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The uncertain analysis of fixed solar compound parabolic concentrator (CPC) collector system is investigated for use in combination with solar PV cells. Within solar CPC PV collector systems, any radiation within the collector acceptance angle enters through the aperture and finds its way to the absorber surface by multiple internal reflections. It is essential that the design of any solar collector aims to maximize PV performance since this will elicit a higher collection of solar radiation. In order to analyze uncertainty of the solar CPC collector system in the optimization problem formulation, three objectives are outlined. Seasonal demands are considered for maximizing two of these objectives, the annual average incident solar energy and the lowest month incident solar energy during winter; the lowest cost of the CPC collector system is approached as a third objective. This study investigates uncertain analysis of a solar CPC PV collector system using fuzzy set theory. The fuzzy analysis methodology is suitable for ambiguous problems to predict variations. Uncertain parameters are treated as random variables or uncertain inputs to predict performance. The fuzzy membership functions are used for modeling uncertain or imprecise design parameters of a solar PV collector system. Triangular membership functions are used to represent the uncertain parameters as fuzzy quantities. A fuzzy set analysis methodology is used for analyzing the three objective constrained optimization problems.

Solar CPC PV collector systems are capable of dealing with general situations under concentrated sunlight and issues resulting from higher cell operating temperatures; that is essential in utilizing concentrating systems as solar PV systems. Solar CPCs of PV concentrations have been considered for use in combination with solar cells.

Within solar CPC PV systems, any radiation within the collector acceptance angle enters through the aperture and finds its way to the absorber surface by multiple internal reflections. Improving the efficiency and reducing the cost of these solar collectors is a hot research topic in the field of solar collectors. Winston et al. (1975) [

This CPC concentrator is used with a combination of a photovoltaic and thermal (PV-T) collector to be a solar CPC PV-T system. To improve energy efficiency, the concept of a dual-fluid concentrating photovoltaic thermal (PV-T) solar collector was first introduced by Trupanagnostopoulos (2007) [

In many real-world problems, the design data, objective functions, and constraints are presented in vague and linguistic terms. However, the optimization problem should be stated in precise mathematical terms. It seems that it is more reasonable to describe a transition progress from absolute possibility to absolute impossibility. The role of fuzzy logic is to establish a bridge between qualitative and quantitative modeling. In this view, as the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance become almost mutually exclusive characteristics. For a nonlinear engineering problem, fuzzy set theory is very helpful, and a tool that transforms this linguistic control strategy into a mathematical control method in modeling complex and vague systems. Therefore, the fuzzy logic approach is intended to streamline the decision analysis process and produce an evaluation according to the decision-maker's value system and judgment, while maintaining simplicity and tractability. The fuzzy set theory was introduced by Zadeh over half a century ago (1965) [

In the collection of the maximum amount of solar energy, the most important considerations in a solar energy system are elevation, declination, and azimuth angles. Although the horizontal face of the solar energy system absorbs the solar energy for maximum performance in summer, the amount of solar radiation is not always maximized due to the particular location and the season as shown Figure

Comparison of solar radiation and length in terms of month on horizontal surface by region.

Variation of solar radiation density with respect to month

Variation Solar length of days with respect to months

The angle of incidence of the sun’s ray on the concentrator is a main concern in collecting as much sunlight as possible. As a high concentration ratio, it is possible to use a multijunction photovoltaic cell with maximum efficiency. Reflector technology can be applied to low concentration photovoltaic module systems to collect sunlight via a solar cell. Determining the angle of the mirrors is dependent on the direction of a photovoltaic module system, which is fixed, and includes inclination of installation and location.

The main concern in a solar CPC PV collector is to maximize amounts of collected solar radiation by the collectors as shown in Figure

Cross section of compound parabolic concentrator and enlarged schematic of receiver.

Compound parabolic concentrators (CPCs) are a part of concentrating collectors, which contain parabolic reflectors and planar receivers as shown in Figure

Design process for maximization of a solar CPC PV collector system.

