An opportunistic maintenance strategy for offshore wind turbine based on accessibility evaluation

Abstract

The maintenance costs of offshore wind turbines operated under the irregular, non-stationary conditions limit the development of offshore wind power industry. Unlike onshore wind farms, the weather conditions (wind and waves) have greater impacts on the operation and maintenance of offshore wind farm. Accessibility is a key factor related to the operation and maintenance of offshore wind turbine. Considering the impact of weather conditions on the maintenance activities, the Markov method and dynamic time window are applied to represent the weather conditions, and an index used to evaluate the maintenance accessibility is then proposed. As the wind turbine is a multi-component complex system, this article uses the opportunistic maintenance strategy to optimize the preventive maintenance age and opportunistic maintenance age for the main components of the wind turbine. Taking the minimum expectation cost as objective function, this strategy integrates the maintenance work of the key components. Finally, an offshore wind farm is taken for simulation case study of this strategy; the results showed that the maintenance cost of opportunistic maintenance strategy is 10% lower than that of the preventive maintenance strategy, verifying the effectiveness of the opportunistic maintenance.

Keywords

Offshore wind turbine accessibility operation and maintenance opportunistic maintenance Markov method

Introduction

In recent years, the global wind power industry has been developing rapidly. Due to the abundant offshore wind energy resources, the installed capacity and power generation of offshore wind power are growing rapidly every year all over the world (Perveen et al., 2014; Shafiee, 2015; Wang et al., 2009). The operation and maintenance cost is a key factor restricting the development of offshore wind power, which usually accounts for 20%–35% of the total cost of power generation in offshore wind farms (Lyding et al., 2010; Polinder et al., 2009). The special operating environment of offshore wind turbines, in which the equipment is susceptible to be affected by natural conditions, such as typhoon, tides, and waves, can cause the accelerating failure of unit components, and the increase in failure rate of electrical and mechanical systems, finally leading to the lower reliability of wind farms. On the contrary, the accessibility of offshore wind farms is greatly limited by the sea wind and wave conditions, which hinders the operation and maintenance of the unit, thus resulting in an increase in the outage time of the wind turbines, a decrea se in the availability of the units, and an increase in the operation and maintenance costs. As a consequence, it is necessary to consider the impact of sea weather conditions on unit operation and maintenance when operating and maintaining offshore wind turbines.

Up to now, the impact of weather conditions on wind turbine maintenance strategies has been investigated by previous researchers. In China, Huang et al. (2013) proposed a wind turbine reliability evaluation method suitable for time on the weather accessibility window. Monbet et al. (2007) make a survey of stochastic models for sea state and different offshore wind farms according to the different maintenance schemes of offshore wind turbines, combined with the failure rate and maintenance time of various components of the unit. Considering the special operating environment of offshore wind turbines, Liu et al. (2016) analyzed the impact of weather accessibility and maintenance time on the maintenance strategy of offshore wind turbines combined with the maintenance waiting problems during operation and maintenance. Zheng et al. (2014) established an offshore wind turbine maintenance strategy with the minimum maintenance cost and maximum reliability, considering the influence of wind speed and maintenance waiting wind time series and conducted a simulation study on different stochastic models of weather conditions at sea, also propose an original statistical method based on Monte Carlo goodness-of-fit tests. Matha et al. (2012) proposed a novel method to model wind and wave conditions for offshore sites and indicated that the persistence of weather windows for significant wave height values could be captured reasonably well by this approach. All tests show promising results and the weather model can be deemed accurate enough for simulations of offshore wind parks. Moreover, as short-duration events are overestimated, Scheu et al. (2012a) established a Markov model based on a statistical analysis of the wind speed and wave height at sea, and finally made predictions for the duration and maintenance waiting time. Although there have been some related studies on the effect of weather conditions on offshore wind turbine maintenance strategies, up to now, researchers have made contrapuntal study on wind weather; however, the wind speed together with the wave height has not yet been considered to evaluate the accessibility of the offshore wind turbine maintenance.

In this study, considering the constraints of accessibility during maintenance of offshore wind turbines, the Markov method and dynamic time window are used to describe sea wind speed and wave height weather conditions, and the impact of weather factors such as wind waves on the accessibility of offshore wind farms is studied. Based on the economic correlation of component maintenance, the opportunistic maintenance strategy combining failure maintenance and preventive maintenance is adopted. And the minimum expected maintenance cost in the cycle is taken as the objective function to optimize the opportunity maintenance age and preventive maintenance service age. Through simulation analysis, it was found that the opportunistic maintenance is more cost-effective than traditional age-based preventive maintenance, thus verifying the scientific value of the opportunistic maintenance strategy applied to offshore wind turbines.

