1 Introduction

Solar-photovoltaic (PV) systems have emerged as one of the world's most commonly contributed clean energy sources. At the end of 2016, they had contributed nearly 303 GW of deployed power globally. The accelerated growth rate of these networks has piqued the appetite of investors, operators, and stakeholders for financial investment, which can be jeopardized by unintended disruptions leading to prolonged downtime times. As a result, more concentrated efforts are expected to ensure that a PV device produces the projected amount of energy. To ensure a reliable forecast of photovoltaic energy generation, reliability, availability, maintainability, and dependability (RAMD) evaluation is conducted for grid-connected solar-PV system planning by Balcioglu et al. (2017) and Baschel et al. (2018). The frequency of system failures in grid-connected PV systems can influence the operation of other interconnected systems, which is why such systems should undergo a detailed reliability reviewed by Kececioglu et al. (1991). The reliability analysis also enables Crow LH to create the most effective maintenance schedule for both the customer and the machine (2004), as well as Crow (2003)

The examined integrated PV system is described as a PV application that hires a customer for a particular purpose (see Fig. 1). Chapman–Kolmogorov differential equations for each of the sub-systems have been derived using the Markov birth–death approach for mathematical simulation of the PV system. Table 1 indicates that both failure and repair rates for each subsystem are considered to be constant. For each subsystem, a state transition diagram has been developed. In each subsystem, unit performance metrics, such as maintainability, availability, durability, mean time to failure (MTTF), mean time to repair (MTTR), and reliability ratio, were obtained by solving corresponding Chapman–Kolmogorov differential equations in steady state using normalizing conditions. The simulation of solar cell operation on an accelerated test basis (under the influence of three crucial meteorological parameters: solar irradiance, temperature, and humidity) permitted the usual running cycle under normal/stress conditions to be demonstrated, determining the initial depreciation time of stressed solar cells. The Weibull most probable statistical repartition was also detected using proper solar cell testing; on this basis, the operating period at which a significant concentration of defects is reached was determined. Gupta et al. (2007) measured the dependability and availability of a plastic pipe manufacturing facility in which all modules were configured in sequence. Using dynamic block diagrams, Distefano and Puliafito (2009) assessed the availability and reliability of dynamic-based architectures (RBDs). FMEA analyses were conducted by Sharma et al. (2011) to increase the reliability of an aircraft engine's rotor support system. Carazas et al. (2011) investigated the availability of a heat recovery steam generator used in thermal power plants. They used failure mode analysis and effect analysis for each variable (FMEA). Upadhya and Srinivasan (2012) used simulation to predict the availability of military systems. Adhikary et al. (2012) conducted a RAM analysis on a coal-fired thermal power plant and calculated a preventive maintenance cycle dependent on efficiency. RAMD technique was recently used by Saini et al. (2020) and Goyal et al. (2019) to identify the most susceptible components of serial processes, such as sugar evaporation systems and water treatment plants. The durability and availability of many industrial systems are investigated by Saini and Kumar (2020), Saini et al. (2020), and Goyal et al. (2019). Craciunescu et al. (2017) study the Optimized management for photovoltaic applications based on LEDs by fuzzy logic control and maximum power point tracking. Nearly zero energy communities. The study regarding the Long-term performance degradation of various kinds of photovoltaic modules under moderate climatic conditions was carried out by Ishii et al. (2011).

Fig. 1
figure 1

The pictorial representation of components of PV system

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Table 1 Failure and repair rates of component of the photovoltaic system
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Dependability is a measure of a system's availability, reliability, maintainability, and, in some cases, other characteristics, such as durability, safety, and security, in systems engineering. Dependability in real-time computing refers to the ability to provide services that can be relied on within a given time frame. Even if the system is subject to attacks or natural failures, the service guarantees must be maintained.

Due to the lack of PV system data, the current paper introduced a reliability modeling approach based on RAMD to study the overall performance of the PV system. In this paper, we present a new photovoltaic system model composed of four subsystems: panel, inverter, battery bank, and control charger.

The objectives of the present study are threefold: First is to develop expressions of reliability, availability, maintainability and dependability so as to predict some reliability metrics of testing the system strength and effectiveness based on the based on Markovian technique and the equations of the subsystems are derived using the Chapman–Kolmogorov method. The second to capture the effect of failure and repair rates on reliability, availability, maintainability and dependability. Third is to determine the most critical subsystem for proper maintenance action to avoid failure. The RAMD technique establishes a relationship between the system's reliability and the reliability of the individual subsystems. According to the analysis presented in this study, subsystem 4 is the most critical. This clearly shows that, of the subsystems in this study, subsystem 4 requires more attention in terms of maintenance to avoid failure.

