Reliability Evaluation of Multi-State Solar Energy Generating System with Inverters Considering Common Cause Failures

Abstract

To ensure the efficient functioning of solar energy generation systems, it is crucial to have dependable designs and regular maintenance. However, when these systems or their components operate at multiple working levels, optimizing reliability becomes a complex task for models and analyses. In the context of reliability modeling in solar energy generation systems, researchers often assume that random variables follow an exponential distribution (binary-state representation) as a simplification, although this may not always hold true for real-world engineering systems. In the present paper, a multi-state solar energy generating system with inverters in series configuration is investigated, in which unreliable by-pass changeover switches, common cause failures (CCFs), and multiple repairman vacations are also considered. Furthermore, the arrivals of CCFs and the repair processes of the failed system due to CCFs are governed by different Markovian arrival processes (MAPs), and the lifetimes and repair times of inverters and by-pass changeover switches and the repairman vacation time in the system have different phase-type (PH) distributions. Therefore, the behavior of the system is represented using a Markov process methodology, and reliability measures for the proposed system are derived utilizing aggregated stochastic process theory. Finally, a numerical example and a comparison analysis are presented to demonstrate the findings.

1. Introduction

Solar energy resources are frequently utilized as a sustainable source of power in contrast to conventional power plants due to their ability to convert unlimited sunlight into energy without emitting carbon dioxide or any other pollutants into the air. Therefore, solar energy generation technology has attracted great attention and investment in the past few decades [1]. The solar energy source’s power outputs heavily rely on external natural resources, specifically the level of solar irradiation, and these resources are random, thus it is a considerable challenge to propose an effective reliability assessment method for complicated solar energy generating systems.

In general, there are four different types of methods for reliability assessment of renewable energy generation systems: (i) Monte Carlo simulation, (ii) state enumeration method, (iii) universal generating function (UGF) method, and (iv) Markov model method. The application of Monte Carlo simulation in different aspects of a power network, such as generation systems, has been extensive [2], as well as transmission networks [3], distribution networks [4], etc. However, Monte Carlo simulation may be ineffective to simplify binary state system models because it has the drawbacks of unstable accuracy and long calculation time [5,6]. State enumeration techniques prove to be efficient when applied to power networks of limited size, such as those found in local distribution systems. For large power networks, state enumeration methods can lead to the problem of the curse of dimension [7,8]. Although the UGF method can model and analyze the reliability of large power networks, it cannot obtain the dynamic reliability indexes of the system over time, that is, the derived reliability indexes are static. Li and Zio [9] investigated the reliability of a distributed generating system, and the reliability indexes in a static situation were formulated by employing UGF techniques. According to the research conducted by Ding et al. [10], UGF techniques were employed for evaluating the dependability of large-scale photovoltaic systems. Other reliability analyses and studies of energy systems have been carried out by Fotopoulou et al. [11] and Murphy et al. [12]. The stochastic behavior of a complex system can be effectively captured by the Markov model, enabling the derivation of both transient and steady-state reliability measures. Nevertheless, as the number of electronic components like photovoltaic modules and inverters increases, the system’s state space expands exponentially. Vast space results in many reliability models becoming an unsolved problem, for example, binary-state representations of the working/failure processes and transient reliability indexes cannot be obtained [13]. Goel et al. [14] studied the reliability of an $n$-unit solar energy system by using Markov process theory, in which failure rates were assumed to be constant, that is, the lifetime of cells was exponentially distributed. Cheng et al. [15] conducted an analysis on the reliability of a solar energy generation system, with a focus on the configuration of inverters connected in series, in which the random variables were also assumed to be exponentially distributed. While the exponential distribution’s memoryless property simplifies analysis, its application in engineering practice for reliability modeling is significantly limited [16,17,18]. Unfortunately, the research conducted on the incorporation of multi-state and vacation policy in ensuring reliability of solar energy generating systems has been quite limited.

To avoid the disadvantage of exponential distribution, Neuts [19] proposed the PH distribution, which on the one hand maintains the easy-to-analyze characteristics of the exponential distribution and on the other hand includes more new stochastic distributions. Meanwhile, due to the partially memoryless advantage of the PH distribution, it is easy to handle in analysis and algorithms in Markov models. The set of PH distributions is widely applicable as it encompasses all distributions defined on the non-negative real line, allowing for the approximation of any such distribution. This class extends beyond the exponential distribution and incorporates a matrix algebraic structure. The matrix representation of PH distribution is convenient for computer solving. Specifically, different states of multi-state systems/multi-state components can be represented by different phases in PH distributions. The main reliability measures associated with a system can be modeled and calculated in an algorithmic form. In recent times, various disciplines including queuing theory, risk assessment, statistical analysis of signals, optimization of traffic systems, inventory management theory, and reliability analysis have frequently employed PH distributions to evaluate their operational efficiency [20,21,22,23,24,25,26,27]. Proper use of maintenance personnel is also another key factor in solar power systems [28]. In reality, the majority of small and medium-sized enterprises (SMEs) face financial constraints when it comes to hiring a full-time repair technician. As a result, these businesses often rely on a single repairman who takes on dual responsibilities—one for equipment maintenance and another for miscellaneous tasks. The repairman conducts regular system checks, leaving it unattended if everything is functioning properly; however, in case of any issues, immediate repairs are carried out once the vacation period concludes. Some additional profits are generated during the repairman’s vacation. In a study conducted by Yu et al. [29], a repair model for PH geometric process was examined, incorporating the implementation of a policy that allows multiple vacations for repairmen. Similarly, Yuan [30] investigated a repairable system with a k-out-of-n: G system and also considered the inclusion of a multiple vacation policy for repairmen in their research.