Markov prediction theory

The Markov method, which was first proposed by the famous Russian mathematician Markov (1856–1922), is a statistical analysis method based on probability theory and stochastic process. This method uses a mathematical model to analyze of the evolution of objective objects (Lu, 1986). In the Markov process, the time series is regarded as a stochastic process. By studying the initial probability of different states of things and the state transition matrix, the state change trend is determined and the future state of things can be predicted. This method has been widely used in communications, biology, social sciences, and other fields (Liu, 2008).

Markov process

If the random process ${X (t), t \in T}$ satisfies the following conditions (Liu, 2005; Zhang, 2005):

If the state of the stochastic process ${X (t), t \in T}$ at time t is known, and the state at time t + 1 is only related to the state at time t and independent of the state before t time, then it is considered that the stochastic process ${X (t), t \in T}$ has Markov property, that is, “Markov property.”

When the state space of the stochastic process ${X (t), t \in T}$ is S, if for any n ⩾ 2, and any $t_{1} < t_{2} < \dots < t_{n} \in T$ in the condition $X (t_{i}) = x_{i}, x_{i} \in S, i = 1, 2, \dots, n - 1$ , the conditional probability distribution function of $X (t_{n})$ is equal to its probability distribution function under the condition $X (t_{n - 1}) = x_{n - 1}$ , which is

\begin{array}{l} P (X (t_{n}) \leq x_{n} | X (t_{1}) = x_{1}, X (t_{2}) = x_{2}, \dots, X (t_{n - 1}) = x_{n - 1}) \\ = P (X (t_{n}) \leq x_{n} | X (t_{n - 1}) = x_{n - 1}) \end{array}

(1)

Then the stochastic process ${X (t), t \in T}$ is called the Markov process.

Markov chain

As one of the Markov processes, the Markov chain has attracted wide attention from researchers all over the world due to its more mature and more applicable theory. The Markov chain is currently mainly used in the fields of medicine, economy, disaster prediction, and so on.

Markov chain is a discrete-time stochastic process with Markov property and the model is usually expressed as $λ = (S, P, Q)$ , where the meaning of each element is as follows (Sheldon, 1996):

S represents a non-empty set of all possible states in a random process. State is the result of a random process occurring at a certain moment. Depending on the random process and the predicted target, the states can have different partitions, which can be finite, countable, or arbitrary non-empty sets, generally represented by i and j.

Ρ is a one-step state transition probability matrix. Where conditional probability can be expressed as follows

P {X (t + 1) = j | X (t) = i} = P_{i j}

(2)

indicating the probability that the stochastic process will transition to state j at time t + 1 under the condition that it is at state i at time t. If the state set of the stochastic process is S = {1, 2, …, n}, then the one-step transition probability matrix consisting of the one-step transition probability P_ij is

P = P_{i j} = [\begin{matrix} p_{11} & p_{12} & \dots & p_{1 n} \\ p_{21} & p_{22} & \dots & p_{2 n} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ p_{n 1} & p_{n 2} & \dots & p_{n n} \end{matrix}]

(3)

This matrix satisfies the following equation

{\begin{cases} 0 \leq p_{i j} \leq 1 i, j = 1, 2, \dots, n \\ \sum_{j = 1}^{n} p_{i j} = 1 \end{cases}

(4)

Q is the initial state probability distribution vector, Let $Q = [q_{1}, q_{2}, \dots, q_{n}]$ denote the probability that the stochastic process is in state x_i at time t = 0 is q_i, then

\sum_{i = 1}^{n} q_{i} = 1

(5)

Data processing based on Markov method

When using the Markov chain model for predictive analysis, the random variables need to be processed as follows: (1) state division of random variables, (2) calculation of state transition probability, and (3) test of “Markov property.”

State division of random variables

For the state division of random variables, there are usually the following methods (Qian, 1998): (1) sample mean value–mean variance partition method, (2) ordered clustering method, and (3) fuzzy clustering method. In this article, only the first method is used and will be briefly described below.