Tilted installation of single CPC PV Collector considering elevation, declination, and azimuth

Multiple-row compound parabolic collectors with shading effects in a given area

This study investigates uncertain analysis of a solar CPC PV collector system using fuzzy set theory. The fuzzy analysis methodology is considered suitable for ambiguous problems to predict variations and provides a more complete set of uncertainty information as part of the solution as well as uncertainty only associated with the initial uncertain input data. The fuzzy set theory is concerned with membership of precisely defined sets and is a mechanism for describing objective matters with countable events. Fuzzy set theory is beneficial in three clear ways. First, the theory contributes to realistic analysis of uncertainty since it provides tight bounds on the solutions of possible optimization and presents a method to compute a risk (deviation) as given by the level of the interval uncertainty. Second, a membership function can be established for each of these linguistic values using a certain shape on a certain range as fit for given conditions using triangles, which have been the most popular set shape for approximating nonlinear systems. Last, the fuzzy set theory provides a useful framework for better representing the information about desired projects and explicit benefits in risk modelling for better depicting situational reality. Graphically, a membership function can be represented by a variety of shapes, but it is usually convex. These membership functions can be determined subjectively; the closer an element to satisfy the requirements of a set, the closer its grade of membership is to 1 and vice versa. The fuzzy theory can help design engineers to predict and analyze the performance of objectives with variations in design parameters and/or uncertain input values for uncertainty and design and behavior constraints. For example, the statement “the solar CPC PV collector system generates 100 kW with a probability of 0.8” is imprecise because of randomness in the input of uncertain parameters with material properties, manufacturing processes, and environmental conditions of the system. Similarly, in the optimum design of the width of a flat receiver of

Concentrator CPC PV array systems use reflectors to concentrate sunlight onto PV cells. This technique leads to a reduction in the cell area required for generating a desired amount of power. In a mathematical optimization problem, two different types of values are used for finding the solution: design parameters and variables for design and behavior constraints. They are summarized in Table

Design parameters and variables for design and behavior constraints.

Design parameters | Variables for design and behavior constraints | ||
---|---|---|---|

CPC unit design | Seasonal characteristic | Cost | |

| | | |

| | | |

| | | |

| | | - |

| | | - |

| | | - |

| | A | - |

| | - | - |

- | | - | - |

In the case of a flat receiver, the geometry of a CPC unit is designed in accordance with two main factors: the acceptance angle of

In examining the role of cost objective function in the optimization problem, there are three components to cost effects: CPC reflectors for concentrating light energy, solar PV cells, and a given installation area. Especially, since the cost of an installation area varies by location, the cost of land of ^{2}, respectively. Thus, the cost objective function of the CPC collector system can be minimized and given as

The optimal design of a solar CPC PV collector system is designed to efficiently collect and concentrate the sun’s rays with the acceptance angle. Once the acceptance angle is adjusted, solar CPC PV collector systems are able to concentrate sunlight on the solar cells.

When we use a mathematical optimization technique, we need to provide an initial guess as a starting point for the algorithm. Optimization iteratively improves the initial guess in an attempt to converge to an optimal solution. Consequently, the initial guess of design parameters determines how initial guess converges to a solution within the algorithm. The results of the single-objective optimization problems of the maximization of the annual average incident solar energy

In order to find the single objective optimization of solar CPC collectors, lower and upper bounds, predefined values are illustrated as follows:

_{min} = 15m,_{max} = 30m,_{min} = 0.5m,_{max} = 2m,_{min} = 0.8m,_{max} = 150,_{max} = 150, P = 80%,_{1} = 100 $/m^{2},_{2} = 20 $/m^{2},_{3} = 1/20/50 $/m^{2},

The solar collector is assumed to be installed in a specific location, Miami, Florida (USA) and the stating initial design vectors are given as_{1}, f_{2}, and f_{3} are found using the MATLAB program

Initial design and single-objective optimization results (design variables)

Objectives and other outputs | |||||||||
---|---|---|---|---|---|---|---|---|---|

