Markov Modeling and Reliability analysis of solar photovoltaic system Using Gumbel Hougaard Family Copula

The present work illustrated the reliability analysis of solar photovoltaic systems and the efficiency of medium grid-connected photovoltaic (PV) power systems with 1-out of-2 PV panels, one out of one charge controller, 1-out of 3 batteries, 1-out of 2 inverters and one out one Distributor. The units that comprise the solar were studied. Gumbel Hougaard Family Copula method was used to evaluate the performances of solar photovoltaics. Other reliability metrics were investigated, including availability, mean time to failure, and sensitivity analysis. The numerical result was generated using the Maple 13 software. The numerical results were presented in tables, with graphs to go along with them. Failure rates and their effects on various solar photovoltaic subsystems were investigated. Numerical examples are provided to demonstrate the obtained results and to assess the influence of various system characteristics. The current research could aid companies, and their repairers overcome some issues that specific manufacturing and industrial systems repairers face.


Introduction
The increase in linked PV's percentage growth can be attributed to various factors. Examples include low installation costs, quick energy, investment payback, and consumer stimulation. In this case, continuous output energy production must be demonstrated to satisfy the cost-benefit analysis of PV systems. As a result of the rapid expansion of PV system capacity on global grid systems, PV system technology is maturing and becoming more competitive in the power market. As a result, PV system engineers will prioritize PV operations in terms of reliability, efficiency, maintenance, and fault management. The sun's energy is one of the most ancient and cost-effective primary energy sources and has long been used for preservation and fabric drying. Agricultural commodities are dried, which is still done in most impoverished countries today (Solar energy as thermal). System Reliability is a metric that assesses how well a system performs under adverse conditions. Most complex systems are composed of components and subsystems linked in series, parallel, standby, or a combination of these, according to the specifications. In social, political, commercial, and technological settings, dependability terms express faith/trust in a person, firm, or piece of equipment. An analysis of a solar system can assist users in making timely decisions to ensure the system's optimal performance. The subject of dependability theory evolved as a result of operational research in the context of military studies. The terms "reliable" and "reliability" have been used interchangeably since antiquity. In reality, they are frequently used in the social, political, economic, and practical sectors to demonstrate the efficacy of a person or a piece of mechanical equipment. The word "reliability" was given a mathematical structure later that year, in 1950, in conjunction with its scientific use for military goals. Dependability theory was developed in the Western world due to its importance. The history of India's dependability technology development will be informative and exciting for academics. Almost every problem we encounter daily is influenced by dependability theory, either directly or indirectly. Power, transportation, medical services, steel, and communication networks are just a few examples of systems whose resiliency directly impacts society. System failures can occur in any discipline, according to modern engineering history.

Literature Review
We studied the following material to better understand modeling, photovoltaics, and the Gumbel Haugaard Family copula. The Gumbel-Hougaard Family Copula was used to analyze the reliability and performance of a series-parallel system by Maihulla et al. [1]. Maihulla and Yusuf [2] studied Performance Analysis of Photovoltaic Systems Using (RAMD) Analysis. Goyal et al. [3] studied the Reliability, maintainability, and sensitivity analysis of the physical processing unit of the sewage treatment plant. The Reliability assessment in electrical power systems: the Weibull-Markov stochastic model was studied by Casteren et al. [4], Ogaji et al. [5] investigated the Reliability of the Afam electric power generating station, Nigeria. Ebelin [6] studied the introduction to reliability and maintainability engineering. Gupta and Tewari [7] established the Simulation modeling and analysis of the complex system of the thermal power plant. Tsarouhas et al. [8] Studied Reliability and maintainability analysis of strudel production line with experimental data. Carazas and Souza [9] studied the Availability analysis of gas turbines used in power plants. Lado and Singh [10] studied the Cost assessment of a complex repairable system consisting of two subsystems in the series configuration using the Gumbel Hougaard family copula. Singh et al. [11] studied the Performance analysis of a complex repairable system with two subsystems in a series configuration with an imperfect switch. Yusuf et al. [12] studied the Performance Analysis of Multi-computer System Consisting of Three Subsystems in Series Configuration Using Copula Repair Policy. Raghav [13] studied the Reliability Prediction of Distributed System with Homogeneity in Software and Server using Joint Probability Distribution via Copula Approach. The Gumbel-Hougaard family copula was investigated for reliability modeling and performance evaluation of solar photovoltaic systems by Maihulla et al. [14]. Reliability and Performance Analysis of Two Unit Active Parallel System Attended by Two Repairable Machines was studied by Yusuf et al. [15]. The Reliability, availability, maintainability, and dependability analysis of photovoltaic systems was studied by Maihulla and Yusuf [16]. Gumbel-Hougaard Family Copula Reliability Analysis of Multi-Workstation Computer Network Set Up as a Series-Parallel System [17].
From all the above literature, Markov modeling for the reliability analysis of the solar photovoltaic system was not addressed. Also, sensitivity analysis regarding the study of repairable solar Photovoltaic was very little or non in the existing literature.