Suppose a random observation set of random variable X is $x_{1}, x_{2}, \dots, x_{n}$ , the sample mean is $\bar{x}$ , and the sample mean variance is s. If the absolute value $| r |$ of the auto-correlation coefficients of this set is ⩽0.30, it can be approximated as an independent and identically distributed sequence. According to the central limit theorem

{\begin{cases} P {\bar{x} - s \leq \bar{x} \leq \bar{x} + s} = 2 Φ (1.0) - 1 \approx 0.68 \\ P {\bar{x} - 1.5 s \leq \bar{x} \leq \bar{x} + 1.5 s} = 2 Φ (1.5) - 1 \approx 0.87 \end{cases}

(6)

where

\begin{array}{l} \bar{x} = \frac{1}{n} \sum_{i = 1}^{n} x_{i} \\ s = \sqrt{\frac{1}{n - 1} \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}} \end{array}

(7)

The sample mean value $\bar{x}$ is centered, and the sample mean squared s is grouped into criteria. In general, it can be divided into five groups: $(- \infty, \bar{x} - 1.0 s)$ , $(\bar{x} - 1.0 s, \bar{x} - 0.5 s)$ , $(\bar{x} - 0.5 s, \bar{x} + 0.5 s)$ , $(\bar{x} + 0.5 s, \bar{x} + 1.0 s)$ , and $(\bar{x} + 1.0 s, + \infty)$ .

This method simply take the sample mean value as the center of the observation sequence, from the statistical point of view but regardless of the influence of physical genesis on variables. Although lacking in rationality and scientificity, this method is convenient and widely applicable to classify random variables.

Calculation of state transition probability

Before using the Markov chain for prediction, it is necessary to estimate the transition probability of the Markov chain from the historical observation sequence.

Suppose a set of observation sequences in a random variable X is $x_{1}, x_{2}, \dots, x_{m}$ , which contains n states, namely the state space $S = {1, 2, \dots, n}$ . Use f_ij to indicate the frequency of the observation sequence from state i to state j through one step, $i, j \in S$ . A matrix ${(f_{i j})}_{i, j \in S}$ composed of $f_{i j} (i, j \in S)$ is called a transfer frequency matrix. The value obtained by dividing the element of the ith row and the jth column in the transfer frequency matrix by the sum of the elements of the ith row is called the maximum likelihood estimation of the transition probability, ${\hat{P}}_{i j}, i, j \in S$ , as follows

{\hat{P}}_{i j} = \frac{f_{i j}}{\sum_{j = 1}^{n} f_{i j}}

(8)

Actually, ${\hat{P}}_{i j}$ is the frequency that is transferred from state i to state j through one step. It can be known from the frequency stability that when n → ∞, the transfer frequency is approximately equal to the transition probability, so it can be used to estimate the transition probability. One-step transition probability matrix can be expressed as follows

P = {({\hat{P}}_{i j})}_{i, j \in S}

(9)

Test of “Markov property”

It is necessary to test whether the random variable sequence has “Markov property” before applying the Markov chain model to analyze and solve practical problems. Usually a discrete sequence of the Markov chain can be tested using the $χ^{2}$ statistic.

Suppose the random variable sequence contains n states, and use f_ij to indicate the frequency of the random sequence $x_{1}, x_{2}, \dots, x_{m}$ from state i to state j, $i, j \in S$ . The value obtained by dividing the sum of the jth column of the transfer frequency matrix by the sum of the lines and columns is called the marginal probability and is denoted as $P_{. j}$ , as follows

P_{• j} = \frac{\sum_{i = 1}^{n} f_{i j}}{\sum_{i = 1}^{n} \sum_{j = 1}^{n} f_{i j}}

(10)

Then when n is sufficiently large, the statistics obeys the $χ^{2}$ distribution with a degree of freedom of ${(n - 1)}^{2}$ , where P_ij is the transition probability

χ^{2} = 2 \sum_{i = 1}^{n} \sum_{j = 1}^{n} f_{i j} | \log \frac{P_{i j}}{P_{• j}} |

(11)

Given the significance level $α$ , we can get the value of $χ_{α}^{2} {(n - 1)}^{2}$ from the look-up table, and finally calculate the value of the available statistic $χ_{α}^{2}$ . If $χ_{α}^{2} > χ_{α}^{2} {(n - 1)}^{2}$ , it can be considered that the random sequences $x_{1}, x_{2}, \dots, x_{m}$ have “Markov property”; if not, this sequence cannot be treated as a Markov chain (Luo, 1987).

Markov prediction model

The operation and maintenance of offshore wind turbines are affected by weather conditions such as sea wind speed and wind direction, and sea conditions such as wave height and cycle. Among them, wind speed and wave height are the two most important factors affecting the accessibility of wind turbines (Van Bussel et al., 2001; Van Bussel and Zaayer, 2001). As a consequence, in this article, the accessibility of offshore wind turbines is studied mainly considering the wind speed and wave height.

Offshore wind speed and wave height are characterized by time-varying and randomness. Commonly used research methods include Gaussian statistics, Markov method, and auto-regressive moving average method. Markov method was used to build the model since the method can describe the long-term and seasonal wind speed and wave height distribution accurately (Hagen et al., 2013; Scheu et al., 2012b).

