles model and ray tracing analysis.