Figure 2 outlines the type 2 fuzzy hidden Markov model for individual offshore power equipment. D i , i = 1,2 , … n . D 1 denotes the “as good as new” state, D 2 , D 3 ,…, D n are the states with different levels of deteriorations, and D f is the failed state. The transition rates among different states form the matrix Λ .

Different from the regular Markov model, the state D i is invisible, but the output O i depending on D i is visible. Therefore, the hidden Markov model can be assumed as a regular Markov model with unobserved states. The visible output sequence provides some information about possible invisible states. In the hidden Markov model (Equation 1), the difference ΔΛ ( T ) between the transition matrix of the invisible state D i and the transition matrix of the observed state O i can express the change and uncertainty of the operation as Δ Λ ( t ) = f T 2 ( C ( t ) ) ,(1) where C ( t ) is the operating condition of a single component in the time interval t and f T 2 represents the mapping function from C ( t ) to Δ Λ ( t ) .

C ( t ) is designed for each type of component, and its combination will seriously affect the reliability of components. In this work, C ( t )   t r f represents the operation condition of the transformer, including life, load, previous maintenance time, and working environment. C ( t )   c b is the operating condition of the circuit breaker, including the previous maintenance life and time. The operational variables of C ( t ) are mapped with each other by fuzzy language rules. Once the rules are established, the fuzzy system can be regarded as a mapping function f T 2 from input to output. Fuzzy language rules are derived from expert knowledge and mathematical strategies (Mendel, 2001; Zadeh, 1975; Mendel et al., 2006). Therefore, Equation 2 can be used to obtain the transfer matrix of D i , and then the reliability index, mean time to failure (MTTF), and failure probability of a single component can be calculated according to the standard steps of the Markov model (Papoulis and UnnikrishnaPillai, 2002): Λ D ( t ) = Λ O ( t ) + ΔΛ ( t ) . (2)

The configuration of the power system with suitable protection solutions would directly affect the reliability of related renewable energy systems. In this study, the minimum cut set method is used to analyze the impact of configuration on system reliability. According to the definition, the minimal cut set belongs to a group of irreducible components whose failure could definitely lead to system failure. The method from Yang et al. (2008) has been applied to the reliability estimation of complex systems, which cannot be simplified to a simple configuration.

From the perspective of reliability, the reliability of a system can be expressed as the block diagram of the minimum cut set that causes system failure, as shown in Figure 3. Here, F _mn represents the fault n in the minimum cut set m. As can be seen from Figure 3, all faults within a minimum cut set can be regarded as parallel, and all minimum cut sets are in series. Therefore, the reliability of the system can be easily evaluated according to the rules for a simple configuration.

It should be noted that the failure modes include the first-order fault events, second-order passive fault, and main protection fault. In a substation with multiple loads, each load will be assigned a different priority to ensure that the load can be transferred to a more important load first. After meeting the higher priority load requirements, the excess load will be transferred to other loads.

The unavailable energy E within the power system can be calculated in the following way: { D i = ∑ j = 1 M D i j , E = ∑ i = 1 p E i • D i ,         (3) where D i j represents the duration of the minimum cut set j of load point i and D i is the failure duration of load point i .

In a fuzzy logic system, the rule-based expert system can capture the overall impact of uncertainty on reliability, and the method of spreading uncertainty between rules is very important for the reasoning engine and is accomplished by the experts who are well acquainted with the operational characteristics of power systems. The input and output of the fuzzy logic system can be combined by experts using the “if-then” rule given by the fuzzy reasoning engine to obtain fuzzy output. Then, the output is defuzzified to get a clear value.

The format of the rule is as follows: If the input is [(the working environment is good) and (the load factor is low), and (the time from previous maintenance is short) and (the equipment age is old)], then the output is [(the transition rates will be decreased by a related percentage value)].

We take an example to show how this fuzzy logic works. A transformer with the operation hours less than 8,760*16 h is considered “young,” while the one with operation hours between 8,760*12 and 8,760*40 is considered “middle-aged.” Therefore, a transformer with the operation hours of 8,760*14 is considered both “young” and “middle-aged” with different levels of confidence. And this particular component is more reliable because of its slower deterioration rate than the older ones if the other operational variations are the same.

Unlike type 1 fuzzy sets with a single membership value, type 2 fuzzy sets are specially designed to deal with secondary uncertainties by introducing a membership value range associated with each value of the main variable. However, due to the significant increase of computational complexity, their implementations are limited.