In the present paper, we study a multi-state solar energy generating system with inverters in a series configuration and a repairman, in which the inverters are equipped with changeover switches and the multiple vacation policy of the repairman is adopted. The lifetimes and maintenance times of inverters and changeover switches are represented using PH distributions. The consecutive vacation times follow distinct PH distributions. The system’s potential failure can also be attributed to adverse weather conditions and a lack of technical support, as numerous systems are deployed in high-altitude regions, arid landscapes, and remote locations. The phenomenon is called common cause failure (CCF). Because CCFs may lead to disastrous consequences in systems, the reliability analysis of repairable systems with CCFs was investigated in references [31,32,33]. A MAP is considered to model the arrival of CCFs in the system, and another MAP is dedicated to governing the completion of the repairs due to CCFs. In the system, the correlation among the interarrival times of CCFs and completed repair due to CCFs is considered for the first time. There is a Markovian dependence between the repair times due to CCFs. The entire system reliability can be computed by employing multi-dimensional Markov process and aggregated stochastic process theory because of the extensive practicality offered by PH distributions and MAPs. The reliability indexes for the proposed systems in transient and stationary regimes can be calculated easily by employing the aggregated stochastic process theory, which effectively solves the curse of dimension. The aggregated stochastic process theory was proposed by Colquhoun and Hawkes [34] to deal with ion-channel modeling theory. Thereafter, the aggregated stochastic process was successfully applied in the reliability modeling of repairable systems [35,36,37].

Furthermore, the arrivals of CCFs and the repair processes of the failed system due to CCFs are governed by different Markovian arrival processes (MAPs). MAPs are a mathematical model used to describe the change in a system’s state over time, where the probability of a system moving from one state to another depends only on the current state and not on past state history. This property makes the Markov process widely used in physics, economics, and other fields. MAPs are defined based on a collection of state spaces, where each element represents a state that the system may be in. Whereas the transition between states is described by the transition probabilities, which define the likelihood of moving from one state to another in a given time. MAPs make the dynamic behavior of a system easier to analyze and understand by simplifying complex system behavior into transitions between states. Through MAPs, the state distribution of the system at a certain point in time in the future can be predicted, which is of great significance for system planning and decision making. At the same time, this process can effectively deal with random events in the system, making the model closer to real life.

In comparison to existing studies, the main contributions of the present study are as follows: (1) The multiple vacation policy is adopted for reliability assessment of solar energy generating systems, making the human resources in the system fully utilized and reducing the system’s costs, which represents a new exploration; (2) the various states of multi-state systems/multi-state components in the field of reliability can be effectively represented by distinct phases in PH distributions. Consequently, it is more practical to govern the multi-state feature of solar energy generating systems using PH distributions, as this approach overcomes the limitations associated with binary-state representations highlighted in Cheng et al. [15]; (3) the aggregated stochastic process theory is employed to calculate exact system transient and stationary performance metrics; (4) by comparing reliability measures for systems with a repairman taking multiple vacations and the repairman without vacations, it reveals that a well-designed strategy for multiple vacations can effectively optimize the balance between system maintenance costs and operational efficiency.

The rest of the paper is organized as follows. Section 2 describes the structure of the system model and derives the infinitesimal generator of the system. Section 3 calculates the transient and static performance metrics of the system, respectively. Section 4 gives a numerical example and comparative analysis. Finally, Section 5 gives concluding remarks and some suggestions for future work. In this paper, the symbol $e_k$ represents a column vector consisting of all 1s with a size of $k$, while $I_k$ refers to the identity matrix with a size of $k$.

2. Descriptions of the Model

In this section, we present a solar energy generation system with inverters that operate in multiple states. We provide a detailed explanation of the proposed multi-state system and its corresponding generator matrix within the context of Markov processes.

2.1. Assumptions of the Model

In this subsection, we examine a solar energy system consisting of $N$ units and a repairman who is responsible for maintaining the system. The solar energy system comprises $N$ units connected in series, each equipped with an inverter and an unreliable by-pass changeover switch (refer to Figure 1). To enhance system dependability and minimize expenses, the implementation of a repairman’s multiple vacation strategy is contemplated. The details of the system are as follows.

Assumption 1. 

At the initial moment $t=0$, all $N$ inverters function concurrently, resulting in the by-pass changeover switches being in a state of inactive readiness while the repairman commences his first vacation. When a failure occurs in one of the inverters, the corresponding by-pass changeover switch is triggered to take over its functions. The system is subject to failure when one of the online by-pass changeover switches fails. Therefore, there can be at most one failed by-pass changeover switch in the system.

Assumption 2. 

Multiple vacations are adopted for the repairman [22,29,30], and the failed by-pass changeover switch has priority repair privilege.

Assumption 3. 

The restoration process aims to bring the failed by-pass changeover switches and inverters back to their initial state, effectively rejuvenating them. Consequently, these repaired components exhibit a level of performance equivalent to brand new ones. Additionally, adherence to the repair procedure strictly follows the principle of ‘first-in-first-out’.

Assumption 4. 

The lifetime of online inverters and the lifetime of online by-pass changeover switches are random variables following PH$({\alpha }_1,W_1)$ of order 1 and $\mathrm{PH}({\alpha }_2,W_2)$ of order 1, respectively. The repair times of the failed inverters and by-pass changeover switches are distributed according to PH distributions as $\mathrm{PH}({\beta }_1,S_1)$ with 1 repairing phase and $\mathrm{PH}({\beta }_2,S_2)$ with $m$ repairing phases, respectively. The vacation time is a random variable following PH$(\gamma ,T)$ of order $n$.

Assumption 5. 

The system may also be subject to failure with total components due to the cause of common cause failures. The arrival of common cause failures in the online inverters and by-pass changeover switches follows a MAP$(C_0,C_1)$ of order $l_1$, with the initial probability vector $c$ . Once common cause failures arrive in the online inverters and by-pass changeover switches, all of them will be affected. Also, the flow repair completion of common cause failures follows a MAP$(D_0,D_1)$ of order $l_2$, with the initial probability vector $d$.

Assumption 6. 

All considered times are mathematically independent.

