An Optimal Number-Dependent Preventive Maintenance Strategy for Offshore Wind Turbine Blades Considering Logistics

Abstract

In offshore wind turbines, the blades are among the most critical and expensive components that suffer from different types of damage due to the harsh maritime environment and high load. The blade damages can be categorized into two types: the minor damage, which only causes a loss in wind capture without resulting in any turbine stoppage, and the major (catastrophic) damage, which stops the wind turbine and can only be corrected by replacement. In this paper, we propose an optimal number-dependent preventive maintenance (NDPM) strategy, in which a maintenance team is transported with an ordinary or expedited lead time to the offshore platform at the occurrence of the 𝑁th minor damage or the first major damage, whichever comes first. The long-run expected cost of the maintenance strategy is derived, and the necessary conditions for an optimal solution are obtained. Finally, the proposed model is tested on real data collected from an offshore wind farm database. Also, a sensitivity analysis is conducted in order to evaluate the effect of changes in the model parameters on the optimal solution.

1. Introduction

Wind energy has become an attractive source of renewable energy in the European energy market because it is free, abundant, and perceived as having a low impact on the environment. Over the past five years (2008–2012), the wind energy industry has been the fastest growing renewable energy source with an annual average growth rate of 28% [1]. For instance, in Sweden, 763 MW of wind power (onshore and offshore) was installed in 2011 which increased the wind power capacity to 2906 MW—about 2% of the total electricity consumption [2]. Certain forecasts indicate that the share of wind power in Sweden’s electricity generation will reach up to 20% by 2020.

Nowadays more and more wind turbines are being installed offshore due to the high potential of wind energy, less visual disturbance, and larger potential areas for installation. Presently, there are five offshore wind farms in the sea waters of Sweden (Lillgrund, Vanem, Utgrunden, Yttre Stengrund, and Bockstigen) with a total operating capacity of 163.7 MW [3]. However, a wind power system located at sea comes with higher installation costs and more difficult maintenance conditions compared to an onshore system. Furthermore, an offshore wind turbine has undesirable features like a higher failure rate, lower reliability, and higher operation and maintenance (O&M) costs. The O&M costs of onshore wind turbines account for around 20–25% of the wind energy generation cost, whereas in offshore wind farms, they represent a larger portion of the costs—up to 30% [4]. Therefore, there is a critical need to optimize the maintenance management of offshore wind turbines in order to reduce the O&M costs.

The literature on the optimization of maintenance strategies (opportunistic maintenance, reliability centered maintenance, and condition-based maintenance) in the wind energy industry is significant. Byon et al. [5] develop an optimization model to derive an optimal preventive maintenance strategy such that the expected average cost of a wind turbine under stochastic weather conditions is minimized. Byon and Ding [6] propose a season-dependent optimal maintenance strategy for a multistate deteriorating wind turbine in order to reduce the operation costs and enhance the marketability of wind power. Tian et al. [7] apply an artificial neural network approach to optimize the condition monitoring strategy for a wind farm assuming the same lead times for all maintenance activities. Nielsen and Sørensen [8] propose an optimal risk-based maintenance strategy for an offshore wind turbine with a single critical component. Ding and Tian [9, 10] develop three opportunistic optimization strategies based on perfect, imperfect, and two-level actions for wind turbines and apply a simulation optimization method to evaluate the maintenance cost of the proposed strategies.

Among the wind turbine components, the blade is one of the most critical and expensive components, whose function is to convert the kinetic energy into mechanical energy. According to the 2003 Netherlands wind energy report, 34% of the total number of failures in offshore wind farms was due to blades failure [11]. Also, a recently conducted study on wind farms in India indicates that the blades system is the most critical cause of turbine failure during the first five years of operation [12].

The blades are usually made from composite materials such as fibreglass reinforced plastic (see Figure 1) with an expected useful lifetime of twenty years. However, in offshore wind farms, the useful lifetime of a blade is significantly shorter than its expected lifetime. The reason is that the blades are “stressed” in a harsh maritime environment and extreme weather conditions and suffer from different types of damages (such as wear, fatigue, deterioration, crack, corrosion, and erosion) [13].

Figure 1 

A wind turbine blade made of fibreglass reinforced plastic.

Basically, the damages of wind turbine blades can be categorized into two types. One is a minor damage (such as microscopic cracks arising from fatigue) which is usually detected by online condition monitoring techniques (for more details see [14]). This type of damage weakens the system and causes a loss in wind capture without resulting in any turbine stoppage. The other one is a major (catastrophic) damage (such as a metre long fracture) which leads to a turbine system failure and stops the whole wind turbine.

In this paper, we assume that a maintenance team is transported to the offshore platform at the occurrence of the 𝑁th minor damage (to perform preventive maintenance) or the first major damage (to perform corrective replacement), whichever comes first. In practice, 𝑁 is the damage threshold for performing (PM) action which is considered to be a decision variable under the control of the wind farm manager and should be optimized. This strategy, namely, number-dependent preventive maintenance (NDPM), was first considered by Makabe and Morimura [15–17] and later extended by Morimura [18] and Park [19]. Since then, it has been considered from many different aspects such as when the system is subject to two types of minor and major failures (see Nakagawa [20], Sheu and Griffith [21]); with age-dependent repair costs (see Sheu [22], Sheu et al. [23]); under a repair-cost limit (see Park [24], Chang et al. [25], and Sheu et al. [26]); and with spare parts consideration (see Sheu and Chien [27], Chien [28, 29]).