Objectives | | | W | D | K | N | | ||

(m) | (deg) | (m) | (deg) | (m) | |||||

Initial | 0.10 | 40.00 | 25.00 | 40.00 | 1.000 | 70 | 10 | 0.5000 | |

Min | 0.11 | 89.54 | 30.00 | 52.98 | 0.806 | 100 | 9 | 0.0036 | |

Min | 0.17 | 89.82 | 29.98 | 53.79 | 0.807 | 101 | 6 | 0.4160 | |

| |||||||||

Min | | 0.11 | 25.62 | 28.81 | 21.69 | 0.802 | 76 | 4 | 0.6136 |

| 0.14 | 25.03 | 29.37 | 21.26 | 0.800 | 69 | 4 | 0.2622 | |

| 0.25 | 89.97 | 29.48 | 20.35 | 0.801 | 64 | 4 | 0.0012 |

Initial design and single-objective optimization results (objective functions and other outputs)

Objectives and other outputs | ||||||
---|---|---|---|---|---|---|

Objectives | cpc ratio | | | | | |

( | ( | ( | ($/ | |||

Initial | 1.4450 | -1.2675 | -1.0230 | 0.5517 | 0.4353 | |

Min | 1 | -1.3790 | -1.1462 | 0.7190 | 0.5214 | |

Min | 1 | -1.3778 | -1.1480 | 0.7257 | 0.5267 | |

| ||||||

Min | | 2.1921 | -1.1032 | -0.8421 | 0.3063 | 0.2776 |

| 1.7963 | -1.1032 | -0.8833 | 0.4007 | 0.3632 | |

| 1 | -1.1032 | -0.8927 | 0.4689 | 0.4250 |

In the case of the maximization of annual monthly average incident solar energy (f_{1}), in order to collect the maximum average incident solar energy during all seasons, the geometric design parameters of the CPC collectors such as_{2}), the design parameters have a similar role in ensuring an increase in the amount of solar energy, excluding the value of the truncation ratio (_{3}), the objective function depends on the CPC PV receivers and reflectors and installation land with three different values, 1/20/50 $/m^{2}. Therefore, we could observe the value transitions from initial guess to optimum point of each objective finding the feasible solution while satisfying the behavior constraints for each objective function.

The uncertain input parameters of the CPC collector unit have a length of receiver

The mapping of uncertain input onto an uncertain response is called fuzzy set analysis. Fuzzy set theory provides gradual membership from the domain of quantitative and precise phenomena to vague, qualitative and imprecise conceptions. A fuzzy member can be represented using the concept of a range of interval confidence. The fuzzy set theory allows for gradual membership functions in relation to the set. This gradual membership is explained by a membership function. Membership in a classical subset A of X can be defined as a characteristic function

Triangular fuzzy number.

In the case of

Variation of deviations of

Uncertainty in

Uncertainty in

Uncertainty in

The uncertain input parameters of

In the case of

In the case of

Variation of triangular shapes of

Results of the influence on

Results of influence on

Results of influence on

Influence on cost of solar CPC PV collector with respect to uncertain input parameters.

Influence on average month incident solar energy of solar CPC PV collector with respect to uncertain input parameters

Influence on lowest month incident solar energy of solar CPC PV collector with respect to uncertain input parameters

Influence on cost of solar CPC PV collector with uncertain input parameters

Fuzzy set analysis techniques used in solar CPC PV collector systems have been estimated. The results of fuzzy set analysis are gained by applying values of

From a practical standpoint, variation values of design parameters of CPC units and PV collectors predict how variations from the absolute values of the optimal design parameters affect various deviations of the crisp values of objective performances. The variations can graphically describe multiple points along the transition range, from absolute possibility to absolute impossibility with a variety of shapes derived from fuzzy set theory descriptions on uncertainty. The results from the uncertain analysis of the design variables of single objectives contribute to the existing research in an impactful way by arming designers with a more economical and robust design approach based on customer requirements when they endeavor to design more efficient solar CPC PV collectors.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.