2.2. Infinitesimal Generator

The continuous-time Markov process $\{X(t),t\ge 0\}$ represents the solar energy system with $N$ units and the presence of a repairman. The state space of $\{X(t),t\ge 0\}$ can be depicted as follows.

$$\Omega =\{S_0^V,S_1^V,S_2^V,\dots ,S_{N-1}^V,S_1^R,S_2^R,\dots ,S_{N-1}^R,S_N^V,S_N^R,S_{1F}^V,S_{2F}^V,\dots ,S_{N-1,F}^V,S_{1F}^R,S_{2F}^R,\dots ,S_{N-1,F}^R,S_{CCF}^V,S_{CCF}^R\}$$

For the purpose of facilitating subsequent discussion, we establish definitions for the phases

$$\{(k_1,k_2,v,j_1,i_1,i_2,j_2):k_1=k_2=1,1\le v\le n,1\le j_1\le l_1,i_1=1,1\le i_2\le m,1\le j_2\le l_2\},$$

in which $k_1$ and $k_2$ represent different phases related to operational time for online inverters and by-pass changeover switches. Additionally, there is a separate phase denoted as $v$ which corresponds to vacation time for repairmen. Furthermore, we have another phase represented as $j_1$ that signifies when common cause failures occur in both online inverters and by-pass changeover switches. There are two distinct phases denoted as $i_1$ and $i_2$ which correspond to repair times specifically for failed inverters and by-pass changeover switches, and $j_2$ denotes the phase of the repair time for the system due to common cause failures. Meanwhile, $n_I$ and $n_s$ denote the number of remaining operational inverters and the number of remaining operational by-pass changeover switches in the system, respectively. Each macro-state within the state space $\Omega$ is defined in the following manner.

Macro-state $S_0^V=\{(N,0;k_1,v,j_1):k_1=1,1\le v\le n,1\le j_1\le l_1\}$: The system is new and there is no failed inverter or by-pass changeover switch. Therefore, the repairman is currently on vacation while all of the by-pass changeover switches remain in a state of cold standby. The MAP associated with common cause failures in the system is operational. Therefore, $n_I=N$, $n_s=0$.

Macro-state $S_i^V=\{(N-i,i;k_1,k_2,v,j_1):k_1=k_2=1,1\le v\le n,1\le j_1\le l_1\}$, $1\le i\le N-1$: The system currently experiences $i$ failed inverters, while the repairman is on vacation. Consequently, the corresponding by-pass changeover switches seamlessly take over the duties of these malfunctioning inverters, while other by-pass changeover switches remain in a state of cold standby. The MAP associated with common cause failures in the system is operational. In this situation, $n_I=N-i$, $n_s=i$.

Macro-state $S_N^V=\{(0,0;v):1\le v\le n\}$: The system has encountered $N$ instances of inverter failure, coinciding with the repairman’s vacation period. This state indicates system failure and is waiting for repairing. The MAP associated with common cause failures in the system does not function. Therefore, $n_I=0$, $n_s=0$.

Macro-state $S_h^R=\{(N-i,i;k_1,k_2,j_1,i_1):k_1=k_2=1,1\le j_1\le l_1,i_1=1\}$, $1\le i\le N-1$: The system has a total of $i$ inverters that have experienced failures, and currently, one of the failed inverters is being repaired by the repairman following the repair schedule. The corresponding by-pass changeover switches shift to take over the work of the failed inverters and other by-pass changeover switches are maintained in a state of cold standby. The MAP associated with common cause failures in the system is operational. In this situation, $n_I=N-i$, $n_s=i$.

Macro-state $S_N^R=\{(0,0;i_1):i_1=1\}$: The system has encountered $N$ failed inverters, and a repairman is currently repairing one of the malfunctioning inverters based on the repair schedule. This state indicates system failure. The two considered MAPs do not function. Therefore, $n_I=0$, $n_s=0$.

Macro-state $S_{iF}^V=\{(0,0;v):1\le v\le n\}$, $1\le i\le N-1$: The system has encountered $i$ failed inverters, and unfortunately, one of the remaining operational by-pass changeover switches has also malfunctioned. Therefore, the system is subject to failure and is waiting for repairing because the repairman is taking his vacation. The MAP associated with common cause failures in the system does not function. In these states, $n_I=0$, $n_s=0$.

Macro-state $S_{iF}^R=\{(0,0;i_2):1\le i_2\le m\}$, $1\le i\le N-1$: The system has encountered $i$ failed inverters, and unfortunately, one of the remaining operational by-pass changeover switches has also malfunctioned. Currently, the repairman is diligently working on fixing the faulty by-pass changeover switch in accordance with the established repair schedule. The MAP associated with common cause failures in the system does not function. This state indicates system failure. Therefore, $n_I=0$, $n_s=0$.

Macro-state $S_{CCF}^V=\{(0,0;v):1\le v\le n\}$: In these states, the system is subject to failure due to the cause of common causes failure. The repairman is currently on vacation, causing a delay in the system’s maintenance. Therefore, $n_I=0$, $n_s=0$.

Macro-state $S_{CCF}^R=\{(0,0;j_2):1\le j_2\le l_2\}$: In these states, the system is subject to failure due to the cause of common cause failures. The repairman is repairing the system according to the MAP associated with flow repair completion. Therefore, $n_I=0$, $n_s=0$.

Based on the macro-states defined above, there exist $2N-1$ functional macro-states denoted as $U=\{S_0^V,S_1^V,S_2^V,\dots ,S_{N-1}^V,S_1^R,S_2^R,\dots ,S_{N-1}^R\}$, while the set of states indicating failure is referred to as $F=\{S_N^V,S_N^R,S_{1F}^V,S_{2F}^V,\dots ,S_{N-1,F}^V,S_{1F}^R,S_{2F}^R,\dots ,S_{N-1,F}^R,S_{CCF}^V,S_{CCF}^R\}$.