In addition, most maintenance optimization models in the literature are based on the assumption that the maintenance team is immediately transported to the offshore platform. However, this assumption is unrealistic, as it is often the case that the maintenance team is sent only when the transportation means (ships, helicopters), specialized equipment (lifting cranes), and spare parts become available. Also, in some cases, the maintenance team has to wait for a number of days due to bad weather and sea conditions and then travel to the offshore platform. Since the offshore wind turbines are less accessible and/or are subject to very high costs for transport, an NDPM strategy possesses a huge potential for reducing the maintenance costs and enhancing the reliability of wind turbines, if applied properly.

In this paper, we propose an optimization model to determine the optimal number of minor damages allowed to occur in a blades system before sending a maintenance team to perform preventive maintenance actions. Our objective function is to minimize the long-run expected cost which includes all costs due to corrective replacement, preventive maintenance, logistics, and production loss. To our knowledge, this paper is the first attempt to optimize the NPDM strategy for offshore wind turbine blades subject to damages.

The rest of this paper is organized as follows. In Section 2, we present the problem definition. In Section 3, we describe the model components and use these in Section 4 to formulate our optimization model. In Section 5, the proposed model is tested using real data from an offshore wind farm database. Finally, in Section 6, we conclude this study with a brief discussion on topics for future research.

2. Problem Definition

Suppose that the lifetime 𝑇 of a wind turbine blade has the density function 𝑓∶𝑅₊↦[0,1], cumulative distribution function 𝐹∶𝑅₊↦[0,1], defined by 𝐹(𝑡)=∫^𝑡₀𝑓(𝑥)𝑑𝑥, and survival function 𝐹∶𝑅₊↦[0,1], defined by 𝐹(𝑡)=1−𝐹(𝑡) with 𝐹(0)=1.

We consider a preventive maintenance model of a blades system, in which the system consists of 𝑛_𝑏 i.i.d. blade lifetimes in series and is subject to damages. Suppose that the number of damages arriving in the interval [0,𝑡) in each blade is a nonhomogenous Poisson process (NHPP) with an intensity function ℎ(𝑡), defined by ℎ(𝑡)=𝑓(𝑡)/𝐹(𝑡), and a mean value function 𝐻(𝑡), defined by 𝐻(𝑡)=∫^𝑡₀ℎ(𝑥)𝑑𝑥 with 𝐻(0)=0. The assumption of NHPP has been considered extensively in shock and damage modelling in various domains (see, e.g., [30]) as well as wind energy industry (see [31]).

Further, it is assumed that the probability that a certain damage is a minor (major) damage is𝑞[𝑝=1−𝑞], where 0≤𝑝,  𝑞≤1. The case 𝑝=0 (𝑝=1) indicates that the damages are always minor (major). In this paper, the probability 𝑝 is considered as a constant value. However, this is unrealistic in some situations, especially in offshore environments. Then, a possibly more flexible model could be suggested by assuming 𝑝 as a weather-dependent function, which will be considered in our future work.

In the model to be developed in this article, the maintenance team is transported with an ordinary or expedited lead time to the offshore platform at the occurrence of the 𝑁th minor damage or the first major damage, whichever comes first. More precisely, the decision to repair or replace the blades system is made according to the following scheme (see Figure 2).

(a)

(b)

(c)

Figure 2 

Three cases of an NDPM cycle.

(a) No Major Damage Occurs before the 𝑁th Minor Damage or during the Ordinary Logistics Lead Time following the 𝑁𝑡ℎ Minor Damage (Figure 2(a)). In this case, the maintenance expeditions are initiated immediately after the 𝑁th minor damage, and the maintenance team is transported as soon as weather and sea conditions permit. Let 𝐿_𝑜>0 [𝑐_𝑜>0] denote the ordinary logistics lead time (cost), where 𝐿_𝑜 is the length of the interval between the time at which the maintenance decision is made and the time when the maintenance is performed and 𝑐_𝑜 is the total cost of hiring the service vessels, ordering spare parts, and transporting the maintenance team. Here, the maintenance strategy is to perform a perfect or complete preventive maintenance action on each blade at a cost 𝑐_𝑝, which returns the blades system to an “as-good-as-new” state.

(b) A Major Damage Occurs before the 𝑁th Minor Damage (Figure 2(b)). In this case, the blades system will be nonoperational and the wind turbine is shut down. Since such an event is costly, an expedited logistics plan is initiated at a shorter-than-ordinary lead time, but higher-than-ordinary cost [32]. We define the expedited logistics lead time, 𝐿_𝑒, as follows: 𝐿_𝑒=𝐿_𝑜−𝜉_𝐿, (1) where 𝜉_𝐿 (≥0) is the “expedited lead time term”, and the expedited logistics cost, 𝑐_𝑒 is 𝑐_𝑒=𝑐_𝑜+𝜉_𝑐, (2) where 𝜉_𝑐 (≥0) is the “expedited cost term”. Here, the maintenance strategy is to perform a corrective replacement on the failed blade at a cost 𝑐_𝑟 (>𝑐_𝑝) and a perfect PM action on each un-failed blade.