The paper is organized as follows: Sect. 2 provides the assumptions and description of the system. Section 3 deals with the formulation of the RAMD indices. Discussion of the results obtained and concluding remark are provided on Sect. 4.

2 System description and assumptions

2.1 System description

In this section, a brief description of PV system plant and its subsystem has been given. PV system mainly consists of four components, namely PV modules, controller, batteries, and inverter. All components are arranged in series configuration. The pictorial representation of components is appended in Fig. 1 below.

2.1.1 Subsystem A (solar module)

There are five units of solar panel which are connected to the following unit in series. Out of which three are consecutively in operation. Failure of two units in this subsystem leads to system failure.

2.1.2 Subsystem B (charge controller)

There is one unit of charge controller which is connected to the following unit in series. Failure of this unit leads to system failure.

2.1.3 Subsystem C (battery)

This subsystem has one unit of battery which is connected in series to other subsystem. This unit's failure causes complete system failure.

2.1.4 Subsystem D (inverter)

It consists of one unit of inverter. This unit's failure causes complete system failure as it is connected to the following unit in series.

2.2 Assumptions

  1. 1.

    Each subsystem's failure and repair rates are distributed exponentially.

  2. 2.

    The Failure and repair rates are statistically independent of one another.

  3. 3.

    There are no concurrent faults in the subsystem.

  4. 4.

    There are plenty of maintenance and replacement options. Repairmen are still present in the factory, and the restored machine performs as well as new.

  5. 5.

    The standby subsystem switchover systems are flawless.

The RAMD indices for subsystems of PV system plant are computed as.

3 RAMD indices for solar components

3.1 RAMD indices for solar panels S1

This subsystem has two units only. Failure of the two units leads to complete system failure. The transition diagram and Chapman–Kolmogorov differential equations associated with it is are given as (Fig. 2):

$$\frac{d}{dt}P_{0} \left( t \right) = - 3\lambda_{1} P_{0} + \xi_{1} P_{1}$$
(1)
$$\frac{d}{dt}P_{1} \left( t \right) = - \left( {2\lambda_{1} + \xi_{1} } \right)P_{1} + 3\lambda_{1} P_{0} + \xi_{1} P_{2}$$
(2)
$$\frac{d}{dt}P_{2} \left( t \right) = - \left( {\lambda_{1} + \xi_{1} } \right)P_{2} + 2\lambda_{1} P_{1} + \xi_{1} P_{3}$$
(3)
$$\frac{d}{dt}P_{2} \left( t \right) = - \xi_{1} P_{2} + \lambda_{1} P_{1}$$
(4)
Fig. 2
figure 2

Transition diagram of PV panels

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Under steady state, Eq. (1–4) reduces to

$$P_{1} = \frac{{\lambda_{1} }}{{ \xi_{1} }}P_{0}$$
(5)

Substituting (5) into (2)

$$P_{2} = \frac{{\lambda_{1}^{2} }}{{\xi_{1}^{2} }}P_{0}$$
(6)

Substituting (5) into (4)

$$P_{3} = \frac{{\lambda_{1}^{3} }}{{\xi_{1}^{3} }} P_{0}$$
(7)

Using normalization condition

\(P_{0} + P_{1} + P_{2} + P_{3} = 1\).

Substituting (5) and (6) into (7), we have

$$P_{0} + = \frac{{\lambda_{1} }}{{ \xi_{1} }}P_{0} + \frac{{\lambda_{1}^{2} }}{{\xi_{1}^{2} }}P_{0} + \frac{{\lambda_{1}^{3} }}{{\xi_{1}^{3} }} P_{0} = 1$$
$$P_{0} = \frac{{\xi_{1}^{3} }}{{\xi_{1}^{3} + 6\lambda_{1}^{3} + 3\xi_{1}^{2} \lambda_{1} + 6 \xi_{1} \lambda_{1}^{2} }}$$
(8)

Table 4 contains important device output metrics that have been extracted.