The matrix $\mathit{Q}$, which represents the infinitesimal generator and contains the rates of transition within state space $\Omega$, is presented below.

$$Q=\begin{array}{cc}& \begin{array}{cccccccccccccccccccccc}{S}_0^V& S_1^V& S_2^V& \cdots & S_{N-1}^V\hspace{1em} & S_1^R& S_2^R& \cdots & S_{N-2}^R& S_{N-1}^R& S_N^V& S_N^R& S_{1F}^V& S_{2F}^V& \cdots & S_{N-1,F}^V& S_{1F}^R& S_{2F}^R& \cdots & S_{N-1,F}^R& S_{CCF}^V& S_{CCF}^R\end{array}\\ \begin{array}{c}{S}_0^V\\ S_1^V\\ S_2^V\\ ⋮\\ S_{N-1}^V\\ S_1^R\\ S_2^R\\ ⋮\\ S_{N-2}^R\\ S_{N-1}^R\\ S_N^V\\ S_N^R\\ S_{1F}^V\\ S_{2F}^V\\ ⋮\\ S_{N-1,F}^V\\ S_{1F}^R\\ S_{2F}^R\\ ⋮\\ S_{N-1,F}^R\\ S_{CCF}^V\\ S_{CCF}^R\end{array}& (\begin{array}{cccccccccccccccccccccc}{A}_{00}& F_{01}& & & & & & & & & & & & & & & & & & & L_0& \\ & A_{11}& F_{12}& & & H_{11}& & & & & & & R_{11}& & & & & & & & L_1& \\ & & A_{22}& \ddots & & & H_{22}& & & & & & & R_{22}& & & & & & & L_2& \\ & & & \ddots & F_{N-2,N-1}& & & \ddots & & & & & & & \ddots & & & & & & ⋮& \\ & & & & A_{N-1,N-1}& & & & & H_{N-1,N-1}& F_{N-1,N}& & & & & R_{N-1,N-1}& & & & & L_{N-1}& \\ I_{1,0}& & & & & B_{11}& G_{12}& & & & & & & & & & O_{11}& & & & & M_1\\ & & & & & I_{2,1}& B_{22}& \ddots & & & & & & & & & & O_{22}& & & & M_2\\ & & & & & & \ddots & \ddots & G_{N-1,N-2}& & & & & & & & & & \ddots & & & ⋮\\ & & & & & & & & B_{N-2,N-2}& G_{N-2,N-1}& & & & & & & & & & & & M_{N-2}\\ & & & & & & & & I_{N-1,N-2}& B_{N-1,N-1}& & G_{N-1,N}& & & & & & & & O_{N-1,N-1}& & M_{N-1}\\ & & & & & & & & & & C_{11}& H_{NN}& & & & & & & & & & \\ & & & & & & & & & I_{N,N-1}& & C_{22}& & & & & & & & & & \\ & & & & & & & & & & & & D_{11}& & & & J_{11}& & & & & \\ & & & & & & & & & & & & & D_{22}& & & & J_{22}& & & & \\ & & & & & & & & & & & & & & \ddots & & & & \ddots & & & \\ & & & & & & & & & & & & & & & D_{N-1,N-1}& & & & J_{N-1,N-1}& & \\ & & & & & K_{11}& & & & & & & & & & & E_{11}& & & & & \\ & & & & & & K_{22}& & & & & & & & & & & E_{22}& & & & \\ & & & & & & & \ddots & & & & & & & & & & & \ddots & & & \\ & & & & & & & & & K_{N-1,N-1}& & & & & & & & & & E_{N-1,N-1}& & \\ & & & & & & & & & & & & & & & & & & & & C_{33}& M_{11}\\ M_{00}& & & & & & & & & & & & & & & & & & & & & C_{44}\end{array})\end{array},$$

(1)

where

$$A_{00}=N{W}_1\oplus T\oplus C_0+I_1\otimes T^0\gamma \otimes I_{l_1}, A_{ii}=(N-i)W_1\oplus i{W}_2\oplus T\oplus C_0, 1\le i\le N-1;$$

$$B_{ii}=(N-i)W_1\oplus i{W}_2\oplus C_0\oplus S_1, 1\le i\le N-1;$$

$$C_{11}=T, C_{22}=S_1, C_{33}=T, C_{44}=D_0;$$

$$D_{ii}=T, 1\le i\le N-1; E_{ii}=S_2, 1\le i\le N-1;$$

$$F_{0,1}=N{W}_1^0\otimes {\alpha }_2\otimes I_n\otimes I_{l_1}, F_{i,i+1}=(N-i)W_1^0\otimes I_1\otimes {\alpha }_2\otimes I_n\otimes I_{l_1}1\le i\le N-2, F_{N-1,N}=W_1^0\otimes I_n\otimes e_{l_1};$$

$$G_{i,i+1}=(N-i)W_1^0\otimes {\alpha }_2\otimes I_1\otimes I_{l_1}\otimes I_1, 1\le i\le N-2, G_{N-1,N}=W_1^0\otimes e_{l_1}\otimes I_1;$$

$$H_{ii}=I_1\otimes I_1\otimes T^0\otimes I_{l_1}\otimes {\beta }_1, 1\le i\le N-1, H_{NN}=T^0\otimes {\beta }_1;$$

$$I_{1,0}={\alpha }_1\otimes I_1\otimes e_1\otimes \gamma \otimes I_{l_1}\otimes S_1^0, I_{i,i-1}={\alpha }_1\otimes I_1\otimes I_1\otimes I_{l_1}\otimes S_1^0{\beta }_1, 2\le i\le N-1,I_{N,N-1}={\alpha }_1\otimes c\otimes S_1^0{\beta }_1;$$

$$J_{ii}=T^0\otimes {\beta }_2, 1\le i\le N-1; K_{ii}={\alpha }_1\otimes {\alpha }_2\otimes c\otimes S_2^0{\beta }_1, 1\le i\le N-1;$$

$$L_i=e_1\otimes e_1\otimes I_n\otimes C_1{e}_{l_1}, 0\le i\le N-1;$$

$$M_i=(e_1\otimes e_1\otimes C_1)e_{l_1}\otimes d, 1\le i\le N-1, M_{00}={\alpha }_1\otimes \gamma \otimes c\otimes D_1{e}_{l_2}, M_{11}=T^{0}d;$$

$$O_{ii}=e_1\otimes i{W}_2^0\otimes e_{l_1}\otimes e_1\otimes {\beta }_2, 1\le i\le N-1; R_{ii}=e_1\otimes i{W}_2^0\otimes I_2\otimes e_2, 1\le i\le N-1.$$

The infinitesimal generator matrix $\mathit{Q}$ has a size of $n(l_1+2)+(N-1)(n{l}_1+l_1+n+m)+1$, and Appendix A provides a detailed explanation of the formula for blocks in infinitesimal generative matrices.