(c) No Major Damage Occurs before the 𝑁𝑡ℎ Minor Damage, but a Major Damage Occurs during the Ordinary Logistics Lead Time following the 𝑁𝑡ℎ Minor Damage (Figure 2(c)). In this case, the wind turbine is shut down until the ordinary lead time has elapsed during which the maintenance team performs maintenance activities. Here, the maintenance strategy is the same as for case (b).

Let the random variable 𝑇_𝑖, 𝑖=1,2,…, represent the waiting time until the 𝑖th minor damage occurs in the blades system. Since the minor damages in the blades system occur according to an NHPP with a mean value function 𝑞𝑛_𝑏𝐻(𝑡), the survival function of 𝑇_𝑖 is given by 𝐹_𝑇𝑖(𝑡)=Pr𝑇_𝑖>𝑡=∑^𝑖−1_𝑘=0𝑃_𝑘(𝑡)=𝐹(𝑡)^𝑞𝑛𝑏×∑^𝑖−1_𝑘=0𝑞𝑛_𝑏𝐻(𝑡)^𝑘𝑘!,𝑡≥0, (3) where 𝐹(⋅) and 𝐻(⋅) denote, respectively, the survival and the cumulative distribution function of each blade. 𝑃_𝑘(𝑡) is the conditional probability of 𝑘 minor damages occurring during the time interval [0,𝑡) on knowing that no major damage has occurred within [0,𝑡) and can be expressed as 𝑃_𝑘(𝑡)=𝐹_𝑇𝑘+1(𝑡)−𝐹_𝑇𝑘(𝑡),𝑘=0,1,2,…,with𝐹_𝑇0(𝑡)=0. (4)

Further, let Y denote the time to the first major damage in the blades system. Since the minor damages in the blades system occur at an NHPP with a mean value function 𝑝𝑛_𝑏𝐻(𝑡), the survival and the distribution function of Y, respectively, are given by 𝐹_𝑌𝑦=Pr𝑌>𝑦=e^{−𝑝𝑛𝑏𝐻(𝑦)}=𝐹𝑦^𝑝𝑛𝑏𝑓,𝑦≥0,_𝑌𝑦𝑑=−𝑑𝑦𝐹_𝑌𝑦=𝑝𝑛_𝑏ℎ𝑦𝐹_𝑌𝑦,𝑦≥0. (5)(6)

The following fact will be utilized in the forthcoming section, for the random variables 𝑇_𝑁 and Y, we have ^∞₀𝐹_𝑌(𝑡)𝑑𝐹_𝑇𝑁(𝑡)=−^∞₀𝐹_𝑌(𝑡)𝑑𝐹_𝑇𝑁(𝑡)=−𝐹_𝑌(𝑡)𝐹_𝑇𝑁(𝑡)^∞₀−^∞₀𝐹_𝑇𝑁(𝑡)𝑑𝐹_𝑌(𝑡), (7) and then by using (6) we have ^∞₀𝐹_𝑌(𝑡)𝑑𝐹_𝑇𝑁(𝑡)=𝑝𝑛_𝑏^∞₀ℎ(𝑡)𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)𝑑𝑡. (8)

3. Components of the Model

3.1. Expected Length of an NDPM Cycle

As shown in Figure 2, the length of an NDPM cycle with 𝑁 allowable minor damages in a blades system, CL(𝑁), can be expressed as CL(𝑁)=𝑌+𝐿_𝑒if𝑌<𝑇_𝑁𝑇_𝑁+𝐿_𝑜if𝑌≥𝑇_𝑁. (9) Then the expected length of the cycle, 𝐸[CL(𝑁)], is given by 𝐸[CL(𝑁)]=^∞₀^𝑡₀𝑦+𝐿_𝑒𝑑𝐹_𝑌𝑦𝑑𝐹_𝑇𝑁(𝑡)+^∞₀^∞_𝑡𝑡+𝐿_𝑜𝑑𝐹_𝑌𝑦𝑑𝐹_𝑇𝑁(𝑡). (10) By using (1) and (8), we have that 𝐸[(𝑁=𝐿CL)]_𝑒∞0𝐹_𝑌(𝑡)𝑑𝐹_𝑇𝑁(𝑡)+𝐿_𝑜^∞₀𝐹_𝑌(𝑡)𝑑𝐹_𝑇𝑁(𝑡+_∞0𝑡0)𝑦𝑑𝐹_𝑌𝑦𝑑𝐹_𝑇𝑁(𝑡)+^∞₀𝑡𝐹_𝑌(𝑡)𝑑𝐹_𝑇𝑁(𝑡)=𝐿_𝑜−𝜉_𝐿^∞₀𝐹_𝑌(𝑡)𝑑𝐹_𝑇𝑁(𝑡)+^∞₀^𝑡₀𝐹_𝑌𝑦𝑑𝑦𝑑𝐹_𝑇𝑁(𝑡)=𝐿_𝑜−𝑝𝑛_𝑏𝜉_𝐿^∞₀ℎ(𝑡)𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)+𝑑𝑡_∞0𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)𝑑𝑡. (11)