ASSUMPTIONS
Throughout the model's explanation, the following assumptions are made: 1. At first, all subsystems are in good functioning order. 2. For the system to be operational, two units from subsystems 3 and one from subsystems 1, 2, 4, and 5 must be used consecutively. 3. If one of the units in subsystems 1 and 4 fails, the system will be rendered reduced capacity. 4. The system will be rendered inoperable if all two units from subsystems 1, 3, and 5 fail. 5. A system's failing unit can be fixed when it is in a reduced capacity or failed state. Copula maintenance is required once a unit in a subsystem fails completely. A copula-repaired system is believed to operate like a new system, and no damage occurs during the repair. 6. Once the faulty unit has been fixed, it is ready to execute the task. The number of respective states in the state transition diagram in figure 2 below was illustrated in table 1 above. P 0 : Denote the initial state where the system is working perfectly. P 1 : Denote state with an incomplete failure in subsystem-1 due to failure of first unit and repair machine is busy repairing the failed unit. P 2 : Denote state with a complete failure in subsystem-1 due to failure of the second unit, and Copula repair is busy repairing the failed unit. P 3 : Denote state with a complete failure in subsystem-2 due to failure of the only unit in the subsystem. P 4 : Denote state with a degraded state in subsystem-3 due to failure of the first unit. P 5 : Denote state with an incomplete failure in subsystem 3. Previously first has failed. P 6 : Denote state with a complete failure in subsystem 3. This is due to the failure of the first and second units from the subsystem. The Copula repair is employed for automatic repair of the completely failed unit. P 7 : Denote the incomplete state of the system due to the failure of the first unit from subsystem 4. The repair machine is busy repairing the failed component. P 8 : Denote the complete state of the system due to the failure of the second unit from subsystem 4. The Copula repair is employed for automatic repair of the completely failed unit. P 9 : Denote an incomplete failure state of the system. This is due to the failure of the first units from subsystems 1 and 3. The repair machine is automatically busy repairing the failed component. P 10 : Denote an incomplete failure state of the system. This is due to the failure of the first and second units from subsystems-1 and the first unit from subsystem 3. The repair machine is automatically busy repairing the failed component. P 11 : Denote an incomplete failure state of the system. This is due to the failure of the first units from subsystems 3 and 4. The repair machine is automatically busy repairing the failed component. P 12 : Denote an incomplete failure state of the system. This is due to the failure of the first and second units from subsystems-3 and the first unit from subsystem 4. The repair machine is automatically busy repairing the failed component. P 13 : Denote an incomplete failure state of the system. This is due to the failure of the first units from subsystems 4 and 1. The repair machine is automatically busy repairing the failed component.
Boundary condition By taking the Laplace transform of (1) to (27), we've Substituting the Laplace transformation boundary condition in (43) to (55) into (56) to (69) we obtain the solution of the partial differential equations from (1) to (14) ) ( ) ( It is clear that; Which is the sum of all operational states of the system is therefore   And all the repair rates are set to be equal to 1.

Formulation and Analysis of System Availability
And applying the inverse Laplace transform to (62), the expression for system availability is Taking t = 0, 10,…,100, the availability of the system is obtained and presented in Table 2 and figure 3 below.

Formulation and Analysis of Reliability
Letting all repair rates, = = = = 0 in equation (88), Taking the failure rate values and applying the inverse Laplace transformation, the expression is reliability relation.

Discussion and conclusion
The simulation in Figure 3 shows that as time passes, availability decreases. When the time is less than 60 days, the chart clearly shows that the system's availability is higher. Figure 4 depicts the system's reliability over time in the same way. The graph shows that reliability decreases   50 , 0  t as time t goes from 0 to 100. On the other hand, the time interval has a higher level of trustworthiness. Table 2 and 3 and Figures 3 and 4 respectively show how more units are on standby, perfect repair in the event of an incomplete failure, replacing the affected subsystem with a new one in the event of a complete failure, regular inspection, and preventive maintenance, employing more repair machines. Other measures can improve the system's availability and reliability. Table 5 and corresponding Figure 6 depict a simulation of mean time to failure vs. failure rate . The graph shows that as grows, the MTTF decreases. The MTTF decreases as increases, resulting in a decrease in the system's longevity. To improve the system's MTTF and longevity, fault-tolerant components should be used.
  2 0.01, 0.02, 0.03, 0.04, 0.05 K  Figure 6 depicts the relationship between profit and time t. For any value of K2, the predicted profit decreases with increasing time, as shown in the graph. However, as the value decreases, the predicted profit rises. The expected profit can be increased by implementing the replacement mentioned above and redundancy suggestions. Table 5 and the corresponding figure 7 show the sensitivity analysis results in terms of failure rate.

Conclusion
Due to a lack of data on PV systems, the current study developed a reliability modeling technique to assess the PV system's overall strength, efficiency, and performance. The reliability, availability, MTTF, and profit function of this paper can all be evaluated. We present a novel solar system model with four subsystems: panel, inverter, battery bank, and control charger in this paper.
According to the paper's findings, reliability modeling can be used to assess a PV system's strength, efficiency, and performance. Once the PV system's strength, efficiency, and performance are determined, users can serve the cost of kerosene, gasoline, diesel, and other fuels that expose human hearths to air and land pollution for their household and commercial uses. As a result, the model's graphical representation demonstrates that for any given set of parametric parameters, the future behavior of a complex system can be confidently predicted at any time.
Reducing carbon dioxide emissions from conventional power generation can be achieved by adopting solar energy. Additionally, enterprises struggle with machine failure, slowing down technological growth globally due to power fluctuations.