In this study, a simpler method to realize type 2 fuzzy logic is proposed, that is, type 2 fuzzy logic system with quadratic uncertainty change. Secondary uncertainties are captured by initializing a set of primary membership functions. As shown in Figure 4, for a specific value, the membership function takes the value at the position where it intersects the vertical line. As a result, there are a certain range of membership values at x = x ' ( x ϵ X ) , and each value is given by a specific membership function. The selection of main membership functions is determined by the rules, and the format is as follows:

If the input is secondary uncertainty and Y is B 4 (Figure 4B), THEN the output is a primary membership function and the choice is A 4 (Figure 4A).

With these rules, the number of primary membership functions for each primary linguistic variable equals the number of linguistic variables of its associated secondary uncertainty. As shown in Figure 4, five primary membership functions correspond to the five linguistic variables for secondary uncertainty, respectively.

Membership functions are required for secondary uncertainty, which is the third dimension of type 2 fuzzy logic. In addition, the domain of the secondary membership functions at input x ' is determined by the range of amplitudes of primary memberships u ( x ’ ) , ( 0 < U l o w ( x ' ) < U ( x ' ) < U u p p ( x ' ) < 1 ) , which is depicted in Figure 4.

Although many factors affect reliability, this study mainly focuses on the effects of component aging, load, and different maintenance strategies for individual components, which are “primary variables.” The “primary variables” are taken as input to this type 2 fuzzy logic system. The uncertainty in the primary variables is “primary uncertainty.” Besides, additional uncertainty of variations is treated as “secondary uncertainty.” Taking one of the primary variables, time from previous maintenance, as an example, the secondary uncertainty comes from different maintenance extent (minor, medium, and major maintenance).

Figure 5 is a schematic diagram of the type 2 fuzzy logic system of the second uncertainty change. As shown in Figure 5, the implementation of the fuzzy logic system includes two steps: 1) the choice of primary membership functions and 2) mapping the primary input to output. For example, if the input time from previous maintenance is “short” and the previous maintenance is “secondary maintenance,” the primary membership function is first selected and then sent to fuzzier 1. After that, information 1 will map this primary input to an output based on Rule 1. If we want to consider more operation changes, we can combine their influences by more fuzzy inputs and modify the fuzzy inference engine.

Figures 6, 7 show the membership functions of type 2 fuzzy logic systems for transformers and circuit breakers, respectively. The uncertainty caused by various operation changes will affect the reliability of transformers and circuit breakers (annual MTTF and annual failure probability). Table 1 lists the major and minor uncertainties of operational changes.

Generally, the reliability of the transformer will decrease with the aging of insulation. The discourse range of each fuzzy variable is quantified as many overlapping fuzzy sets. These variables are called linguistic variables, as shown in Figure 6(a-1), and are represented by Gaussian membership functions. For the fuzzy variable “insulation age,” the discourse range is the service life of the transformer about 50 years, which is quantified into three linguistic variables, namely, “young,” “middle-aged,” and “old.” Similarly, the age-specific secondary uncertainty (component condition) is quantified as three linguistic variables, namely, “good,” “normal,” and “poor.” For the “young” transformer, if its state is “good,” the membership degree of “young” is higher than that of “bad” state at the same age, so the reliability is higher. The increase of load will reduce the reliability of the transformer. Therefore, as shown in Figure 6(a-2), the other major variable “load” is quantified into three variables, “light,” “medium,” and “heavy.” There is no quadratic uncertainty associated with this variable.

The reliability of the transformer will decrease with the increase of previous maintenance time. Figure 6(a-3) shows the third major variable “time of previous maintenance,” which is quantified as three variables: "short,” “normal,” and “long.” In addition, compared with small-scale maintenance, the improvement of reliability is more significant in large-scale maintenance. Therefore, the influence of maintenance degree is a secondary uncertainty, which is represented by the Gaussian membership function.

The reliability of the transformer is also affected by the working environment as the fourth main variable. As shown in Figure 6(a-4), the “working environment” is quantified into three variables: "fine,” “normal,” and “adverse.” For offshore power plants, ambient temperature is one of the most important factors. Therefore, weather is selected as the factor to bring the secondary uncertainty into the operating environment. The membership function of weather is also shown in Figure 6(a-4).

The reliability output of the transformer and the change percentage of the transition rate of Markov model ( Δ Λ ( t ) ) are quantified into five variables, namely, “MS,” “SS,” “UC,” “SL,” and “ML,” as shown in Figure 6B. “MS” means much smaller, “SS” means smaller, “UC” means unchanged, “SL” means larger, and “ML” means much larger. The smaller the transition rate, the higher the reliability. Using Equation 2, a transition matrix corresponding to the monitoring condition can be obtained as Λ D ( t ) , and reliability can be evaluated.