3.2 RAMD indices for charge controller S2

This subsystem only has one unit. Its failure causes the whole system to crash. The transformation diagram and Chapman–Kolmogorov differential equations that go with it are as follows (Fig. 3):

$$\frac{d}{dt}P_{0} \left( t \right) = - \lambda_{2} P_{0} + \xi_{2} P_{1}$$
(9)
$$\frac{d}{dt}P_{1} \left( t \right) = - \xi_{2} P_{1} + \lambda_{2} P_{0}$$
(10)
Fig. 3
figure 3

Transition diagram of charge controller

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Under steady state, Eq. (9) and (10) reduce to

$$- \lambda_{2} P_{0} + \xi_{2} P_{1} = - \xi_{2} P_{1} + \lambda_{2} P_{0}$$
(11)
$$2\xi_{2} P_{1} = 2\lambda_{2} P_{0}$$
$$P_{1} = \frac{{\lambda_{2} }}{{ \xi_{2} }} P_{0}$$
(12)

Now using normalization condition

$$P_{0} + P_{1} = 1$$
(13)

Substituting (12) into (13), we have

$$P_{0} + \frac{{\lambda_{2} }}{{ \xi_{2} }} P_{0} = 1$$
(14)
$$P_{0} = \frac{{ \xi_{2} }}{{ \xi_{2} + \lambda_{2} }}$$
(15)

Table 4 summarizes key system performance metrics that have been derived.

3.3 RAMD indices for battery S3

There is only one device in this subsystem. When it fails, the whole machine fails. The corresponding transformation diagram and Chapman–Kolmogorov differential equations are as follows (Fig. 4):

$$\frac{d}{dt}P_{0} \left( t \right) = - \lambda_{3} P_{0} + \xi_{3} P_{1}$$
(16)
$$\frac{d}{dt}P_{1} \left( t \right) = - \xi_{3} P_{1} + \lambda_{3} P_{0}$$
(17)
Fig. 4
figure 4

Transition diagram of battery

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Under steady state, Eq. (16) and (17) reduce to

$$- \lambda_{3} P_{0} + \xi_{3} P_{1} = - \xi_{3} P_{1} + \lambda_{3} P_{0}$$
(18)
$$2\xi_{3} P_{1} = 2\lambda_{3} P_{0}$$
$$P_{1} = \frac{{\lambda_{3} }}{{ \xi_{3} }} P_{0}$$
(19)

Now using normalization condition

$$P_{0} + P_{1} = 1$$
(20)

Substituting (12) into (13), we have

$$P_{0} + \frac{{\lambda_{3} }}{{ \xi_{3} }} P_{0} = 1$$
(21)
$$P_{0} = \frac{{ \xi_{3} }}{{ \xi_{3} + \lambda_{3} }}$$
(22)

Table 4 includes essential computer performance metrics that have been extracted.

3.4 RAMD indices for inverter S4

This subsystem contains only one unit. If it crashes, the whole system fails. The following are the related integration diagram and Chapman–Kolmogorov differential equations (Fig. 5):

$$\frac{d}{dt}P_{0} \left( t \right) = - \lambda_{4} P_{0} + \xi_{4} P_{1}$$
(23)
$$\frac{d}{dt}P_{1} \left( t \right) = - \xi_{4} P_{1} + \lambda_{4} P_{0}$$
(24)
Fig. 5
figure 5

Transition diagram of inverter

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Under steady state, Eq. (23) and (24) reduce to

$$- \lambda_{4} P_{0} + \xi_{4} P_{1} = - \xi_{4} P_{1} + \lambda_{4} P_{0}$$
(25)
$$2\xi_{4} P_{1} = 2\lambda_{4} P_{0}$$
$$P_{1} = \frac{{\lambda_{4} }}{{ \xi_{4} }} P_{0}$$
(26)

Now using normalization condition

$$P_{0} + P_{1} = 1$$
(27)

Substituting (12) into (13), we have

$$P_{0} + \frac{{\lambda_{4} }}{{ \xi_{4} }} P_{0} = 1$$
(28)
$$P_{0} = \frac{{ \xi_{4} }}{{ \xi_{4} + \lambda_{4} }}$$
(29)

Using the following equations

$$R\left( t \right) = \mathop \int \limits_{t}^{\infty } f\left( x \right)dx\quad {\text{Reliability function}}$$
(30)
$${\text{Availability}} = \frac{{\text{Life time }}}{{\text{Total time }}} = \frac{{\text{Life time }}}{{{\text{Life time}} + {\text{Repair time}} }} = \frac{{{\text{MTTF}}}}{{{\text{MTTF}} + {\text{MTTR}}}}$$
(31)
$$M\left( t \right) = 1 - e^{{\left( { - \frac{ - t}{{{\text{MTTR}}}}} \right)}} \quad {\text{Maintainability function}}$$
(32)
$${\text{MTBF}} = \mathop \int \limits_{0}^{\infty } R\left( t \right)dt = \mathop \int \limits_{0}^{\infty } e^{ - \theta t} = \frac{1}{\theta }\quad {\text{Mean time between failure}}$$
(33)
$${\text{MTTR}} = \frac{1}{\xi }\quad {\text{Mean time to repair}}$$
(34)