Markov prediction method

There are three main methods for Markov chain prediction (Xia, 2005):

Markov chain prediction based on absolute distribution. For a set of dependent random sequences of random variables, assuming that they satisfy the “time homogeneity,” the absolute distribution of random variables at a certain moment in the future can be predicted according to the Markov chain model with one-step state transition probability and the initial distribution analysis.

Superimposed Markov chain prediction. For a set of dependent random sequences of random variables, assuming that the state transition probabilities of the various orders have the same effect, the absolute distribution of the random variable at a certain moment in the future can be analyzed using the Markov chain model of each order state transition probability and the initial distribution prediction.

Weighted Markov chain prediction. For a set of dependent random sequences of random variables, using the correlation analysis of state transition probability and dependent random variables, the absolute distribution probabilities of variables at a certain moment in the future can be obtained by weighted summation according to the strength of the dependent relationship.

Model building process

In the case of unrestricted visual conditions, the operation and maintenance of offshore wind farms basically only needs to consider the effects of wind speed and wave height. On the contrary, according to the monitoring of coastal sea conditions by the National Ocean Forecasting Station, the relationship between the average wind speed at sea and the effective wave height can be approximately expressed as equation (12) (Feng et al., 2009)

H_{s} = 0.038 V^{2.0} + 0.1950

(12)

where H_s is the effective wave height and V represents the wind speed. Based on this, the article simplifies the weather conditions at sea.

Due to the fact that the wind speed and wave height at sea have obvious periodicity and seasonality and the state transition probability satisfies the “time homogeneity” characteristic, therefore, this article uses the Markov chain prediction method based on absolute distribution to model the weather of offshore wind farms. The detailed method and steps for modeling are as follows:

State division of historical observation data of wind seed and wave height. Considering the current data characteristics of sea speed and wave height, combined with the sample mean-mean variance grouping method, in this article, ∆H_s = 0.1 m, ∆V = 1 m/s or ∆H_s = 0.4 m, ∆V = 1 m/s were used as the standard interval grouping method to classify the state of the sea wind and waves.

According to the state group established by step 1, the corresponding state of the historical observation data is determined, and the results are statistically calculated. Thus, a frequency transfer matrix ${(f_{i j})}_{i, j \in S}$ and a Markov chain state transition probability matrix ${({\hat{P}}_{i j})}_{i, j \in S}$ having a step size of one can be obtained.

Test of “Markov property.”

If it is known that the state at time t at which the wind speed and the wave height is i, the initial distribution can be considered to be as follows

P (0) = (0, \dots, 0, 1, 0, \dots, 0)

(13)

where, P(0) is a unit row vector, and the ith component is 1, and the remaining components are 0. Therefore, the absolute distribution of wind speed and wave height at t + 1 is

P (1) = P (0) P = (p_{i 1}, p_{i 2}, \dots, p_{i n})

(14)

The predicted state j at time t + 1 satisfies equation (15)

p_{i j} = \max {p_{i j}, j \in S} S = {1, 2, \dots, n}

(15)

Then, the wind speed and the wave height at the time t + k can be obtained.

Duration prediction of accessibility and inaccessible state. According to the current ship performance and sea level in China, referring to the domestic- and foreign-related literature (Hagen et al., 2013), combined with the weather and ocean conditions in the sea area where the offshore wind farm is located, the operation and maintenance of offshore wind farms are generally operated at wind speed V ⩽ 10 m/s and wave height H_s ⩽ 2 m.

Suppose H is a set of accessibility states that contains multiple wind speeds and wave height states, according to equation (16), the one-step state transition probability matrix $\tilde{P}$ for H and its complement $S - H$ (denoted as $\tilde{H}$ ) can be obtained. The steady-state distribution vector is $\tilde{π} = {{\tilde{π}}_{H}, {\tilde{π}}_{\tilde{H}}} = {{\tilde{π}}_{1}, {\tilde{π}}_{2}}$ , where ${\tilde{π}}_{H}, {\tilde{π}}_{\tilde{H}}$ represent the probability of state set H and $\tilde{H}$ , respectively

\tilde{P} = [\begin{matrix} {\tilde{p}}_{H H} & {\tilde{p}}_{H \tilde{H}} \\ {\tilde{p}}_{\tilde{H} H} & {\tilde{p}}_{\tilde{H} \tilde{H}} \end{matrix}] = [\begin{matrix} {\tilde{p}}_{11} & {\tilde{p}}_{12} \\ {\tilde{p}}_{21} & {\tilde{p}}_{22} \end{matrix}]

(16)