For circuit breakers, the main variables we focus on are the time and duration of previous maintenance because the reliability of circuit breakers is not sensitive to weather or load. Similar to transformers, age is quantified as three variables: “young,” “middle-aged,” and “old.” The secondary uncertainty of age is the component condition, i.e., “good,” “normal,” or “bad,” as shown in Figure 7(a-1). The second “main variable,” the time of previous maintenance, is quantified as “short,” “normal,” and “long.” The secondary uncertainty factor is “maintenance degree,” i.e., “minor maintenance,” “medium maintenance,” or “major maintenance” [Figure 7(a-2)]. The output variables are the same as the transformer variables but are represented by triangular membership functions, as shown in Figure 7B.

Figure 8 shows the ring configuration of bus 07 in the IEEE-RTS, which can be regarded as an offshore substation. The reliability of the load point is affected by the reliability of the transformer and circuit breaker in the substation. The adaptive maintenance consultant first obtains the initial maintenance plan from the system maintenance optimizer. The research period is set at 20 years.

Table 2 lists the basic fault data of transformers and circuit breakers without any maintenance during the first maintenance interval, which are obtained from the existing work (Billinton et al., 1985). Different priorities are assigned to each load point to reflect the importance of the load they transmit. In this study, load point 2 has priority 1 because it transfers the load back to the medium-sized system to which it is connected, while load point 1 has lower priority because it provides the load to individual customers.

Taking the fuzzy logic system designed for the transformer as an example, it is shown that the proposed type 2 fuzzy logic system is easy to implement. Several items including the age, load, last maintenance time, working environment, load factor, and maintenance information have been utilized as the inputs of the transformer. In this type 2 fuzzy system, each input has three membership functions. Three additional uncertainties are superimposed on three inputs, which are represented by three fuzzy sets. In a word, the two kinds of fuzzy systems generate rules. Class 1 fuzzy systems with rules can represent the same uncertainty. Therefore, in dealing with other uncertainties, type 2 fuzzy logic is superior to type 1 fuzzy logic in computational complexity.

Figures 9, 10 are used as the inputs of quadratic uncertainty type 2 fuzzy logic to calculate the reliability index of each component. When maintenance is not carried out, the Markov model is used to calculate the annual MTTF and failure probability.

When calculating the MTTF and failure probability, the operation change and its secondary uncertainty should be considered and then compared with the MTTF and failure probability under average conditions (Table 3). Figures 11, 12 show the changes of failure probability and MTTF of the transformer under different service life, load, time from previous maintenance, and working environment, respectively. Considering the secondary uncertainty, these two reliability indexes are different from those only considering the main operation differences. The results show that the second fuzzy logic system successfully captures the quadratic uncertainty. Similar results for circuit breakers are shown in Figures 13, 14.

These conditions may vary from time to time due to component aging, weather conditions, load requirements, and previous maintenance times. Therefore, it is not enough to carry out maintenance according to the schedule set at the beginning of the long-term maintenance plan. The purpose of the maintenance plan is to achieve the best reliability at the lowest operating cost under any different conditions. The maintenance plan is carried out under the same three conditions: 1) average condition; 2) considering operational changes without secondary uncertainties; and 3) considering secondary uncertainty.

In order to evaluate the reliability of the whole system, it is necessary to determine the minimum cut-off points of each load point, which are listed in Table 4.

The effects of operational variations on maintenance scheduling are shown in Figures 15, 16. As can be seen in Figure 16, the positions of the two Pareto fronts with varied operation conditions are allocated above the one with average conditions. As a result, in order to achieve the same reliability or budget, the maintenance schedule should be changed according to the operation variations. For example, in order to maintain the operation cost at $6*105 when the operation conditions vary from average conditions, the maintenance schedule A2 should be chosen instead of A1, and the energy not served will be 480 MWh/Yr, which is 9 MWh/Yr higher than the one with average conditions. The higher energy not served also indicates that the monitored conditions which are more severe than average will cause worse reliability. A similar change also happens on the Pareto fronts in Figure 16.

Furthermore, the secondary uncertainty of the operational variations can also be well handled in the maintenance scheduling problem by this type 2 maintenance advisor. Figure 16 shows that the secondary uncertainty of the operational conditions also leads to the change of maintenance schedules. For example, in Figures 15, 16, the schedule A2 will be replaced by A3 because of the secondary uncertainty in the operational conditions.