\(\xi = {\text{repair rate}}\), λ = failure rate

$$d = \frac{\xi }{\lambda } = \frac{{{\text{MTBF}}}}{{{\text{MTTR}}}}$$
$$D_{\min } = 1 - \left( {\frac{1}{d - 1}} \right)\left( {e^{{ - \frac{\ln d}{{d - 1}}}} - e^{{ - \frac{d \ln d}{{d - 1}}}} } \right)$$

System reliability

$$R_{{{\text{sys}}}} \left( t \right) = R_{s1} \left( t \right) \times R_{s2} \left( t \right) \times R_{s3} \left( t \right) \times R_{s4} \left( t \right) = e^{{ - (\lambda_{1} + \lambda_{2} + \lambda_{3} + \lambda_{4} )t}}$$
(35)

System availability

$$A_{{{\text{sys}}}} = A_{s1} \times A_{s2} \times A_{s3} \times A_{s4}$$
(36)

System maintainability

$$M_{{{\text{sys}}}} \left( t \right) = M_{s1} \left( t \right) \times M_{s2} \left( t \right)M_{s3} \left( t \right) \times M_{s4} \left( t \right)$$
(37)
$$= \left( {1 - e^{{ - \xi_{1} \left( t \right)}} } \right) \times \left( {1 - e^{{ - \xi_{2} \left( t \right)}} } \right) \times \left( {1 - e^{{ - \xi_{3} \left( t \right)}} } \right) \times \left( {1 - e^{{ - \xi_{4} \left( t \right)}} } \right)$$
(38)
$$1 - e^{{ - (\xi_{1} + \xi_{2} + \xi_{3} + \xi_{4} )t}}$$
(39)

System dependability

$$D_{{\min \left( {sys} \right)}} = D_{{\min \left( {s1} \right)}} \times D_{{\min \left( {s2} \right)}} \times D_{{\min \left( {s3} \right)}} \times D_{{\min \left( {s4} \right)}}$$
(40)

System availability

$$A_{sys} = A_{s1} \times A_{s2} \times A_{s3} \times A_{s4}$$
$$A_{{{\text{sys}}}} = \left( {\frac{{\xi_{1}^{3} }}{{\xi_{1}^{3} + 6\lambda_{1}^{3} + 3\xi_{1}^{2} \lambda_{1} + 6 \xi_{1} \lambda_{1}^{2} }}} \right) \times \left( {\frac{{ \xi_{2} }}{{ \xi_{2} + \lambda_{2} }}} \right) \times \left( {\frac{{ \xi_{3} }}{{ \xi_{3} + \lambda_{3} }} } \right) \times \left( {\frac{{ \xi_{4} }}{{ \xi_{4} + \lambda_{4} }}} \right)$$
$$\begin{aligned} A_{{{\text{sys}}}} & = \left( {\frac{{\left( {0.5} \right)^{3} }}{{\left( {0.5} \right)^{3 } + 6\left( {0.001} \right)^{3} + 3\left( {0.5} \right)^{2} \left( {0.001} \right) + 6\left( {0.5} \right) \times \left( {0.001} \right)^{2} }}} \right) \\ & \times \left( {\frac{0.7}{{0.7 + 0.002}}} \right) \times \left( {\frac{0.9}{{0.9 + 0.003}}} \right) \times \left( {\frac{1.1}{{1.1 + 0.004}}} \right) \\ \end{aligned}$$
$$A_{{{\text{sys}}}} = 0.99401 \times 0.99715 \times 0.99668 \times 0.99638$$
$$A_{sys} = 0.{98431}$$

System maintainability

$$M_{sys} \left( t \right) = M_{s1} \left( t \right) \times M_{s2} \left( t \right) \times M_{s3} \left( t \right) \times M_{s4} \left( t \right)$$
$$\left( {1 - e^{{ - \xi_{1} \left( t \right)}} } \right) \times \left( {1 - e^{{ - \xi_{2} \left( t \right)}} } \right) \times \left( {1 - e^{{ - \xi_{3} \left( t \right)}} } \right) \times \left( {1 - e^{{ - \xi_{4} \left( t \right)}} } \right)$$

System dependability

Reliability of series–parallel system is given by

$$R_{{{\text{System}}}} = R_{{{\text{subsystem}}\;{1}}} \cdot \,R_{{{\text{subsystem}}\,{2}}} \cdot R_{{{\text{subsystem}}\,{3}}} \cdot R_{{{\text{subsystem}}\,{4}}}$$

Sequel to the above definition, Dependability of the system is

$$D_{{{\text{System}}}} = D_{{{\text{subsystem}}\,{1}}} \cdot D_{{{\text{subsystem}}\,{2}}} \cdot D_{{{\text{subsystem}}\,{3}}} \cdot D_{{{\text{subsystem}}\,{4}}}$$

