1. Introduction
There is no doubt about the important role that energy plays in our societies. Its implications go beyond mere technical aspects. Social structuring, economic development, and the environment, among others, are aspects that make up the complex implication of energy in the global agenda [
1]. In this context, renewable energies have been experiencing sustained growth in recent years. Thus, the International Energy Agency (IEA) foresees a record increase of 218 GW in the year 2021 in the net capacity of renewable electricity installed in the world, in an average scenario, that could reach up to 266 GW, in an accelerated scenario [
2]. In fact, during the first quarter of 2020, renewable energies were the only source of electricity whose demand increased despite the 2.5% decrease in global electricity demand caused by the blockades implemented by different governments to curb the spread of COVID-19 [
3].
Within the field of renewables, solar energy, in general, and photovoltaic (PV), in particular, are candidates to satisfy a large part of the global energy demand in the coming years due to their abundance and competitiveness [
4]. In fact, in the IEA predictions for the year 2021, solar PV accounts for 54% of the growth in the world’s installed net renewable electricity capacity [
2]. The remarkable progress that the technology associated with the implementation of photovoltaic energy has been experiencing has not only driven its boom in recent years [
5,
6] but this growth is expected to accelerate during the 2023–25 period [
2].
One of the fundamental factors in the energy production of PV plants is the incident solar irradiance on the collectors. Among the various strategies that exist to increase this irradiance in solar collectors and, therefore, energy production in PV plants, is solar tracking, which is a technological niche in which there are still possible improvements that can contribute to such an increase [
5]. This strategy is in contrast with fixed panel structures that have a constant orientation towards the sun depending on the latitude of the place where the PV installation is located. Thus, in the case of solar trackers, the PV modules move while looking for an orientation that generates more energy, either by capturing solar energy for as long as possible [
7] or by capturing maximum solar irradiance [
8,
9].
The most common classification of trackers is established based on the number of axes used to move the modules. Thus, we speak of tracking systems on one axis (movement in azimuth or elevation) or on two axes (movement in azimuth and elevation). Improving the technology of solar tracking systems is an important objective considering the high demand of energy resources, being the research niche for many authors. For example, in [
7], different tracker motion control systems are studied with regard to their economic evaluation. In [
8], the different solar trackers are studied in depth, performing a comparison between them in terms of efficiency, performance, advantages, and disadvantages, while other studies perform an in-depth review of the related literature to define the advantages of the applicability of solar tracking [
10,
11].
In the scientific literature, there are studies that question the use of trackers with one or two axes versus fixed systems or one axis versus two. Specifically, various studies are being developed that question the various possibilities that arise in terms of efficiency, cost, location, production, etc., which is a real scientific challenge. For example, in [
12,
13,
14], authors analyze the effect of different solar trackers strategies considering the location. Other authors provide profits of the tracking photovoltaic systems in comparison with fixed photovoltaic systems [
15,
16]. An interesting result is the one presented in [
17], where a method for optimal storage capacity was calculated under the power-curtailment and storage/discharge requirements. Hence, although solar tracking systems have a higher cost than fixed systems, their maintenance is more complex and their exposure to environmental conditions is greater [
18]; the performance of dual-axis trackers are greater than those of fixed systems [
19,
20] and single-axis trackers [
21,
22,
23]. Bahrami states in [
21] that a dual-axis tracking system would result in greater irradiance than a single-axis due to its ability to minimize losses associated with cosine effect. Authors in [
22] highlight their conclusion that using the two-axis sun tracker system enables the PV panel to collect and produce higher amounts of electrical energy than using single-axis and fixed structures; their study considered five configurations of sun tracking systems and two traditional fixed panels. The results presented in [
23] show that the optimal trajectories for the tilt and azimuth angle depend on the available solar radiation, solar cell efficiency, tracking system consumption, and the optimization bounds. Therefore, based on these studies, it can be affirmed that the choice of the type of strategy depends on several factors, it being necessary to delve into the technological and economic components to reach useful conclusions [
5].
Regarding the tracking mechanism, different driver methods can be distinguished to achieve the objectives of the collector movement [
8]. Among them, the sensor driver systems stand out, whose operation is based on the variation of light received by optical sensors that cause the movement of the collectors looking for the position of the sun. Additionally, there are microprocessor driver systems that incorporate small processors with movement strategies programmed through mathematical models to locate the position of the sun. Within these in the bibliography are the open-loop driver systems that modify the movement of the actuators of the modules from mathematical equations that fix the position of the sun from the day and the hour. In contrast, closed-loop driver systems modify the movement of the actuators based on the information provided by position sensors, recalculating the position of the sun. Finally, intelligent driver systems incorporate artificial intelligence techniques to control the movement of collectors [
24].