3. System Performance Metrics

In this section, the transient and stationary metrics of the system are presented, respectively.

3.1. Transient Regime

The proposed Markov process $\{X(t),t\ge 0\}$ is constructed on the state space $\Omega$, where $P\{X(t)=a\}$ denotes the occupancy probability of macro-state $a$ at time $t$. $\left(P_{ab}(t)\right)$ is the matrix representing the transition probabilities of a continuous-time Markov process $\{X(t),t\ge 0\}$, where $P_{ab}(t)$ denotes the transition probability from macro-state $a$ to macro-state $b$ at any given time $t$, $a,b\in \Omega$. The transition probability function can be derived from the exponential function matrix as $\mathit{P}\left(t\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathit{Q}t\right)$, s.t. $\mathit{P}\left(0\right)=\mathit{I}$.

3.1.1. Transient Availability

At the start $t=0$, all PH distributions and MAPs commence by utilizing their initial probability vectors. Consequently, the system’s initial probability vector is determined by $\underset{n}{\underbrace{{\alpha }_1\otimes {\alpha }_1\otimes \cdots \otimes {\alpha }_1}}\otimes \gamma \otimes c$, and the operational macro-states are $U=\{S_0^V,S_1^V,S_2^V,\dots ,S_{N-1}^V,S_1^R,S_2^R,\dots ,S_{N-1}^R\}$. Hence, the transient availability of the system is as follows.

$$A(t)=(\underset{n}{\underbrace{{\alpha }_1\otimes {\alpha }_1\otimes \cdots \otimes {\alpha }_1}}\otimes \gamma \otimes c)[{\displaystyle \sum _{i=0}^{N-1}{P}_{S_0^V{S}_i^V}(t)e_{n{l}_1}}+{\displaystyle \sum _{i=1}^{N-1}{P}_{S_0^V{S}_i^R}(t)e_{l_1}}]$$

(2)

3.1.2. Reliability and Mean Time between Consecutive System Failures

Reliability refers to the probability of the system functioning properly throughout a given time period $[0,t]$. Let

$$Q_{UU}=\begin{array}{cc}& \begin{array}{cccccccccc}{S}_0^V& S_1^V& S_2^V& \cdots & S_{N-1}^V\hspace{1em} & S_1^R& S_2^R& \cdots & S_{N-2}^R& S_{N-1}^R\end{array}\\ \begin{array}{c}{S}_0^V\\ S_1^V\\ S_2^V\\ ⋮\\ S_{N-1}^V\\ S_1^R\\ S_2^R\\ ⋮\\ S_{N-2}^R\\ S_{N-1}^R\end{array}& \left(\begin{array}{cccccccccc}{A}_{00}& F_{01}& & & & & & & & \\ & A_{11}& F_{12}& & & H_{11}& & & & \\ & & A_{22}& \ddots & & & H_{22}& & & \\ & & & \ddots & F_{N-2,N-1}& & & \ddots & H_{N-1,N-1}& F_{N-1,N}\\ & & & & A_{N-1,N-1}& & & & & \\ I_{1,0}& & & & & B_{11}& G_{12}& & & \\ & & & & & I_{2,1}& B_{22}& \ddots & & \\ & & & & & & \ddots & \ddots & G_{N-1,N-2}& \\ & & & & & & & & B_{N-2,N-2}& G_{N-2,N-1}\\ & & & & & & & & I_{N-1,N-2}& B_{N-1,N-1}\end{array}\right).\end{array}$$

(3)

According to the well-known Colquhoun and Hawkes [34], the reliability is

$$R(t)=(\underset{n}{\underbrace{{\alpha }_1\otimes {\alpha }_1\otimes \cdots \otimes {\alpha }_1}}\otimes \gamma \otimes c,0)\exp (Q_{UU}t)e_{(Nn+N-1)l_1}$$

(4)

The periods between the consecutive system failures are random variables following PH distribution as PH$((\underset{n}{\underbrace{{\alpha }_1\otimes {\alpha }_1\otimes \cdots \otimes {\alpha }_1}}\otimes \gamma \otimes c,0),Q_{UU})$ with order $(Nn+N-1)l_1$, thus the mean time between consecutive system failures can be computed by $\mu =-(\underset{n}{\underbrace{{\alpha }_1\otimes {\alpha }_1\otimes \cdots \otimes {\alpha }_1}}\otimes \gamma \otimes c,0)Q_{UU}^{-1}{e}_{(Nn+N-1)l_1}$.

3.1.3. The Idle Probability of the Repairman

The repairman is idle when the system is in macro-states $S_0^V,S_1^V,S_2^V,\dots ,S_{N-1}^V,S_N^V,S_{1F}^V,S_{2F}^V,\dots ,S_{N-1,F}^V,S_{CCF}^V$. Hence, the idle probability of the repairman is as follows.

$$p_{Idle}(t)=(\underset{n}{\underbrace{{\alpha }_1\otimes {\alpha }_1\otimes \cdots \otimes {\alpha }_1}}\otimes \gamma \otimes c)[{\displaystyle \sum _{i=0}^{N-1}{P}_{S_0^V{S}_i^V}(t)e_{n{l}_1}}+P_{S_0^V{S}_N^V}(t)e_n+{\displaystyle \sum _{i=1}^{N-1}{P}_{S_0^V{S}_{iF}^V}(t)e_n}+P_{S_0^V{S}_{CCF}^V}(t)e_n]$$