See for instance, Saini and Kumar (2019), Saini et al. (2020) and Adhikary et al. (2012)

$$D_{{\min \left( {sys} \right)}} = D_{{\min \left( {s1} \right)}} \times D_{{\min \left( {s2} \right)}} \times D_{{\min \left( {s3} \right)}} \times D_{{\min \left( {s4} \right)}}$$
$$D_{\min } = 1 - \left( {\frac{1}{d - 1}} \right)\left( {e^{{ - \frac{\ln d}{{d - 1}}}} - e^{{ - \frac{d \ln d}{{d - 1}}}} } \right)$$
$$d = \frac{\xi }{\lambda } = \frac{{{\text{MTBF}}}}{{{\text{MTTR}}}}$$
$$d_{1} = \frac{{\xi_{1} }}{{\lambda_{1} }} = \frac{0.5}{{0.001}} = 500$$
$$d_{2} = \frac{{\xi_{2} }}{{\lambda_{2} }} = \frac{0.7}{{0.002}} = 350$$
$$d_{3} = \frac{{\xi_{3} }}{{\lambda_{3} }} = \frac{0.9}{{0.003}} = 300$$
$$d_{4} = \frac{{\xi_{4} }}{{\lambda_{4} }} = \frac{1.1}{{0.004}} = 275$$
$$D_{{\min \left( {s1} \right)}} = 1 - \left( {\frac{1}{500 - 1}} \right)\left( {e^{{ - \frac{\ln 500}{{500 - 1}}}} - e^{{ - \frac{500 \ln 500}{{500 - 1}}}} } \right)$$
$$D_{{\min \left( {s1} \right)}} = 1 - 0.00198$$
$$D_{{\min \left( {s1} \right)}} = 0.99802$$
$$D_{{\min \left( {s2} \right)}} = 1 - \left( {\frac{1}{350 - 1}} \right)\left( {e^{{ - \frac{\ln 350}{{350 - 1}}}} - e^{{ - \frac{350 \ln 350}{{350 - 1}}}} } \right)$$
$$D_{{{\text{min}}\left( {s2} \right)}} = 0.99719$$
$$D_{{{\text{min}}\left( {s3} \right)}} = 1 - \left( {\frac{1}{300 - 1}} \right)\left( {e^{{ - \frac{ln300}{{300 - 1}}}} - e^{{ - \frac{300ln 300}{{300 - 1}}}} } \right){ }$$
$$D_{{{\text{min}}\left( {s3} \right)}} = 0.99673$$
$$D_{{{\text{min}}\left( {s4} \right)}} = 1 - \left( {\frac{1}{275 - 1}} \right)\left( {e^{{ - \frac{ln275}{{275 - 1}}}} - e^{{ - \frac{275 ln 275}{{275 - 1}}}} } \right){ }$$
$$D_{{{\text{min}}\left( {s4} \right)}} = 0.98843$$
$$D_{{\min \left( {sys} \right)}} = 0.99802 \times 0.99719 \times 0.99673 \times 0.98843$$
$$D_{{\min \left( {sys} \right)}} = 0.98048$$

4 Discussion and conclusion

To obtain the reliability metrics of the different subsystems and method, an empirical analysis for a case was carried out by assigning numerical values to various parameters as shown in Table 1. Tables 2 and 3 show the results for multiple subsystems' stability and repair habits, respectively. The other RAMD steps are summarized in Table 4. According to the numerical review in Table 2, after 80 months of service, the PV system plant's reliability remains 0.44933, although inverter reliability is very poor (0.72615) across all subsystems and demands special attention. As a result, system builders must devise a maintenance strategy for it. Tables 5, 6, 7, 8 demonstrated the improvements in the reliability behavior of various subsystems with respect to time and differing failure rates. From this analysis, it is concluded that subsystem S4, i.e., inverters are most important and highly sensitive components and need particular attention to improve the reliability of the PV system plant.

Table 2 Variation of reliability of subsystems with respect to time
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Table 3 Variation of maintainability of subsystems with respect to time
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Table 4 RAMD indices for the photovoltaic system
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Table 5 Variation in reliability of system due to variation in failure rate of subsystem 1 (solar panels)
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Table 6 Variation in reliability of system due to variation in failure rate of subsystem 2 (charge controller)
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Table 7 Variation in reliability of system due to variation in failure rate of subsystem 3 (battery)
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Table 8 Variation in reliability of system due to variation in failure rate of subsystem 4 (inverter)
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