Another of the determining factors in the performance of a PV plant is the shading of its modules, since these shadows not only imply a lower incident irradiance [
25] but also give rise to the appearance of hot spots that bring with them overheating and losses in energy production [
26]. In this regard, the behavior of PV modules when they are partially shaded has been widely debated. Several simulation models have been used to find a configuration less susceptible to shadow problems of solar cells [
27,
28,
29,
30]. Specifically, Díaz-Dorado et al. [
31] have analyzed the effect of shading in a PV tracker with partially shaded astronomical tracking based on the exact arrangement of shaded cells and modules. Other authors have analyzed the energy cost in the production of PV plants as a function of the connections between the cells and the modules [
32] of the ground cover ratio (GCR) of the plant that depends on the variables of their design [
33] or of the tracking strategy [
26,
34].
One possible solution to alleviate the effect of shadows in PV plants with solar tracking is back tracking [
35], which consists of modifying the orientation of the collecting surfaces in shading situations between panels in order to eliminate such shadows. Another possible solution is to modify the geometry of the collecting surfaces. Thus, although most of the PV panels found in PV plants connected to grids are rectangular, there are already some installations with dual-axis solar tracking where the collectors have other geometric shapes, such as those developed by the Deger Ibérica company in Tarragona (Spain), Ontario (Canada), or Estonia, with 15.6 kWp, 24 MWp, and 100 kWp installed, respectively [
36]. However, no previous works have been found in the literature aimed at characterizing the geometric shape of the collectors or their degree of modularity in terms of optimizing the performance of a photovoltaic installation in the event of the possible incidence of shading.
In this context, in the present work, the annual radiative uptake has been quantified in a wide set of PV installations with dual-axis monitoring with different geometries in which the shape of the collectors and the design parameters have been systematically varied. For each case, it has been assumed that the collectors follow the optimal solar tracking strategy proposed by Fernández-Ahumada et al. [
37,
38]. Unlike traditional solar tracking methods that search for the position of the sun at each moment using astronomical models [
39,
40,
41], according to this strategy, the collecting planes are oriented at each instant of time towards the direction of space in which the irradiance is maximum, except during the moments when such orientation implies the shading of one collector over another. When this occurs, normally at the beginning and end of the day, the collectors are oriented in the direction in which, without causing shading, the incident irradiance on the collectors is maximum. In this way, the production of each installation, which is considered proportional to annual solar radiation, is calculated under the hypothesis that each installation will follow an optimal tracking strategy adapted to its own geometry. Therefore, this study aims to advance the characterization of the electrical behavior of shaded solar trackers, which is an issue where the scientific community has made a considerable effort, simulating tracking strategies for an improvement in photovoltaic production [
12,
14,
42].
To achieve this objective, after this introduction in which the scientific advances made by the scientific community in the field of solar tracking in PV plants are presented, the following section describes the methodology followed in this study to simulate annual solar radiation incident on collectors with a dual-axis tracking strategy that optimizes radiative uptake while avoiding shading between collectors. Similarly, the methodology established for the study of the influence of the design parameters of a PV plant on this annual solar irradiance is explained. Based on this,
Section 3 presents the results when applying the methodology described to an existing PV plant (“El Molino”, Córdoba, Spain) with its design parameters systematically modified. Similarly, an adjustment model is proposed that represents in a simplified manner the dependence of the annual solar irradiance with respect to the design variables studied, and these dependencies are quantified. Finally, in
Section 4, the main conclusions of the present study are presented.
3. Results
This section describes the incident radiation values on collectors and the results of the study of the influence of the design variables on the solar incidence on the PV plant designs considered. The synoptic values obtained for the annual incident radiation in collectors are summarized in
Table 2.
Figure 10 shows the distribution of values obtained depending on the membership intervals.
In the set of values obtained, it is observed that certain forms of the collector offer the same value. These are the cases in which the set of cuts A, B, C and D of
Figure 11a are permuted as shown in
Figure 11b–d,f–i. A detailed analysis of the procedure followed makes it possible to verify that the shape of the solar collector does not influence directly but rather through its envelope. As a consequence, the collector shapes of the first column (11 (a), 11 (b), 11 (c), 11 (d)) give rise to identical annual radiation results since they all give rise to the same envelope Σ (represented in
Figure 11e). On the other hand, the coincidence of radiation outcomes in the results between the collector shapes of the first (
Figure 11a–d) and the second column (
Figure 11f–i) should be understood as a consequence of the symmetry with respect to the NS axis of the studied configurations and of the symmetry with respect to this plane in the positions of the sun between the hours of the morning and afternoon.
The dependence of the values obtained with respect to the considered design variables was studied using an approximation and simplified model, in which the calculated annual radiation was expressed as a function of the following explanatory variables:
Collector surface .
Distance between trackers in east–west direction,
Distance between trackers in north–south direction,
Discriminatory variable of the type of configuration T (T = 1 for staggered configurations and T = 0 for regular grids).