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3.1.4. The Probability of the Repairman Comes Back to the System

The system is revisited by the repairman during the occurrence of system transitions $S_i^V\to S_i^R(1\le i\le N),S_{jF}^V\to S_{jF}^R(1\le j\le N-1),S_{CCF}^V\to S_{CCF}^R$. Thus, the probability of the repairman coming back to the system at time $t$ is

$$\begin{array}{l}{p}_{back}(t)=(\underset{n}{\underbrace{{\alpha }_1\otimes {\alpha }_1\otimes \cdots \otimes {\alpha }_1}}\otimes \gamma \otimes c)[{\displaystyle \sum _{i=1}^{N-1}{P}_{S_0^V{S}_i^V}(t)(T^0\otimes e_{l_1})}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +P_{S_0^V{S}_N^V}(t)T^0+{\displaystyle \sum _{j=1}^{N-1}{P}_{S_0^V{S}_{jF}^V}(t)T^0}+P_{S_0^V{S}_{CCF}^V}(t)T^0].\end{array}$$

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3.2. Stationary Regime

To study the behavior of the system in the long run, the performance of the system should be measured in the stationary regime. The stationary probability vector of the system is denoted as

$$\pi =({\pi }_{S_0^V},{\pi }_{S_1^V},\dots ,{\pi }_{S_{N-1}^V},{\pi }_{S_1^R},{\pi }_{S_2^R},\dots ,{\pi }_{S_{N-1}^R},{\pi }_{S_N^V},{\pi }_{S_N^R},{\pi }_{S_{1F}^V},{\pi }_{S_{2F}^V},\dots ,{\pi }_{S_{N-1,F}^V},{\pi }_{S_{1F}^R},{\pi }_{S_{2F}^R},\dots ,{\pi }_{S_{N-1,F}^R},{\pi }_{S_{CCF}^V},{\pi }_{S_{CCF}^R}),$$

where each element ${\pi }_i,i\in \Omega$ represents the probability of the system being in macro-state $i$ under stationary conditions. This vector fulfills the matrix equation $\pi Q=0$ and also satisfies the normalization condition $\pi e=1$. Vector $e$ is a column vector whose entries are 1. The presented linear system can be solved easily by employing MATLAB R2020b software when the parameters of the system are given.

3.2.1. Stationary Availability

The steady-state availability expresses the proportion of time that the system is operational. Since $U$ contains the operational macro-states of the system, the stationary availability is the summation of the occupancy probabilities of those macro-states included in the operational set $U$. It is calculated as follows.

$$A={\displaystyle \sum _{i=0}^{N-1}{\pi }_{S_i^V}{e}_{n{l}_1}}+{\displaystyle \sum _{i=1}^{N-1}{\pi }_{S_i^R}{e}_{l_1}}$$

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3.2.2. The Stationary Idle Probability of the Repairman

The repairman is considered idle when he is taking his vacations, resulting in the stationary probability of the repairman being idle being the cumulative sum of occupancy probabilities for specific macro-states $S_0^V,S_1^V,S_2^V,\dots ,S_{N-1}^V,S_N^V,S_{1F}^V,S_{2F}^V,\dots ,S_{N-1,F}^V,S_{CCF}^V$. It is calculated as follows.

$$p_{Idle}={\displaystyle \sum _{i=0}^{N-1}{\pi }_{S_i^V}{e}_{n{l}_1}}+{\pi }_{S_N^V}{e}_n+{\displaystyle \sum _{i=1}^{N-1}{\pi }_{S_{iF}^V}{e}_n}+{\pi }_{S_{CCF}^V}{e}_n$$

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3.2.3. The Stationary Probability of the Repairman Returning to the System

The probability of the repairman returning to the system in a steady state is

$$p_{back}={\displaystyle \sum _{i=1}^{N-1}{\pi }_{S_i^V}(T^0\otimes e_{l_1})}+{\pi }_{S_N^V}{T}^0+{\displaystyle \sum _{j=1}^{N-1}{\pi }_{S_{jF}^V}{T}^0}+{\pi }_{S_{CCF}^V}{T}^0.$$

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4. Numerical Examples

To facilitate the comparison of this paper with prior research, we provide several numerical illustrations. Section 4.1 elucidates the suitability of reliability performance measures employed in this study, while Section 4.2 contrasts the reliability metrics proposed herein with those presented in earlier investigation.

4.1. Performance Evaluation of the Newly Proposed System

A $5$-unit solar energy system and a repairman who is responsible for the maintenance tasks of the system are considered. The system initiates with five inverters online and five by-pass changeover switches in cold standby mode. The lifetimes of online inverters and by-pass changeover switches are modeled with $\mathrm{PH}({\alpha }_1=1,W_1=-0.9)$ and $\mathrm{PH}({\alpha }_2=1,W_2=-0.0005)$, respectively. The repair times of the inverters and by-pass changeover switches are simulated with $\mathrm{PH}({\beta }_1=1,S_1=-0.25)$ and $\mathrm{PH}({\beta }_2=(1,0),S_2=\left(\begin{array}{cc}-1.25& 0.80\\ 0.80& -1.25\end{array}\right))$, respectively.

The arrival of common cause failures in the system is governed by MAP$(C_0,C_1)$. The matrix $C_0$ represents the interarrival time, the matrix $C_1$ indicates the arrival of common cause failures, and $c$ is the initial vector. The parameters of the process are assumed as follows.

$$c=(1,0), C_0=\left(\begin{array}{cc}-1.2& 0.7\\ 0.5& -0.8\end{array}\right), C_1=\left(\begin{array}{cc}0.5& 0\\ 0& 0.3\end{array}\right).$$

The repair process is governed by MAP$(D_0,D_1)$. The matrix $D_0$ represents that the repair is ongoing, the matrix $D_1$ indicates that the repair is completed, and $d$ is the initial vector. The parameters of the process are assumed as follows.

$$d=(1,0), D_0=\left(\begin{array}{cc}-0.30& 0.01\\ 0.01& -0.18\end{array}\right), D_1=\left(\begin{array}{cc}0.29& 0.00\\ 0.00& 0.17\end{array}\right).$$

The vacation times of the repairman are modeled with $\mathrm{PH}(\gamma =(1,0),T=\left(\begin{array}{cc}-1.55& 0.95\\ 0.95& -1.55\end{array}\right))$.

