**1. Introduction**

The continued increase in energy demands worldwide is leading in an emergently unsustainable situation [1], then energy-related greenhouse gas emissions will result in substantial climate change if no decisive action is taken to reduce global warming. With the United Nations, this target to hold means global temperature will rise by the end of the century to at least 2 ◦C [2]. These facts represent an essential driving force for the gradual implementation of safe and feasible alternatives in all power-consuming sectors [3] in addition to policies that help industries implement strategies to improve the efficiency of energy use through innovative technologies. Where they are integrated within national and foreign policies, and with the mechanisms of ecological technological innovation to give demand to energy saving and emissions reduction [4]. Today, not only is it being applied to the industrial sector, but citizen awareness has brought these energy technology innovations into the home. There are examples of the active use of smart technologies with the internet of things for the home, as in [5] with the aim of managing energy performance and optimizing consumption and obtaining net zero energy. The internet and smart phones have enabled real-time monitoring of sensors and actuators that control active consumption in homes.

The society and energy system as a whole need to be more energy efficient, and the development of renewable energy sources can help. One of the problems posed by renewable sources is the balance

between production and consumption, for which an accurate prediction of the load and a correctly sized storage system are desirable. In [6], the authors have studied two situations for distributed photovoltaic production: "those that predict by separating the load portion due to consumption habits from the production portion due to local climatic conditions, and those that try to predict the load as a whole". Predicting the behaviour of the grid and its dependence on climatic conditions largely defines the efficiency of the system. Precise forecasting techniques are tools for integrating renewable energy systems into the power grid [7]. However, it attention should be given to their rate of development, rapidly growing share in energy demand, and impact in the market [8]. Solar energy is a renewable source that is clean, inexhaustible, and allows for local energy independence [9]. The total energy production from the sun is 3.8 × 1020 MW, equivalent to 63 MW/m2 of solar surface; only a small fraction, 1.7 × 1014 kW, of the total emitted radiation is captured by the Earth [10]. Although it is estimated that even with this tiny portion, 30 min of solar radiation reaching the Earth is equal to the worldwide energy requirement for a year [11]. Scientific research has provided a high importance to solar energy in the last three decades, being the second most important renewable resource, with the 26% of the scientific publications [12].

The relatively low flow of solar energy received at the Earth's surface can be surpassed by the use of concentrating solar collectors that transform solar energy into other types of energy, usually thermal in Concentrating Solar Power (CSP) [13] and Central Receiver System (CRS) [14]. The concentration of solar radiation by reflective mirrors on the receptor of a Thermal Conversion System has the important advantage of reducing thermal energy losses compared to unconcentrated systems [15], resulting in increased thermal conversion performance for the operating conditions specified and allowing higher working temperatures to be achieved with appropriate efficiencies [16]. The thermal energy collected on the receiver of the concentrating solar systems is typically used to produce electricity through a conventional power block, in the commonly named concentrating solar power plants. Although the basic technology had been under development for about 140 years, solar thermal electricity (STE) on grid was not achieved until the 1980s [17].

The parabolic-trough solar energy technology is the most tested and the cheapest large-scale solar energy technology available today for the use of thermal solar energy with different types of working fluids [18,19]. The improvements in these parabolic-trough systems are still far from complete. The introduction of internal longitudinal fins and a reflector shield together results in a thermal efficiency improvement of 2.41% compared to the same system without the improvements [20].

The electrical capacity of CSP plants currently in operation worldwide is 5.7 GW by the end of 2020, according to the International Energy Agency's (IEA) forecast for 2050, an 11% of the worldwide energy mix will be provided by CST systems [16]. 84% of this power is produced by plants with parabolic-trough collectors (PTCs) [21]. Although these solar concentration technologies are among the most widely implemented worldwide, alternative systems are still being studied [22,23].

PTCs are integrated by a trough-shaped reflector with a parabolic cross-section that concentrates and focuses the direct solar irradiation in parallel to the axis of the collector in a focal line (see Figure 1). A receiving pipe with a fluid that flows inside it and which absorbs the concentrated solar energy from the pipe walls and increases its enthalpy is placed along the length of the collector at its focus. The tube is typically coated with a selective layer to reduce thermal losses by radiation to the ambient. A cylindrical glass enclosure concentric to the receiver pipe is also employed to reduce thermal heat loss by convection into the environment. A single-axis tracking system turns the collector to be sure that the sun's ray drops parallel to the collector's axis.

**Figure 1.** Scheme of working of a PTC facility. (1) Reflector; (2) Absorber tube; (3) Structure; (4) Solar Field Piping.

The solar installation is intended to be modular in design and consists of several parallel rows of solar collectors [18]. It involves a large number of reflecting surfaces, from 0.6 to 10 ha/MWe, depending on the capacity of the storage and auxiliary systems [24]. The different rows are separated among them to avoid shadowing and permit the access and handling of cleaning devices. Collector shadowing means a reduction in the net aperture area, so reducing the amount of thermal energy that can be supplied by the solar field. In this sense, distance between adjacent collector rows should be as high as possible.

In general, the land occupation factor (that is, the aperture area of the solar field divided by the land surface occupied by the whole plant) is around 0.245 [25]. This means that the land area required to install a plant is around four times the solar field area, partially due to the separation among solar collector rows. Hence, this separation should be minimized to avoid an unreasonable land use. Consequently, an optimization process to calculate the collector-row separation is required, searching for a compromise that maximizes separation to reduce shadowing but minimizes it to use land wisely. The effectiveness of this optimization process depends on the method used to calculate the shadowing between adjacent PTCs.

### **2. Classical Methods for the Sizing of PTC: A Brief Overview**

The designs of solar energy installations should be designed for the most efficient use of energy. In a classical model, the area of the solar collector should be perpendicular to the received sunlight. However, given the Earth's declination, the relative positions of the Earth's hemispheres vary continuously in relation to the sun throughout the year and therefore the day. Therefore, in order to ge<sup>t</sup> the solar rays perpendicular to each PTC, the tilt of a solar installation with respect to the horizon should also change throughout the year. Thus, a common solution to maximize energy generation is getting the solar installation in the most perpendicular position to the sun at the time of the winter solstice.

It is known that the time of the zenithal passing by of the sun or meridian of the location, i.e., the actual 12 h of the solar day, establishes the relationship between latitude (Φ), height of the sun on the horizon (h) and declination angle (δ). See Figure 2, which is provided by the following equation [26]:

$$\gamma \mathbf{s} = (\pi \mathfrak{L}) - |\boldsymbol{\delta}| - \boldsymbol{\Phi} \tag{1}$$

$$
\Theta\_{\rm ZS} = (\pi/2) - \gamma \text{s} = |\boldsymbol{\delta}| + \boldsymbol{\Phi} \tag{2}
$$

where δ is the Earth's decline (at the winter solstice), γs is solar altitude angle and θZS as zenith angle.

**Figure 2.** Geometry on a PTC for the sun's rays (section view).

### **3. Standard Methods for Determining the Spacing between Collectors in PTC Facilities**