3.2. Expected Logistics Costs

Because the expedited logistics cost only occurs in case (b), the expected logistics cost per cycle, 𝐸[𝑐_𝐿(𝑁)], is given by 𝐸𝑐_𝐿(𝑁)=𝑐_𝑒^∞₀^𝑡₀𝑑𝐹_𝑌𝑦𝑑𝐹_𝑇𝑁(𝑡)+𝑐_𝑜^∞₀^∞_𝑡𝑑𝐹_𝑌𝑦𝑑𝐹_𝑇𝑁(𝑡). (12) By using (2) and (8), we have that 𝐸𝑐_𝐿(𝑁)=𝑐_𝑒^∞₀𝐹_𝑌(𝑡)𝑑𝐹_𝑇𝑁(𝑡)+𝑐_𝑜^∞₀𝐹_𝑌(𝑡)𝑑𝐹_𝑇𝑁(𝑡)=𝑐_𝑜+𝜉_𝑐^∞₀𝐹_𝑌(𝑡)𝑑𝐹_𝑇𝑁(𝑡)=𝑐_𝑜+𝑝𝑛_𝑏𝜉_𝑐^∞₀ℎ(𝑡)𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)𝑑𝑡. (13)

3.3. Expected Cost of Production Loss

Let 𝐸[𝑐_𝑃(𝑁)] be the expected cost of production loss per cycle with 𝑁 allowable minor damages in a blades system. We assume the following cost structure (see Figure 3): 𝐸𝑐_𝑃(𝑁)=𝑐_𝑚𝐸𝑁_𝑚(𝑁)+𝑐_𝑀𝐸[DL(𝑁)], (14) where 𝑐_𝑚 is the fixed cost of production loss due to a minor damage, 𝑐_𝑀 is the fixed cost of production loss per unit downtime, 𝐸[𝑁_𝑚(𝑁)] is the expected number of minor damages per cycle, and 𝐸[DL(𝑁)] is the expected downtime within the cycle. In practice, 𝑐_𝑚 can be considered as a fixed penalty cost resulting from a small reduction in efficiency of wind capture and 𝑐_𝑀 is obtained as follows [33]: 𝑐_𝑀=𝑊𝑐_𝐸𝑓, (15) where W is the wind power rating, 𝑐_𝐸 is the cost of energy, and f is the capacity factor of the wind turbine.

Figure 3 

The cost structure for production loss.

As illustrated in Figure 2, the expected number of minor damages per cycle, 𝐸[𝑁_𝑚(𝑁)], can be expressed as 𝐸𝑁_𝑚(𝑁=_ca)_se(a)∞0𝐹_𝑌𝑡+𝐿_𝑜×𝑁+^{𝑡+𝐿𝑜}_𝑡𝑞𝑛_𝑏ℎ(𝑢)𝑑𝑢𝑑𝐹_𝑇𝑁(𝑡)+^case(b)∞₀^𝑡₀^𝑦₀𝑞𝑛_𝑏ℎ(𝑢)𝑑𝑢𝑑𝐹_𝑌𝑦𝑑𝐹_𝑇𝑁(𝑡)+^case(c)∞₀^{𝑡+𝐿𝑜}_𝑡𝑁+^𝑦_𝑡𝑞𝑛_𝑏ℎ(𝑢)𝑑𝑢𝑑𝐹_𝑌(𝑦)𝑑𝐹_𝑇𝑁(𝑡). (16) By using (6) and (8), we have 𝐸𝑁_𝑚(𝑁)=𝑁^∞₀𝐹_𝑌(𝑡)𝑑𝐹_𝑇𝑁(𝑡)+𝑞𝑛_𝑏^∞₀^{𝑡+𝐿𝑜}₀𝐻𝑦𝑑𝐹_𝑌𝑦𝑑𝐹_𝑇𝑁(𝑡)+^∞₀𝐹_𝑌𝑡+𝐿_𝑜𝐻𝑡+𝐿_𝑜𝑑𝐹_𝑇𝑁(𝑡)−^∞₀𝐹_𝑌(𝑡)𝐻(𝑡)𝑑𝐹_𝑇𝑁(𝑡)=𝑁−𝑁𝑝𝑛_𝑏^∞₀ℎ(𝑡)𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)𝑑𝑡+𝑞𝑛_𝑏∫^∞₀𝐹_𝑌𝑡+𝐿_𝑜𝑑𝐹_𝑇𝑁(𝑡)𝑝𝑛_𝑏−^∞₀ℎ(𝑡)𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)𝑑𝑡+𝑝𝑛_𝑏^∞₀ℎ(𝑡)𝐻(𝑡)𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)𝑞𝑑𝑡=𝑁+𝑝𝐹_𝑌𝐿_𝑜−𝑛_𝑏𝑁𝑝+𝑞^∞₀ℎ(𝑡)𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)𝑑𝑡+𝑞𝑛_𝑏^∞₀ℎ𝑡+𝐿_𝑜𝐹_𝑇𝑁(𝑡)𝐹_𝑌𝑡+𝐿_𝑜𝑑𝑡+𝑝𝑛_𝑏^∞₀ℎ(𝑡)𝐻(𝑡)𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)𝑑𝑡. (17)