A constructed table,

Table 1. Occupancy probability of each macro-state for the system.

t	${\mathit{P}}_{{\mathit{S}}_0^{\mathit{V}}}(\mathit{t})$	${\mathit{P}}_{{\mathit{S}}_1^{\mathit{V}}}(\mathit{t})$	${\mathit{P}}_{{\mathit{S}}_2^{\mathit{V}}}(\mathit{t})$	${\mathit{P}}_{{\mathit{S}}_3^{\mathit{V}}}(\mathit{t})$	${\mathit{P}}_{{\mathit{S}}_4^{\mathit{V}}}(\mathit{t})$	${\mathit{P}}_{{\mathit{S}}_1^{\mathit{R}}}(\mathit{t})$	${\mathit{P}}_{{\mathit{S}}_2^{\mathit{R}}}(\mathit{t})$	${\mathit{P}}_{{\mathit{S}}_3^{\mathit{R}}}(\mathit{t})$	${\mathit{P}}_{{\mathit{S}}_4^{\mathit{R}}}(\mathit{t})$	${\mathit{P}}_{{\mathit{S}}_5^{\mathit{V}}}(\mathit{t})$	${\mathit{P}}_{{\mathit{S}}_5^{\mathit{R}}}(\mathit{t})$	${\mathit{P}}_{{\mathit{S}}_{1\mathit{F}}^{\mathit{V}}}(\mathit{t})$
0	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
5	0.0212	0.0205	0.0197	0.0187	0.0174	0.0034	0.0088	0.0220	0.0777	0.0392	0.2946	0*
10	0.0196	0.0190	0.0184	0.0177	0.0169	0.0032	0.0084	0.0224	0.0870	0.0265	0.3505	0*
15	0.0188	0.0182	0.0176	0.0169	0.0160	0.0030	0.0080	0.0222	0.0901	0.0244	0.3723	0*
20	0.0185	0.0179	0.0172	0.0165	0.0156	0.0030	0.0079	0.0220	0.0914	0.0236	0.3812	0*
25	0.0183	0.0177	0.0171	0.0164	0.0155	0.0029	0.0078	0.0220	0.0919	0.0233	0.3849	0*
$\infty$	0.0182	0.0176	0.0170	0.0163	0.0154	0.0029	0.0078	0.0219	0.0923	0.0231	0.3875	0*
t	$P_{S_{2F}^V}(t)$	$P_{S_{3F}^V}(t)$	$P_{S_{4F}^V}(t)$	$P_{S_{1F}^R}(t)$	$P_{S_{2F}^R}(t)$	$P_{S_{3F}^R}(t)$	$P_{S_{4F}^R}(t)$	$P_{S_{CCF}^V}(t)$	$P_{S_{CCF}^R}(t)$	$A(t)$	$p_{Idle}(t)$	$p_{back}(t)$
0	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	1.0000	1.0000	0.6000
5	0*	0*	0*	0*	0*	0*	0*	0.0907	0.3649	0.2096	0.2279	0.1367
10	0*	0*	0*	0*	0*	0*	0*	0.0697	0.3397	0.2127	0.1881	0.1128
15	0*	0*	0*	0*	0*	0*	0*	0.0649	0.3267	0.2108	0.1769	0.1062
20	0*	0*	0*	0*	0*	0*	0*	0.0631	0.3213	0.2100	0.1725	0.1035
25	0*	0*	0*	0*	0*	0*	0*	0.0624	0.3190	0.2096	0.1708	0.1025
$\infty$	0*	0*	0*	0*	0*	0*	0*	0.0619	0.3174	0.2094	0.1695	0.1017

, presents the calculated expressions from Section 3. It provides the occupancy probabilities of each macro-state throughout the mission time. Probabilities denoted as 0* indicate values lower than ${10}^{-4}$. The final row displays these quantities in a stationary regime. Based on Table 1, it can be observed that these probabilities reach a stationary regime shortly after $t=35$. In this steady state, the most frequently occupied macro-state is $S_5^R$, accounting for 38.75% of the system’s time; macro-state $S_{CCF}^R$ follows closely with a probability of 0.3174, and system availability remains stable $A=0.2094$. Therefore, the repairman should improve maintenance efficiency in the considered system. The probability curve of the repairman being idle over time and the probability curve of the repairman returning to the system during the mission time are illustrated in Figure 2. It is evident that, upon reaching a steady state, there is approximately a 16.95% chance that the repairman will be on vacation, that is, $p_{Idle}=0.1695$. The steady-state probability of the repairman returning to the system is $p_{back}=0.1017$.

Since the time between two consecutive system failures follows a PH distribution, the curve of the probability density function (PDF) and cumulative distribution function (CDF) can be obtained easily (see Figure 3). Accordingly, the mean time between two consecutive system failures is $\mu =1.4573$.