[\begin{matrix} \frac{\sum_{i \in H} \sum_{j \in H} π_{i} p_{i j}}{\sum_{i \in H} \sum_{j \in H} π_{i} p_{i j} + \sum_{i \in H} \sum_{m \in \tilde{H}} π_{i} p_{i m}} & \frac{\sum_{i \in H} \sum_{m \in \tilde{H}} π_{i} p_{i m}}{\sum_{i \in H} \sum_{j \in H} π_{i} p_{i j} + \sum_{i \in H} \sum_{m \in \tilde{H}} π_{i} p_{i m}} \\ \frac{\sum_{m \in \tilde{H}} \sum_{j \in H} π_{m} p_{m j}}{\sum_{m \in \tilde{H}} \sum_{j \in H} π_{m} p_{m j} + \sum_{m \in \tilde{H}} \sum_{k \in \tilde{H}} π_{m} p_{m k}} & \frac{\sum_{m \in \tilde{H}} \sum_{k \in \tilde{H}} π_{m} p_{m k}}{\sum_{m \in \tilde{H}} \sum_{j \in H} π_{m} p_{m j} + \sum_{m \in \tilde{H}} \sum_{k \in \tilde{H}} π_{m} p_{m k}} \end{matrix}]

where i, j, m, k represent the wind and wave state, respectively, $i, j \in H$ and $m, k \in \tilde{H}$ .

Assuming that $τ (τ = 1, 2, \dots, l)$ is the duration of the state set H, the probability that the state set H lasts for l duration is

\begin{matrix} p {τ = l} = p {X_{1} \in H, X_{2} \in H, \dots, X_{l} \in H | X_{0} \in \tilde{H}, X_{l + 1} \in \tilde{H}} \\ = p {X_{1} \in H | X_{0} \in \tilde{H}} p {X_{s} \in H, 1 < s \leq l | X_{l + 1} \in \tilde{H}, X_{1} \in H} \\ = {\tilde{π}}_{2} {\tilde{p}}_{21} {({\tilde{p}}_{11})}^{l - 1} {\tilde{p}}_{12} \end{matrix}

(17)

where $X_{1}, X_{2}, X_{s}, X_{l}$ represent the states of the first, second, sth, and lth periods, respectively. Then, the average duration E(τ) can be obtained according to equation (18)

E (τ) = \sum_{i = 1}^{\infty} p {τ = i} i

(18)

Similarly, the probability $\tilde{p} {τ = l}$ of the state $\tilde{H}$ lasting for l periods of time and the average duration $\tilde{E} (τ)$ are

\tilde{p} {τ = l} = {\tilde{π}}_{1} {\tilde{p}}_{12} {({\tilde{p}}_{22})}^{l - 1} {\tilde{p}}_{21}

(19)

\tilde{E} (τ) = \sum_{i = 1}^{\infty} \tilde{p} {τ = i} i

(20)

Maintenance waiting time prediction analysis. The operation and maintenance of offshore wind turbines are affected by climate and tides, resulting in increased waiting time for maintenance and limited actual maintenance time. Assuming that personnel, vessels, spare parts, and so on, are adequately prepared, the waiting time for maintenance is only relevant to weather conditions.

Assuming that the unit components fail at time $t$ , the total time required for failure repair and transportation is $t_{n}$ , then the maintenance waiting time can be estimated using the dynamic time window. The specific steps are as follows:

Determine the season $w$ at the time $t$ , and then calculate the cumulative probability distribution based on the probability $π_{w}$ of each state in the season. A random number between (0, 1) is generated, and the cumulative probability distribution interval in which the random number is located is determined, thereby a state $X_{t, w}$ of the wind speed and the wave height at the moment can be obtained.

If $X_{t, w} \in H$ , then initialize the duration $t_{l o} = Δ t$ of the reachable state and the maintenance waiting time $t_{w a} = 0$ ; if not, let $t_{l o} = 0$ , $t_{w a} = Δ t$ , where ∆t is the Markov model step size.

According to the one-step state transition matrix $P_{w}$ , the probability vector of the state $X_{t, w}$ converted to the next moment state is $P_{X, w}$ , and $X_{t, w} = P_{X, w}$ . The next time state $X_{t + 1, w}$ is obtained according to the method in step 1.

If $X_{t + 1, w} \in H$ , then update the reachable state duration $t_{l o} = t_{l o} + Δ t$ , $t = t + 1$ ; otherwise, $t_{l o} = 0$ , $t_{w a} = t_{l o} + Δ t$ , $t = t + 1$ .

Repeat steps 3 and 4 until $t_{l o} \geq t_{n}$ , where $t_{w a}$ is the waiting time for this maintenance activity.