Because a wind turbine shutdown occurs only in cases (b) and (c), the expected turbine downtime within the cycle, 𝐸[DL(𝑁)], can be expressed as 𝐸[DL(𝑁)]=𝐿_𝑒^∞₀^𝑡₀𝑑𝐹_𝑌𝑦^case(b)𝑑𝐹_𝑇𝑁(𝑡)+^case(c)∞₀^{𝑡+𝐿𝑜}_𝑡(𝑡+𝐿_𝑜−𝑦)𝑑𝐹_𝑌(𝑦)𝑑𝐹_𝑇𝑁(𝑡), (18) and then by using (1), we have 𝐸[DL(𝑁)]=𝐿_𝑒^∞₀𝐹_𝑌(𝑡)𝑑𝐹_𝑇𝑁(𝑡)−𝐿_𝑜^∞₀𝐹_𝑌(𝑡)𝑑𝐹_𝑇𝑁(𝑡)−^∞₀^{𝑡+𝐿𝑜}_𝑡𝐹_𝑌𝑦𝑑𝑦𝑑𝐹_𝑇𝑁(𝑡)=−𝑝𝑛_𝑏𝜉_𝐿^∞₀ℎ(𝑡)𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)+𝑑𝑡_𝐿𝑜0𝐹_𝑌𝑦𝑑𝑦+^∞₀𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)−𝐹_𝑌𝑡+𝐿_𝑜𝑑𝑡. (19)

By substituting (17) and (19) into (14), we have 𝐸𝑐_𝑃(𝑁)=𝑐_𝑚𝑞𝑁+𝑝𝐹_𝑌𝐿_𝑜−𝑛_𝑏𝑁𝑝+𝑞^∞₀ℎ(𝑡)𝐹_𝑇𝑁𝐹_𝑌(𝑡)𝑑𝑡+𝑞𝑛_𝑏^∞₀ℎ𝑡+𝐿_𝑜𝐹_𝑇𝑁(𝑡)𝐹_𝑌𝑡+𝐿_𝑜𝑑𝑡+𝑝𝑛_𝑏^∞₀ℎ(𝑡)𝐻(𝑡)𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)𝑑𝑡+𝑐_𝑀−𝑝𝑛_𝑏𝜉_𝐿^∞₀ℎ(𝑡)𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)𝑑𝑡+^𝐿𝑜₀𝐹_𝑌𝑦𝑑𝑦+^∞₀𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)−𝐹_𝑌𝑡+𝐿_𝑜𝑑𝑡. (20)

3.4. Expected Maintenance Costs

Since a corrective replacement strategy is applied only in cases (b) and (c) and the perfect preventive maintenance strategy is utilized for unfailed blade(s) in all three cases, the total expected maintenance cost per cycle, 𝐸[𝑐_𝑀(𝑁)], is 𝐸𝑐_𝑀(𝑁=)_case(a)𝑛_𝑏𝑐_𝑝^∞₀𝐹_𝑌𝑡+𝐿_𝑜𝑑𝐹_𝑇𝑁(𝑡)+𝑐_𝑟+𝑛_𝑏−1𝑐^case(b)_𝑝^∞₀𝐹_𝑌(𝑡)𝑑𝐹_𝑇𝑁(𝑡)+𝑐_𝑟+(𝑛_𝑏−1)𝑐_𝑝^∞₀𝐹_𝑌(𝑡+^case(c)𝐿_𝑜)−𝐹_𝑌(𝑡)𝑑𝐹_𝑇𝑁(𝑡). (21) By using (8), we have 𝐸𝑐_𝑀(𝑁)=𝑛_𝑏𝑐_𝑝^∞₀𝐹_𝑌𝑡+𝐿_𝑜𝑑𝐹_𝑇𝑁(𝑡)+𝑐_𝑟+𝑛_𝑏−1𝑐_𝑝^∞₀𝐹_𝑌𝑡+𝐿_𝑜𝑑𝐹_𝑇𝑁(𝑡)=𝑛_𝑏𝑐_𝑝+𝑐_𝑟−𝑐_𝑝^∞₀𝐹_𝑌𝑡+𝐿_𝑜𝑑𝐹_𝑇𝑁(𝑡)=𝑛_𝑏𝑐_𝑝+𝑐_𝑟−𝑐_𝑝×𝐹_𝑌𝐿_𝑜+𝑝𝑛_𝑏^∞₀𝐹_𝑇𝑁(𝑡)ℎ𝑡+𝐿_𝑜𝐹_𝑌𝑡+𝐿_𝑜𝑑𝑡. (22)

4. The Model Formulation and Analysis

From the renewal reward theorem (see [34, page 52]), the expected cost rate for an infinite time span is the expected operational cost incurred in a cycle divided by the expected cycle length, that is, Expectedoperationalcostincurredinacycle.Expectedlengthofacycle (23)

