The components of the photovoltaic system are decomposed into different association sets (subsystems) according to their structural relevance, and maintenance decisions are made in the unit of association sets. Concerning the topology of the photovoltaic power generation system network, the association set includes photovoltaic arrays, combiner boxes, inverters, transformers, and other equipment.

Figure 1 shows a schematic diagram of an association set comprising a power generation unit in a photovoltaic power generation system. Among them, T₃ is the transformer, I is the inverter, H is the DC combiner box, G is the photovoltaic array, and Ω i is different association sets. As can be seen from the figure, the power generation unit is a series–parallel hybrid system consisting of multiple subsystems connected in parallel and then connected in series with T₃.

In this study, the decision-making experiment analysis method (DEMATEL) is combined with the interpretation structure model method (ISM) to describe the failure propagation process between the components of the photovoltaic power generation system. Specific ideas are shown in Figure 2. The specific steps are as follows:

The equipment in the photovoltaic power generation system age during use, and there are also sudden failures during actual operation. Such failures are usually caused by external random factors that affect the results of maintenance decisions.

According to the IEEE Standard Guide, the operating status of power equipment is divided into normal, attention, abnormal, and serious status (IEEE, 2008). In the traditional status transition model, the serious status is generally treated as failure. In order to comprehensively consider the degraded faults and sudden faults, this study distinguishes the serious status from the faulty status and establishes an improved Markov status transition model to describe the fault law of the equipment.

Figure 5 shows the process of equipment state transition. In the figure, 0–3 indicate normal, attention, abnormal, and degraded failure states; 4–6 denote sudden failure states. It is to be noted that the transfer rate between states is marked in the figure. In addition, λ is the transfer rate and μ is the repair rate; a 1 , a 2 , and a 3 represent the rate of transition from fault state 6 to states 0–2; b 1 and b 2 represent the rate of transition from fault state 5 to 0.1; and parameters a i and b i satisfy the following relationship: { a 1 + a 2 + a 3 = μ 2 b 1 + b 2 = μ 1 a 1 , a 2 , a 3 , b 1 , b 2 ≥ 0 . (5)

Assuming that the initial state of the device is 0, the Markov state transition rate matrix of the device based on the aforementioned analysis is as follows: T = [ c 1    λ 01                 λ 0           c 2    λ 12                 λ 1                 c 3    λ 23                 λ 2    μ f              - μ f   μ 0                     - μ 0   b 1    b 2                       - μ 1 a 1    a 2    a 3                       - μ 2 ] , (6) where c 1 = − ( λ 01 + λ 0 ) , c 2 = − ( λ 12 + λ 1 ) , c 3 = − ( λ 23 + λ 2 ) . Let P ( t ) = [ P 0 ( t ) , ⋯ , P 6 ( t ) ] be the state matrix. P i ( t ) represents the probability that the device is in state i at time t ; then, the differential equation of C–K is as follows: d P ( t ) / d t = P ( t ) T . (7)

Carrying out the Laplace transform on the aforementioned formula, we obtain: s [ p 0 ( s ) p 1 ( s )   ⋅ ⋅ ⋅   ⋮ p 6 ( s ) ] - [ p 0 ( 0 ) p 1 ( 0 )   ⋅ ⋅ ⋅   ⋮ p 6 ( 0 ) ] = T T [ p 0 ( s ) p 1 ( s )   ⋅ ⋅ ⋅ ⋮ p 6 ( s ) ] . (8)

Obtaining P ( s ) from the aforementioned equation and the initial state p (0) and then performing the inverse Laplace transform to obtain P ( t ) , the equipment availability is as follows: A ( t ) = p 0 ( t ) + p 1 ( t ) + p 2 ( t ) . (9)

When there is a failure correlation between equipment i and j , the failure correlation of components i and j is described by defining the failure correlation coefficient g i j , that is, g i j = ϕ i j a , (10) where ϕ i j is the correlation coefficient of the actual degradation of components i and j , and a is the failure-related strength used to describe the degree of influence of failure correlation on the component failure rate; its value is determined by the failure degree of the equipment in the actual photovoltaic power generation system.

It can be obtained that the comprehensive failure rate of components is as follows: λ j ( t ) = λ I j ( t ) + ∑ i j g i j λ i j ( t ) g , (11) where λ j ( t ) is the comprehensive failure rate of component j , λ I j ( t ) is the independent failure rate of component j, and λ i j ( t ) g is the failure rate of component i that affects the failure of component j .

The Copula theory connects the joint and marginal distribution functions and can systematically describe the correlation characteristics between variables. Therefore, this study proposes the use of this theory and Sklar theorem to obtain the correlation coefficient of the actual degradation. Through the aforementioned analysis combined with the description of the availability model in Fig 3.1, the availability model of nonfailure source equipment that considers the failure correlation can be obtained.

To compare the availability of photovoltaic power generation systems with and without failure correlation, this section analyzes the system availability model from the point of view of the following two aspects:

The operation and maintenance of photovoltaic power generation systems must keep the lowest maintenance cost under certain availability conditions. Therefore, upper and lower optimization models are proposed.

The upper model considers the economic structure relevance, introduces the idea of ​​opportunistic maintenance in the modeling process, and determines the maintenance method, time, and scope with the lowest maintenance cost as the objective function. The lower model is a verification based on the upper model, with the highest system availability as the objective function. The component availability is the constraint, and the maintenance plan that does not satisfy the lower model is substituted into the upper layer and optimized again until the optimal maintenance plan is obtained. In this study, the quantum genetic optimization algorithm is proposed to optimize the upper and lower models. A block diagram is shown in Figure 7.

This study takes the lowest maintenance cost in the entire maintenance time period as the objective function and establishes the upper-level optimization model with maintenance resources and time as constraints.