4.2. Comparison between the Proposed System and Existing Model with Repairman without Vacations

The reliabilities obtained by considering the repairman with vacations and without vacations are compared in this subsection. In the existing model with a repairman without vacations, the state space of the corresponding Markov process $\{Y(t),t\ge 0\}$ can be represented as ${\Omega }^{\prime }=\{S_0,S_1^R,S_2^R,\dots ,S_{N-1}^R,S_N^R,S_{1F}^R,S_{2F}^R,\dots ,S_{N-1,F}^R,S_{CCF}^R\}$, where macro-states $S_1^R,S_2^R,\dots ,S_{N-1}^R,S_N^R,S_{1F}^R,S_{2F}^R,\dots ,S_{N-1,F}^R$ and $S_{CCF}^R$ are the same as above, while macro-state $S_0=\{(N,0;k_1,j_1):k_1=1,1\le j_1\le l_1\}$. It indicates that the system is new. There is no failed inverter or by-pass changeover switch. Thus, all by-pass changeover switches are kept in cold standby mode and the repairman is in idle. The MAP associated with common cause failures in the system is operational. Therefore, $n_I=N$, $n_s=0$. The working states set of the system is $U^{\prime }=\{S_0,S_1^R,S_2^R,\dots ,S_{N-1}^R\}$, and $F^{\prime }=\{S_N^R,S_{1F}^R,S_{2F}^R,\dots ,S_{N-1,F}^R,S_{CCF}^R\}$ is the failure states set. The infinitesimal generator matrix $Q^{\prime }$ of the Markov process $\{Y(t),t\ge 0\}$ is as follows:

$$Q^{\prime }=\begin{array}{cc}& \begin{array}{cccccccccccc}{S}_0& S_1^R& S_2^R& \cdots & S_{N-1}^R& S_N^R& S_{1F}^R& S_{2F}^R& \cdots & S_{N-2,F}^R& \hspace{1em} S_{N-1,F}^R& S_{CCF}^R\end{array}\\ \begin{array}{c}{S}_0\\ S_1^R\\ S_2^R\\ ⋮\\ S_{N-1}^R\\ S_N^R\\ S_{1F}^R\\ S_{2F}^R\\ ⋮\\ S_{N-2,F}^R\\ S_{N-1,F}^R\\ S_{CCF}^R\end{array}& \left(\begin{array}{cccccccccccc}{\mathit{A}}_{00}^{\prime }& M_{22}& & & & & & & & & & M_0\\ {\mathit{I}}_{1,0}^{\prime }& B_{11}& G_{12}& & & & O_{11}& & & & & M_1\\ & I_{2,1}& B_{22}& \ddots & & & & O_{22}& & & & M_2\\ & & I_{3,2}& \ddots & G_{N-2,N-1}& & & & \ddots & & & ⋮\\ & & & \ddots & B_{N-1,N-1}& G_{N-1,N}& & & & & O_{N-1,N-1}& M_{N-1}\\ & & & & I_{N,N-1}& C_{22}& & & & & & \\ & K_{11}& & & & & E_{11}& & & & & \\ & & K_{22}& & & & & E_{22}& & & & \\ & & & \ddots & & & & & \ddots & & & \\ & & & & & & & & & E_{N-2,N-2}& & \\ & & & & K_{N-1,N-1}& & & & & & E_{N-1,N-1}& \\ {\mathit{M}}_{00}^{\prime }& & & & & & & & & & & C_{44}\end{array}\right)\end{array}$$

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where ${\mathit{A}}_{00}^{\prime }=N{W}_1\oplus C_0$, $M_{22}=N{W}_1^0\otimes {\alpha }_2\otimes I_{l_1}\otimes {\beta }_1$, $M_0=e_1\otimes C_1{e}_2\otimes d$,${\mathit{I}}_{1,0}^{\prime }={\alpha }_1\otimes e_1\otimes I_{l_1}\otimes S_1^0$, ${\mathit{M}}_{00}^{\prime }={\alpha }_1\otimes c\otimes D_1{e}_{l_2}$, and other block matrices in $Q^{\prime }$ and the matrix $Q$ have identical corresponding block matrices, with the order of $Q^{\prime }$ being $N{l}_1+(N-1)m+l_2+1$.

According to the current parameter setting, Figure 4 and Figure 5 illustrate the availability and reliability plots for both the proposed system and the existing model with a repairman who does not take vacations. It can be observed from these figures that implementing multiple vacation periods for the repairman may result in decreased reliability compared to when he does not take any vacations. However, it is worth noting that during his vacation periods, the repairman can generate additional profits. Numerical examples and comparative analyses demonstrate that adopting a well-planned strategy for multiple vacations can effectively balance maintenance costs and operational efficiency of the system.

5. Conclusions

This study presents a solar energy generation system that utilizes inverters connected in series, incorporating unreliable by-pass changeover switches, common cause failures, and multiple vacations for repairmen. The occurrence of common cause failures and the subsequent repair processes are modeled using different Markovian arrival processes (MAPs), enhancing the ability to represent dependent events. Moreover, the durations of operation and repairs for inverters and by-pass changeover switches, as well as the vacation periods for repairmen within the system, follow PH distributions. The novelty of this paper lies in the fact that the multiple vacation strategies of repairmen are applied to the reliability evaluation of solar power generation systems, which is closer to the working mode of actual maintenance personnel. Meanwhile, PH distributions and MAPs are used to simulate the random time, and the transient steady state indexes of the system are calculated by using the aggregate stochastic process theory, which solves the problem of dimensionality disaster. The primary benefit of this research lies in the utilization of the density of PH distributions within a collection of distribution functions defined on the positive real line, enabling approximation for various general distributions. Through combining stochastic process theory and matrix analysis techniques, we provide formulations for evaluating system performance metrics. The results of the performance metrics are numerically controllable due to the advantages of the MAP and PH distributions. Finally, numerical examples are presented to demonstrate the usability of the results and a comparative analysis of the repairman’s multiple vacations strategy is presented. The aim of this paper is to improve the stability of the system and reduce the maintenance cost of the system, and the research results provide theoretical guidance and practical reference for the practical maintenance of the solar power generation system.

This study has good guiding significance for solar power generation in engineering practice, but there are also some shortcomings. Firstly, the orders of the PH distributions of lifetimes for inverters and by-pass changeover switches are one, i.e., they are exponentially distributed, which could potentially simplify the intricacies of practical situations. Therefore, the general form of PH distributions for lifetimes of inverters and by-pass changeover switches will be considered in our future work. Secondly, the degradation process of the power supply is easily affected by environmental factors, but environmental factors were not taken into account. In fact, the environment is variable, and the maintenance strategies of the repairman are different in different environments. Finally, this article selects the Markov process, but in engineering practice, a semi-Markov process is better than a Markov process. These aspects point the way for the future.