An opportunistic maintenance strategy based on accessibility evaluation

The wind turbine is a complex multi-component system. And many of the components work closely together. Once a component fails, the functions of other components will also be affected, resulting in overall shutdown and maintenance of the unit, which will cause large costs, including maintenance costs, rental vessel costs, and loss of economic benefits from on-grid power generation.

Opportunistic maintenance strategy

The opportunistic maintenance strategy refers to that when a component in the system is repaired or replaced due to a malfunction or other reasons, it also provides an opportunity for preventive maintenance or replacement of other components, considering the fact that maintenance of multiple parts at the same time can effectively save fixed maintenance costs and increase availability.

The principle of the opportunistic maintenance strategy for failure repair combined with preventive maintenance of components is shown in Figure 1.

Figure 1.

Opportunistic maintenance strategy.

If the service age of component i is in $(0, T_{o}^{(i)})$ , maintenance will be performed if a fault occurs; if the service age of component i is in $(T_{o}^{(i)}, T_{p}^{(i)})$ , normal maintenance will be performed if a fault occurs, and opportunistic maintenance will also be performed when other components fail or this component reaches preventive maintenance; preventive maintenance is performed when the age of the component reaches $T_{p}^{(i)}$ .

Opportunistic maintenance model

Opportunistic maintenance probability density function

According to the opportunistic maintenance strategy of Figure 1, the expected replacement rate of the component consists of three parts, complete replacement, incomplete repair, and minimum maintenance. This article only considers the opportunistic maintenance strategy for a complete replacement. According to the update process theory, after one component going through replacement for some periods, the replacement rate is close to a constant, which is the reciprocal of the expected replacement cycle of the component. According to the opportunistic maintenance strategy showed in Figure 1, the replacement rate $λ_{i}$ of one component consists of three parts, as shown in equation (21).

λ_{i} = \frac{1}{E_{i} (T)} = λ_{c i} + λ_{o i} + λ_{p i}

(21)

where $λ_{i} = P_{i} (f) . λ_{i}$ , $λ_{o i} = P_{i} (o) . λ_{i}$ , and $λ_{p i} = P_{i} (p) . λ_{i}$ are the failure replacement rate, opportunistic replacement rate, and preventive replacement rate of one component, respectively. $P_{i}$ is the replacement probability and $E_{i} (T)$ is the expected life of the component.

Since the opportunistic replacement of the component i is caused by the failure replacement or preventive replacement of other components, the opportunistic replacement probability density function of the component i can be approximated as an exponential distribution function as follows

g_{i} (t, T_{o}^{(i)}) = δ_{i} \exp [- δ_{i} (t - T_{o}^{(i)})]

(22)

δ_{i} = \sum_{j \neq i}^{N} (λ_{c i} + λ_{p i})

(23)

where $g_{i} (t, T_{o}^{(i)})$ represent the opportunistic replacement probability density function of the component $i$ , $δ_{i}$ is the exponential distribution function, N is the number of key components of the unit.

Key component replacement probability of the wind turbine

During each replacement cycle of the components in the unit, component i will perform three replacement strategies, namely, failure replacement, opportunistic replacement, and preventive replacement. According to the opportunistic maintenance strategy showed in Figure 1, the probability of each replacement is as follows

P_{i} (c) = \int_{0}^{T_{o}^{(i)}} f_{i} (t) d t + \int_{T_{o}^{(i)}}^{T_{p}^{(i)}} f_{i} (t) • {\bar{G}}_{i} (t, T_{o}^{(i)}) d t

(24)

P_{i} (o) = \int_{T_{o}^{(i)}}^{T_{p}^{(i)}} {\bar{F}}_{i} (t) • {\bar{g}}_{i} (t, T_{o}^{(i)}) d t

(25)

P_{i} (p) = {\bar{F}}_{i} (T_{p}^{(i)}) • {\bar{G}}_{i} (T_{p}^{(i)}, T_{o}^{(i)})

(26)

{\bar{F}}_{i} (t) = 1 - \int_{0}^{t} f_{i} (u) d u

(27)

{\bar{G}}_{i} (t, T_{o}^{(i)}) = 1 - \int_{T_{o i}}^{t} g_{i} (u, T_{o}^{(i)}) d u

(28)

where $f_{i} (t)$ is the failure probability density function of component i, which obeys the Weibull distribution.

Opportunistic maintenance optimization model

During one replacement cycle of a wind turbine component, the maintenance cost of component i consists of replacement costs, downtime losses, and rental vessel costs. The expected repair cost of component i during a replacement cycle can be expressed as follows

\begin{matrix} E_{i} (C) = (C_{c i} + (t_{w a} + t_{n}) • C_{e}) • P_{i} (c) + C_{o i} • P_{i} (o) \\ + (C_{p i} + (t_{w a} + t_{n}) • C_{e}) • P_{i} (p) + C_{N} \end{matrix}

(29)

where $C_{c i}$ , $C_{p i}$ , $C_{o i}$ represent the costs of failure maintenance, preventive maintenance, and opportunistic maintenance of component i; $C_{e}$ is the loss per unit downtime; $C_{N}$ is the rental vessel cost.