The total maintenance cost of a photovoltaic power station comprises fixed maintenance cost C com i , downtime loss C loss i , and maintenance personnel cost C peo i , that is, C all = ∑ i = 1 n [ ∑ i = 1 m ( C com i + C loss i ) + C peo i ] , (18) where n is the number of subsystems that need to be maintained and m is the number of devices that need to be maintained in the subsystem.

In the upper layer, the maintenance plan obtained with the lowest maintenance cost is brought to the lower layer for verification. The target is the highest availability of the photovoltaic power generation system, and the component availability is the constraint to establish the lower-layer optimization model expressed as follows: { max   A a l l s . t .    A i ≥ A i min , (29) where A i min is the lowest availability of component i in stable operation, assumed to be 0.9 in this study. An improvement factor is proposed to indicate the degree of recovery to describe the impact of different maintenance strategies on component performance: δ i = ( l ⋅ c i c f ) b i , i = 1 , ⋅ ⋅ ⋅ , n , (30) where l is the maintenance cost’s adjustment parameter, b is the maintenance frequency’s adjustment parameter, and i is the maintenance frequency. l and b are considered to be 1 and 0.0005, respectively.

Two options were selected for comparison analysis.

Option 1: performance analysis without considering the failure correlation. option 2: performance analysis including the failure correlation.

Through the simulation, In Figures 9A,B are shown, respectively, the availability comparison curves of the combiner box and the system under the failure-related strength a = 1 and the failure-independent situation.

It can be seen from Figure 9 that when the failure correlation is considered, the degree of decline in the availability of equipment and systems increases. It is determined by the correlation coefficient and strength between its increasing speed and equipment. The larger the correlation coefficient, the faster the availability curve of option 2 declines than that of option 1. It can be seen that ignoring the failure correlation will lead to errors in subsequent maintenance strategies and time, thus affecting the revenue of the photovoltaic power generation system.

In order to compare the effects of different failure-related strengths on the availability of equipment and systems, the availability simulation curves are obtained by setting different failure-related strengths as shown in Figure 10.

It can be seen from Figure 10 that the failure-independent availability curve is always above the failure-related availability curve, and when the failure-related strength is greater, the availability curve is closer to the failure-independent curve.

When 0 < a < 1 , with the decrease of the failure-related strength, the lowering rate of the availability curve is faster, so when the maintenance plan is formulated, the maintenance time and strategy are more obviously different than those of the failure-independent model. The photovoltaic power generation unit is a hybrid system comprising multiple devices in series and in parallel, and the structure of the parallel subsystem is symmetrical. By comparing the two figures, it can be seen that the availability curves of the system and equipment have similar changes with the failure-related strength, but the trend of the system is more evident.

The following four options are used to compare and analyze: Option 1: a double-layer optimization maintenance strategy that considers component relevance and a = 1 . Option 2: a double-layer optimization maintenance strategy that considers component relevance and a = 0.1 . Option 3: a double-layer optimization maintenance strategy that does not consider component relevance. Option 4: a traditional condition maintenance strategy that does not consider component relevance.

The following results were obtained through the simulation analysis: A and B in Figure 11 are the comparison diagrams of the changeable curves of the availability of the combiner box H 2 and the system after the maintenance of the four schemes, respectively.

It can be seen from A in Figure 13 that the availability of the combiner box H 2 in the initial state is 0.98. If the failure is not considered, it will meet the maintenance conditions for the first time when it runs to the 149th day. If the failure is considered, it will meet the maintenance conditions for the first time when it runs to the 103rd day. Through the double-layer optimization model, it is known that it needs to be repaired slightly. After repair, the availability is increased to 0.9910, and then it continues to operate on this basis until the next repair.

B in Figure 13 is the curve of availability change after system maintenance. If the failure correlation is not considered, the transformer needs to be repaired when the system runs to the 206th and 34th days. If the failure correlation is considered, the transformer needs to be repaired when the system runs to the 186th and 34th days, and the system availability is significantly improved after repair.

It can be seen from Figure 11 that the availability of the double-layer optimization maintenance model is significantly improved compared with that of the traditional condition maintenance model. The maintenance time of the double-layer optimization maintenance model considering failure correlation is earlier than that of the double-layer optimization maintenance model with independent failure, and the smaller the value of a, the longer the maintenance time in advance.

Table 5 shows the common maintenance time decision set for photovoltaic power generation unit systems.

It can be seen from the table that the maintenance strategy of failure-independent system and the maintenance strategy of failure correlation are different. When the failure-related strength a is smaller, the maintenance time is earlier; the maintenance times are greater in number in the whole maintenance cycle, and the maintenance cost is higher. This is because when the failure correlation is considered, the probability of failure of the affected component will increase, so the number of repairs will increase, the maintenance cost will increase, and the maintenance strategy will also change. When the failure-related strength a = 1 , the availability of equipment and systems is close to the failure-independent situation, so the maintenance time is not significantly different, and the maintenance strategy is basically the same.

The total maintenance cost of each equipment is calculated by formulas (19–26), resulting in 66,800 yuan, whereas the maintenance cost is 59,200 yuan after adopting the double-layer optimization model. The saving rate reaches 12.8%, and the downtime of each equipment is 72 h for individual maintenance. The double-layer optimization maintenance model that considers opportunistic maintenance takes into account multiple maintenance equipment. The longest maintenance equipment downtime is taken as the common downtime, which is 40 h. As a result, the downtime saving rate can reach 44.4%, so the double-layer optimization model proposed in this study can effectively reduce both downtime and maintenance costs.

The double-layer optimization maintenance strategy proposed in this study is affected by weather accessibility. The maintenance-cost change rate and system availability are different under different values.