In this sense, it is worth highlighting that the model described and used (for each of the 25,000 cases) can be considered a mathematical function of the variables proposed. However, the complexity of the model and the need to aggregate results on the different representative days made it difficult to know the weight or influence of each variable on the final results. Thus, to overcome this difficulty, this paper proposes to replace this complex function with a mathematical function of simple expression reproducing the result of the complex model with the least possible error. The reader should assume that this is not a statistical problem but rather a problem of adjustment or approximation of a simple expression function to a complex function, so that statistical methods are not applicable. To address the fit, the set of simple variables was extended with composite variables obtained as products and ratios of simple variables. The proposed function (29) was selected from the set of fits to linear functions of composite variables. It lacks a clear physical meaning, but it allows for reproduction of the results of the model with an average relative error equal to 2.561 kWh/m2year; therefore, it is considered suitable for the study of the relative weight of the variables.
Equation (29) shows the mathematical expression of this model for which the parameters
a,
b,
c,
d,
e, f,
g,
h,
l,
r, and
w have been obtained by the least squares method (
Table 3), with an adjustment coefficient
= 0.993.
Table 4 shows the synoptic values of the estimation errors of the model ε and
, given by Equations (30) and (31), where
is the annual solar irradiance on the solar collectors estimated according to Equation (28) and
is the one approximated by the model (29). The low value obtained for the mean square error means that the equation can be considered valid for the study of dependence of the annual solar irradiance with respect to the variables
,
,
and
.
It is important to note that the model adjusted in Equation (30) does not consider the geometric shape of the collectors itself since, given that the mean error,
, of the proposed model is 3.3 kWh/m
2year (
Table 4), the geometric shape would not have an explanatory capacity superior to this
. This reasoning allowed us to conclude the little influence of the shape of the collectors in facilities that follow the tracking/ back-tracking policy considered in this work.
For a better interpretation of the adjusted Equation (29), we can consider it as the addition of four terms separated by parentheses in Equation (32).
The first term only depends on the collector surface . Given that , it was found that, as the collectors were larger, the lower the annual incident radiation. This effect was due to the greater possibility of inter-shading as the collectors had more surface area. With the cases studied, the variation interval of the final result due exclusively to this term was 24 kWh/m2 year.
The second term marks the importance of the geometric design of the plant given by and , regardless of whether it is a staggered or grid configuration. In the group of cases studied, the variation interval was 116 kWh/m2 year.
The third term is interpreted as a function of the ground cover ratio (
GCR) parameter defined by Equation (33).
According to this definition, the third term considered in the model given by Equation (29) can be rewritten obtaining the expression (34). Thus, this term showed that the variation in this term was 104 kWh/m
2 year for the GCR values considered in the set of cases studied.
Finally, given that the parameter w was negative, the term {wT} implied a small difference of 1.6 kWh/m2 year to the detriment of the installations that were arranged in a staggered pattern compared to those with a regular grid.
4. Conclusions
This work presents a novel methodology for the productive study of PV solar collectors mounted on dual-axis trackers. The study, applied to multiple cases, generated as variations with respect to the design adopted in an existing PV Plant (“El Molino”, located in Córdoba) allows for identification of the design variables that fundamentally influence the annual incident irradiation on the solar collectors and, therefore, on the energy production of the PV plant. Specifically, by systematically varying the geometry of the collectors, the distance between them, and their spatial distribution, 25,000 case studies were simulated. For all of them, the annual incident solar radiation on the solar collectors was calculated, using for this the irradiance estimation model of Perez [
43] and assuming that they were governed by a tracking strategy that optimized radiative capture while avoiding inter-shading between collectors [
37,
38]. Although a comprehensive methodology was used to study the case, it was difficult with the data set to understand the influence of each variable on the final result. Therefore, a simple function was fitted, which accurately reproduced the result of the complex model
. From the irradiance data obtained for the different PV plant designs, a simple mathematical model has been obtained, Equation (29) with a high level of adjustment
that represents the dependence of the annual solar irradiance on the PV plant with respect to design variables such as collector surface,
, NS and EW distances between collectors
and
, and spatial distribution of the collectors: regular or staggered grid. Thus, the proposed model has made it possible to identify that the main variables that influence the annual incidence of irradiance are, in order of greatest influence, the geometric design of the plant as a function of the distances between its collectors,
, the GCR of the installation and the surface of its collectors
which can lead to variations in the electrical production of the PV plant of up to 116 kWh/m
2year, 104 kWh/m
2year, and 24 kWh/m
2year. However, for practical purposes, with regard to the spatial distribution of the collectors, it is not appropriate to assume a better behaviour of the regular grid arrangement with respect to the staggered shape since the margin of 1.6 kWh/m
2year for this variable falls within the uncertainty margin of the radiation prediction models. Similarly, the geometric shape of the collectors does not exert a significant influence on the irradiance received since it gives rise to variations of 3.3026 kWh/m
2year, which, therefore, are lower than the uncertainty margin of the estimation model itself. Future works will study the influence of the shape in confluence with the orientation of the terrain.