During one replacement cycle of a wind turbine component, the expected life of component i can be expressed as follows

\begin{array}{l} E_{i} (T) = \int_{0}^{T_{o}^{(i)}} t \cdot f_{i} (t) d t + T_{p}^{(i)} \cdot \bar{F_{i}} (T_{p}^{(i)}) \cdot \bar{G_{i}} (T_{p}^{(i)}, T_{o}^{(i)}) \\ + \int_{T_{o}^{(i)}}^{T_{p}^{(i)}} t [f_{i} (t) \cdot \bar{G_{i}} (t, T_{o}^{(i)}) + \bar{F_{i}} (t) \cdot g_{t} (t, T_{o}^{(i)})] d t \end{array}

(30)

According to the above analysis, by optimizing the opportunistic maintenance age and preventive maintenance service age of each key component of the unit to minimize the average cost rate, the following optimization model was established

\begin{array}{l} \min Z (T_{o}, T_{p}) = \sum_{i = 1}^{N} \frac{E_{i} (C)}{E_{i} (T)} \\ s . t 0 < T_{o} < T_{p} \\ t_{l o} \geq t_{n} \end{array}

(31)

where $Z (T_{o}, T_{p})$ is the average cost rate of the unit.

Simulation analysis

Accessibility evaluation simulation analysis

Due to lack of weather and ocean data of offshore wind farms, this article takes the historical observation data of an observation station in the east sea area of the Yangtze River estuary (30°N, 128°E) in the East China Sea as an example to model the sea weather.

The data of wind speed and wave height in the article were recorded every half hour from 0:00 on 1 January 2014. A total of 17,306 data points recorded in 2014 were used to make a forecast analysis of wind speed and wave height in 2015.

Wind speed and wave height state prediction

Using the Markov method, the wind speed and wave height can be divided into n states, and the state set is S = {1, 2, …, n}. In this article, $H_{s} = 0.1, 0.2, 0.3, and 0.4 m$ are used as grouping criteria to predict the 100 data points of wind speed and wave height in January 2015. The results are shown in Figures 2 and 3.

Figure 2.

Wave height prediction curve with ∆Hs = 0.1 m.

Figure 3.

Wave height prediction curve with ∆H_s = 0.4 m.

The difference between the predicted data and the observed data and the correlation coefficient are used as a cafeteria of the quality of the predicted data. The correlation coefficients of the observed and predicted data of average wind speed and average wave height are calculated separately, as shown in Table 1.

Table 1.

Comparison of observation and prediction data.

$Δ H_{s} (m)$	$\bar{V} (m / s)$	${\bar{H}}_{s} (m)$	$d (m)$	$r$
0	9.9300	4.5618	0	1.0000
0.1	9.2756	3.9360	16.0764	0.8742
0.2	9.7762	4.1960	17.7252	0.8235
0.3	8.1154	3.6840	18.6377	0.8712
0.4	8.7624	3.5240	19.3732	0.8525

In Table 1, the data of the wind speed and wave height when $Δ H_{s} = 0$ are observed data; r is the correlation coefficients of the observed and predicted data; d is the absolute error of prediction data and observation data of the wave height, being expressed as equation (32)

d = \sum_{i = 1}^{100} | {(H_{s})}_{p r e d i c t i v e v a l u e} - {(H_{s})}_{o b s e r v e d v a l u e} |

(32)

Taking the average error of the wave height prediction data as the standard to measure the advantages and disadvantages of different state grouping methods, the average error corresponding to $Δ H_{s} = 0.1, 0.2, 0.3, and 0.4 m$ is shown in Figure 4.

Figure 4.

Average error of the four grouping methods.

From Figure 4, when $Δ H_{s} = 0.1 m$ , the average error of wave height prediction data and observation data reached the minimum, which means the state grouping method of $Δ H_{s} = 0.1 m$ is superior to the other three groups. As a consequence, this state grouping method is used as a grouping criteria for the state in the following research.

Duration prediction of wind speed and wave height

Markov models are established for the spring, summer, autumn, and winter seasons, respectively. The occurrence probability of the accessible state H and the inaccessible state $\tilde{H}$ in different seasons is shown in Figure 5.

Figure 5.

The probability of each state occurring in different seasons.

The duration of each state under different seasons is shown in Table 2. One duration is the step size of Markov model, which is 0.5 h.