The expected operational cost over a cycle is the sum of the expected logistics costs, the expected cost of production loss, and the expected maintenance costs. Thus, the expected cost rate for an NDPM cycle with N allowable minor damages in a blades system, CR(𝑁), can be expressed as (𝑁)=𝐸𝑐CR^𝐿(𝑁)+𝐸𝑐_𝑃(𝑁)+𝐸𝑐_𝑀(𝑁)𝐸[(𝑁,CL)] (24) where 𝐸[CL(𝑁)] is given by (11), 𝐸[𝑐_𝐿(𝑁)] is given by (13), 𝐸[𝑐_𝑃(𝑁)] is given by (20), and 𝐸[𝑐_𝑀(𝑁)] is given by (21). For an infinite-horizon case, under the criterion of the expected long-term cost per unit time, the optimal policy is to determine the value of 𝑁=𝑁^∗ which minimizes CR(𝑁). Therefore, the optimization model can be formulated as follows: minimize𝛿₁₁(𝑁)+^∞₀𝛿₁₂(𝑡,𝑁)𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)+𝑑𝑡_∞0𝛿₁₃(𝑡)𝐹_𝑇𝑁(𝑡)𝐹_𝑌𝑡+𝐿_𝑜×𝐿𝑑𝑡_𝑜+^∞₀𝛿₂₁(𝑡)𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)𝑑𝑡⁻¹, (25) where 𝛿₁₁(𝑁)=𝑐_𝑜+𝑐_𝑚𝑁+𝑐_𝑚1−𝑝𝑝+𝑐_𝑟−𝑐_𝑝𝐹_𝑌𝐿_𝑜+𝑐_𝑀^𝐿𝑜₀𝐹_𝑌𝑦𝑑𝑦+𝑛_𝑏𝑐_𝑝,𝛿₁₂(𝑡,𝑁)=𝑛_𝑏1−𝑐_𝑀𝑝𝜉_𝑐+𝑐_𝑚𝑝1−𝑝𝑛_𝑏𝐻(𝑡)(𝑡)+1−𝑁−1×ℎ+𝑐_𝑀,𝛿₁₃(𝑡)=𝑛_𝑏𝑐_𝑚+𝑝𝑐_𝑟−𝑐_𝑝−𝑐_𝑚ℎ𝑡+𝐿_𝑜−𝑐_𝑀,𝛿₂₁(𝑡)=1−𝑝𝑛_𝑏𝜉_𝐿ℎ(𝑡),𝑝∈[0,1],𝑁∈{1,2,…}. (26)

If a finite integer value of 𝑁^∗ satisfies the inequalities CR(𝑁^∗+1)≥CR(𝑁^∗) and CR(𝑁^∗)<CR(𝑁^∗−1), then 𝑁=𝑁^∗ is the optimal solution. From (25), we can show that CR(𝑁+1)≥CR(𝑁) whenever 𝑄(𝑁)≥0, where 𝑄(𝑁)(𝑁)×𝐿=𝐴_𝑜+^∞₀𝛿₂₁(𝑡)𝐹_𝑇𝑁+1(𝑡)𝐹_𝑌(𝑡)×𝐿𝑑𝑡_𝑜+^∞₀𝛿₂₁(𝑡)𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)𝑑𝑡⁻¹, (27) and 𝐴(𝑁)=𝐿_𝑜𝑐_𝑚+^∞₀𝛿₁₁(𝑁+1)𝛿₂₁(𝑡)−𝐿_𝑜𝛿₁₂(×𝑡,𝑁)𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)−𝑑𝑡_∞0𝛿₁₁(𝑁)𝛿₂₁(𝑡)−𝐿_𝑜𝛿₁₂(×𝑡,𝑁+1)𝐹_𝑇𝑁+1(𝑡)𝐹_𝑌(𝑡)+𝑑𝑡_∞0𝐹_𝑇𝑁+1(𝑡)𝛿₁₂(𝑡,𝑁+1)𝐹_𝑌(𝑡)+𝛿₁₃(𝑡)𝐹_𝑌𝑡+𝐿_𝑜×𝑑𝑡_∞0𝛿₂₁(𝑡)𝐹_𝑇𝑁(𝑡)𝐹_𝑌(𝑡)−𝑑𝑡_∞0𝐹_𝑇𝑁(𝑡)𝛿₁₂(𝑡,𝑁)𝐹_𝑌(𝑡)+𝛿₁₃(𝑡)𝐹_𝑌𝑡+𝐿_𝑜×𝑑𝑡_∞0𝛿₂₁(𝑡)𝐹_𝑇𝑁+1(𝑡)𝐹_𝑌(𝑡)𝑑𝑡. (28) It should be noted that if 𝛿₂₁(𝑡)≥0 or ℎ(𝑡)≤(𝑝𝑛𝜉_𝐿)⁻¹ for all 𝑡≥0, then the condition 𝑄(𝑁)≥0 will become 𝐴(𝑁)≥0.

As the proposed optimization model is fairly complicated (the variable of interest is included in the integrands), obtaining a general analytical solution is probably impossible. We have chosen to use Algorithm 1 to find the optimal solution through a numerical procedure.

