Performance and Cost Assessment of a Small Solar Photovoltaic System Using Gumbel-Haugaard Family Copula Analysis

The main objective of the present study is to analyze the availability of solar photovoltaic system. The solar photovoltaic system in this paper is simple one consisting of four subsystems namely, solar panel subsystem, charge controller subsystem, batteries subsystem and inverter subsystem. Through the schematic diagram of state of the system, availability model is formulated and Chapmen Kolmogorov differential equations are developed and solved using Gumbel Haugaard family Copula technique. The numerical values for availability, reliability, mean time to failure (MTTF), cost analysis as well as sensitivity analysis are presented. The effects of failure rates to various solar photovoltaic subsystems were developed.


Introduction
Reliability modeling and performance evaluation of solar photovoltaic system using Gumbel-Hougaard family copula was studied by Maihulla et al. (2021). Review of failures of photovoltaic modules was carried out by Köntges M (2014). Long-term field test of solar PV power generation using one-axis 3-position sun tracker was analysed by Huang B J et al. (2011). In (2003) the partial shadowing of photovoltaic arrays with different system configurations was studied by Woyte A et al. Nishioka K. et al. (2003) investigated Field-test analysis of PV system output characteristics focusing on module temperature. Kinsey G S.(2015) analyzed the Spectrum sensitivity, energy yield, and revenue prediction of PV modules. Xing Zheng. (2015) carried out a research titled the Modelling dependence structures of soil shear strength data with bivariate copulas and applications to geotechnical reliability analysis. Mao-X. (2020) study the subset simulation for efficient slope reliability analysis involving copula based cross-correlated random fields. Yusuf I. et al. (2020) study the reliability modelling and analysis of client-server system using Gumbel-Hougaard family copula. V. V singh and Monika G (2021) study the Reliability analysis of (n) clients system under star topology and copula linguistic approach. Adebayo Cristaldi, et al. (2015). The study pertaining accurate Sizing of Residential Stand-Alone Photovoltaic Systems Considering System Reliability was carried out by E. . Also Feasibility study of renewable energy-based microgrid system in Somaliland ‫׳‬ s urban centers is a research carried out by A. M. Abdilahi et al. (2014). To address the concerns mentioned in earlier work on grid-connected PV system reliability, this study presents a full thorough Copula analysis for all sub-assemblies of grid-connected solar PV systems with a low dependability grid, taking failure specifics and repair intervals into consideration (period of identification and replacement of the PV system). Furthermore, the goal of this work is to explain the dependability of each sub-assembly of grid-connected PV systems. The scope of this study has also been expanded to determine the optimum probability density function for each solar-PV device subassembly's failure rate.
Because data for the PV system is not readily available, the current work uses a reliability modeling technique to investigate the PV system's overall performance. In this work, we provide a novel solar system model that consists of four subsystems: panel, inverter, battery bank, and control charger. The units in each subsystem are considered to have exponential failure and repair times, according to Ismail et al. (2021). The authors looked at a variety of systems that are related to solar photovoltaic systems. Typically, they haven't paid much attention to their operations when using k-out-of-n: systems, which can be seen in a variety of real-world scenarios. However, in many locations, such as banks, factories, schools, and other communication channels, we see redundancy in subsystems, particularly solar panels, there is provision for another panel to continue to function even when others fail. We have examined this home based modest scaled photovoltaic, with redundancy in the solar panels and batteries alone, in light of this outstanding construction. The setup is series-parallel with a k-out-of-n: G operation scheme. A flawless state, a degraded state, and a failing state are the three states of the system. When there are k excellent states in the system, the entire system is functioning, but when there are less than k good clients, the system is on the verge of failing completely. The failure of the primary panel is considered as a partial failure, whereas the failure of the redundant ones is treated as a full failure before the primary ones are repaired. Charge controller and inverter failures are total system failures, copula repair is used to quickly restore the system. For varied values of failure and repair rates, the system was evaluated using the supplementary variable technique, and various reliability indices were produced.

S5
The state S5 is complete failed state due to the failure of subsystem 2.

S6
The state S6 is complete failed state due to the failure of subsystem 1.

S7
The state S7 is complete failed state due to the failure of subsystem 3.

S8
The state S8 is complete failed state due to the failure of subsystem 4.

ASSUMPTIONS
The following assumption are taken throughout the discussion of the model: 1) Initially, both subsystems are in good working condition.
2) One unit from subsystem, subsystem 2, subsystem 3 and subsystem 4 in consecutive are necessary for operational mode.
3) The system will be inoperative if three units from subsystem 1 failed. Also if two units from sub system 3 failed.

4)
The system will also be inoperative if one unit failed from either of subsystem 2 and 4 respectively.

5)
Failed unit of the system can be repaired when it is inoperative or failed state. 6) Copula repair follows a total failure of a unit in subsystem.
7) It is assumed that a repaired system by copula works like a new system and no damage appears during repair.
8) As soon as the failed the failed unit gets repaired, it is ready to perform the task.

FORMULATION AND SOLUTION OF MATHEMATICAL MODEL
By the probability of considerations and continuity of arguments, the following set of differencedifferential equations are associated with the above mathematical model.
P8(x,t) (1) Initial condition ; ( ) = 1 and other transition probability at t=0 are zero (20) Taking Laplace transformation of equation ( . Taking t = 0, 10,…, 100, availability of the system is obtained and presented in Table 1 below  And then taking inverse Laplace transform, one may have the expression for reliability for the system. Expression for reliability of the system is given as;      The availability of the system diminishes with time and eventually stabilizes at the value. As a result, the graphical representation of the model shows that one may reliably portray the future behavior of a complex system at any moment for any given set of parametric parameters. The addition of copula increases the system's dependability substantially, as seen in Table 3 and Figure   4. As the model's graphical depiction demonstrates, any collection of parametric values may be used to forecast the future behavior of a complex system at any moment. Figure 4 of the investigation focused on the system's reliability while a fix is unavailable. When the availability and reliability numbers in Tables 2 and 3 Table 4 and Figure 5 give the system's mean-time-to-failure (MTTF) with respect to variation in failure rates, Ϙ # , Ϙ $ , Ϙ % , and Ϙ & . Color graphs (blue, green, pink, and yellow) are used to display the information. The Gumbel-Hougaard family copula is also used to evaluate the system. The study found that including copula substantially enhances the system's reliability.
The paper's analytic section includes a sensitivity analysis of the system. The fluctuation in sensitivity with variation in parameter values is shown in Table 5 and Figure 6.
Fuzzy methods will be used in the future to analyze the reliability and performance of multi-unit solar systems for small and large-scale industrial usage.