Table 2.

The duration of each state under each season.

State	Persistence
State	Spring	Summer	Autumn	Winter
$H$	0.624	0.786	0.570	0.455
$\tilde{H}$	0.376	0.214	0.430	0.545

Take the inaccessible state $\tilde{H}$ in January 2015 as an example, the observed and predicted values of the probability distribution and the cumulative distribution of the state duration are presented in Figure 6, which shows the accuracy of the Markov chain model.

Figure 6.

The duration prediction of state $\tilde{H}$ .

According to Figure 5 and Table 2, the occurrence probability of accessibility in summer is larger than that in any other seasons, and the average duration is the longest, indicating that the wind speed and wave height in summer are more suitable for sea. On the contrary, the occurrence probability of the inaccessible state in winter is the largest, indicating that the wind speed and wave height are not suitable for sea, and the bad weather lasts for a long time.

Maintenance waiting time estimate

Taking the maintenance waiting time in January 2015 as an example, the relationship between the maintenance waiting time and the state of the reachable state is shown in Figure 7. One duration is set to be 0.5 h. It can be seen that the waiting time for maintenance almost increases with the duration of the accessible state.

Figure 7.

Maintenance waiting time and duration diagram.

As can be seen in Figure 7, at the initial moment, there is a large difference between the predicted value and the observed value when $Δ H_{s} = 0.1 m$ . As the duration increases, the observations are gradually approached to the predicted value, and finally only a small gap is observed. When $Δ H_{s} = 0.4 m$ , the predicted value is far from the observed value at the initial moment; as the duration increases, the gap decreases, but in the end it still maintains a large gap.

Simulation analysis of opportunistic maintenance strategy

In this article, one 3 MW wind turbine in an offshore wind farm in China is used as an example to optimize the opportunistic maintenance strategy for the critical components of the unit, including the gearbox, the main bearing and the generator. According to the analysis of the statistical data of failure record of the unit, the failure time of the component follows the Weibull distribution, and the specific parameters are shown in Table 3, where numbers 1, 2, 3, and 4 represent the blade, gearbox, main bearing, and generator, respectively; α and β are the shape parameters and scale parameters of the components, respectively; unit downtime loss $C_{e}$ = 720 yuan/h, ship chartering fee $C_{N}$ = 150,000 yuan/day.

Table 3.

Weibull distribution parameters and maintenance costs of critical components.

Component number	$α_{i} / d$	$β_{i}$	$C_{1 i}$ (yuan)	$C_{2 i}$ (yuan)
1	3000	3	672,000	168,000
2	2400	3	912,000	228,000
3	3750	2	360,000	90,000
4	3300	2	600,000	150,000

When the traditional age-preventive maintenance strategy is adopted, the preventive maintenance age of the critical components of the unit is optimized to minimize the average cost rate. The results after optimization are shown in Table 4.

Table 4.

Optimization results of age maintenance strategy.

Component	Preventive maintenance age $T_{p} / d$	Minimum cost rate ( $¥ • d^{- 2}$ )	Total minimum cost rate ( $¥ • d^{- 1}$ )
Blade	1816	184.8898	782.6744
Gearbox	1423	300.5747
Main bearing	2822	108.3843
Generator	2284	188.8256

When using a multi-component opportunistic maintenance strategy, the opportunistic maintenance age and preventive maintenance age of the components are optimized to minimize the average cost rate. The results are shown in Table 5.

Table 5.

Optimization results of opportunistic maintenance strategy.

Component	Opportunistic maintenance age $T_{o} / d$	Preventive maintenance age $T_{p} / d$	Total minimum cost rate ( $¥ • d^{- 1}$ )
Blade	721	2709	702.5834
Gearbox	639	1812
Main bearing	1001	3000
Generator	897	2430

From the comparison of the results in Tables 4 and 5, it can be seen that the maintenance cost of the wind turbine using the opportunistic maintenance strategy is 10% lower than that of the preventive maintenance strategy.

Conclusion

This article considers the accessibility constraints in the operation and maintenance of offshore wind turbines, and Markov method and dynamic time window were used to describe the weather conditions at sea. In the multi-component opportunistic maintenance strategy combining failure maintenance and preventive maintenance, the opportunistic maintenance age and preventive maintenance service age are optimized to make the average cost rate of the unit in the cycle lowest.

An offshore wind farm in China is taken as an example for simulation case study, the simulation analysis results of this strategy showed that the maintenance cost of opportunistic maintenance strategy is 10% lower than that of the preventive maintenance strategy, verifying the scientific value of the opportunistic maintenance applied to offshore wind turbines.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Xie Lubing

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