Input: 𝑛𝑏,𝑓(⋅),𝑝,𝐿𝑜,𝐿𝑒,𝑐𝑜,𝑐𝑒,𝑐𝑟,𝑐𝑝,𝑐𝑚,𝑊,𝐸,𝑓. Step  1. Set 𝑁∶=1,CR(0)∶=∞. Step  2. Compute 𝐹𝑌(⋅), 𝛿11(1), 𝛿12(⋅,1), 𝛿13(⋅) and 𝛿21(⋅). Step  3. Compute 𝐹𝑇𝑁(⋅), 𝛿11(𝑁+1) and 𝛿12(⋅,𝑁+1). Step  4. Compute 𝑄(𝑁). Step  5. If the condition 𝑄(𝑁)≥0 in (27) is satisfied, then 𝑁∗∶=𝑁, CR(𝑁∗)∶=CR(𝑁) and go to the output, otherwise, let 𝑁=𝑁+1, and go to Step  3. Output: 𝑁∗,CR(𝑁∗) Stop. End.

Algorithm 1 

Find the optimal allowable number of minor damages in a blades system.

In addition, a flow sheet representing the computing procedure is presented in Figure 4.

Figure 4 

The flow sheet for computation of the optimal solution.

5. An Illustrative Case

In order to illustrate the proposed model, we present a case study composed by real data gathered from an offshore wind database. The field failure parameters have been used as in [35], and some real logistic parameters have been gathered from the Opti-Owecs offshore wind farm project, which has also been studied in [36]. Consider a deteriorating wind turbine blades system with 𝑛_𝑏∈{1,2,3} blades (Figure 5), in which the time to damage follows a two-parameter Weibull distribution function; that is, 𝐹(𝑡)=1−exp−(𝜆𝑡)^𝛽,ℎ(𝑡)=𝜆𝛽(𝜆𝑡)^𝛽−1with𝜆>0,𝛽>1, (29) where 𝜆 is the scale parameter and 𝛽 is the shape parameter. Table 1 summarizes the values of the model parameters.

Table 1 

Summary of the parameter values.

(a)

(b)

(c)

Figure 5 

A blades system with (a) one blade, (b) two blades, and (c) three blades.

A MATLAB program for the minimization of the long-run average cost per unit time in (25) has been written. Algorithm 1 is iterated until the optimal value, 𝑁^∗, is found. However, when 𝑝 tends to zero (i.e., when the damages tend to target only minor failures), the value of 𝑁^∗ tends to infinity, which decreases the performance of Algorithm 1.

Table 2 presents the optimal number 𝑁^∗ of allowable minor damages in a blades system and the corresponding long-run expected cost, CR(𝑁^∗), for a range of values of 𝑝∈{0.1,0.2,…,0.9} and 𝑛_𝑏∈{1,2,3}. Table 2 shows that as the probability, 𝑝, of a major damage increases, the optimal value of 𝑁^∗ decreases; however, the optimal long-run expected cost, CR(𝑁^∗), increases. Moreover, as shown in Figure 6, the long-run expected costs per blade in two- and three-bladed systems are very close to each other, but they have a noticeable difference compared with the long-run expected cost in a single-blade system.

Table 2 

𝑁^∗ and CR(𝑁^∗) for 𝑝∈{0.1,0.2,…,0.9} and 𝑛_𝑏∈{1,2,3}.

Figure 6 

CR(𝑁∗)/𝑛𝑏 for 𝑝∈{0.1,0.2,…,0.9} and 𝑛𝑏∈{1,2,3}.

The long-run expected cost, CR(𝑁) for a range of values of 𝑁∈{1,2,…,10}, 𝑝=0.5, and 𝑛_𝑏∈{1,2,3} is shown in Figure 7.

(a)

(b)

(c)

Figure 7 

CR(𝑁) for 𝑁∈{1,2,…,10}, 𝑝=0.5 and 𝑛𝑏∈{1,2,3}.

Since most wind turbines have three blades, we conduct a sensitivity analysis to evaluate the effect of changes in the values of the model parameters on the optimal solution for three-bladed wind turbine systems.

(i) Sensitivity Analysis for 𝜆. The optimal number of allowable minor damages in a three-bladed system and the corresponding long-run expected cost for 𝑝∈{0.1,0.3,…,0.9} and 𝜆∈{0.01,0.03,0.05} are given in Table 3.

Table 3 

Sensitivity analysis for 𝜆.

Table 3 shows that as the scale parameter of Weibull distribution 𝜆 grows (i.e., the mean time to arrive a damage decreases), the optimal value of 𝑁^∗ and the optimal long-run expected cost, CR(𝑁^∗), increase.

(ii) Sensitivity Analysis for 𝑐_𝑚. The optimal number of allowable minor damages in a three-bladed system and the corresponding long-run expected cost for 𝑝∈{0.1,0.3,…,0.9} and 𝑐_𝑚∈{2000,5000,8000} are given in Table 4.

Table 4 

Sensitivity analysis for 𝑐_𝑚.

Table 4 shows that as the fixed cost of production loss due to a minor damage 𝑐_𝑚 becomes larger, the optimal value of 𝑁^∗ decreases; however, the optimal long-run expected cost, CR(𝑁^∗), increases.

(iii) Sensitivity Analysis for 𝑐_𝑒. The optimal number of allowable minor damages in a blades system and the corresponding long-run expected cost for 𝑝∈{0.1,0.3,…,0.9} and 𝑐_𝑚∈{25000,30000,35000} are given in Table 5.

Table 5 

Sensitivity analysis for 𝑐_𝑒.

Table 5 shows that the expedited logistics cost does not have any significant effect on the optimal value of 𝑁^∗; however, as the expedited logistics cost reduces to the ordinary logistics cost (i.e., the expedited cost term 𝜉_𝑐 becomes zero), the optimal long-run expected cost, CR(𝑁^∗), also decreases.

(iv) Sensitivity Analysis for 𝐿_𝑒. The optimal number of allowable minor damages in a three-bladed system and the corresponding long-run expected cost for 𝑝∈{0.1,0.3,…,0.9} and 𝐿_𝑒∈{0.5,1.0,1.5} are given in Table 6.

Table 6 

Sensitivity analysis for 𝐿_𝑒.

Table 6 shows that as the expedited logistics lead time gets closer to the ordinary logistics lead time (or, the expedited lead time term, 𝜉_𝐿, becomes smaller), the optimal value 𝑁^∗ and the optimal long-run expected cost, CR(𝑁^∗) increase.

6. Conclusions and Topics for Future Research

In this paper, a cost-rate optimization model is developed in order to determine the optimal number of minor damages allowed to occur in a blades system before sending a maintenance team to the offshore platform in order to perform preventive maintenance actions. There is a wide scope for future research in the area of maintenance optimization for offshore wind turbine blades. Some of the possible extensions are (a) investigating the optimal solution for a case that the probability 𝑝 is a random variable that depends on the weather conditions; (b) formulating and analyzing the model when the blade damages form a non-Poisson arrival stream; (c) developing an age-based preventive maintenance strategy for offshore blades systems with 𝑛≥2 blades, in which in the event of a blade failure a PM action will be performed on the un-failed blade if its operational age exceeds a certain threshold 𝑎>0; and finally (d) providing a cost comparison of our proposed maintenance strategy with other common strategies (such as age-based PM and opportunistic maintenance).

We have worked on some of these extensions and our findings will be reported in the near future.

Notations

𝑛𝑏:	The number of blades in a wind turbine system; 𝑛𝑏∈{1,2,…}
𝑇:	The lifetime of a blade (day)
𝑓(⋅)[𝐹(⋅)]:	The probability density [cumulative distribution] function of 𝑇
ℎ(⋅)[𝐻(⋅)]:	The failure rate [cumulative failure rate] function of 𝑇
𝑝:	Pr{a certain damage is a major damage}; 𝑝∈[0,1]
𝑞:	Pr{ a certain damage is a minor damage}; 𝑞=1−𝑝
𝑖:	The index number of minor damages in the blades system; 𝑖∈{1,2,…}
𝑁:	The allowable number of minor damages before transporting a maintenance team to offshore platform [decision variable]
𝑇𝑖:	The waiting time until the 𝑖th minor damage occurs (day)
𝑌:	The time to the first major damage (day)
𝐹𝑇𝑖(⋅)[𝐹𝑇𝑖(⋅)]:	The cumulative distribution [survival] function of 𝑇𝑖
𝐹𝑌(⋅)[𝐹𝑌(⋅)]:	The cumulative distribution [survival] function of 𝑌
𝐿𝑜[𝐿𝑒]:	The ordinary [expedited] logistics lead time (day)
𝑐𝑜[𝑐𝑒]:	The ordinary [expedited] logistics cost (€)
𝜉𝐿[𝜉𝑐]:	The expedited lead time (day) [cost (€)] term
𝑐𝑟:	The expected replacement cost of a blade (€)
𝑐𝑝:	The cost of performing a perfect PM on an un-failed blade (€)
𝑐𝑚:	The fixed cost of production loss due to a minor damage (€)
𝑐𝑀:	The fixed cost of production loss per unit downtime (€/day)
𝑊:	The wind turbine power rating (MW)
𝑐𝐸:	The cost of energy (€/MW)
𝑓:	The capacity factor of each wind turbine
𝐸[𝐶𝐿(𝑁)]:	The expected length of an NDPM cycle (day)
𝐸[𝐷𝐿(𝑁)]:	The expected downtime within the cycle (day)
𝐸[𝑐𝐿(𝑁)]:	The expected logistics cost per cycle (€)
𝐸[𝑐𝑃(𝑁)]:	The expected cost of production loss per cycle (€)
𝐸[𝑁𝑚(𝑁)]:	The total expected number of minor damages within the cycle (€)
𝐸[𝑐𝑀(𝑁)]:	The total expected maintenance cost per cycle (€)
𝐶𝑅(𝑁):	The long-run expected cost per unit time (€/day).

Acknowledgments

The authors would like to thank the Editor and two anonymous reviewers for several comments that made it possible to substantially improve the presentation of the paper. This research was supported by the Swedish Energy Agency through Chalmers Energy Area of Advance and Grant no. 2010-004592.