Wind energy is one of the lowest-priced renewable energy technologies available today; see . A large part of the total cost associated with wind turbines is due to operation and maintenance, amounting to 34 % for the fixed-bottom offshore wind turbines, according to . To reduce the maintenance cost, one can improve the design of the components, making them more reliable. One can also reduce the maintenance costs by means of an improved scheduling of the maintenance activities for still-functioning components depending on their current age. The latter task is the main concern of this paper, which proposes an optimization model for preventive maintenance (PM) scheduling of a wind turbine or even a farm of wind turbines. Notice that, in this paper, by PM activities we do not mean the practice of regular inspection of the component's condition. Our concern is the optimal planning of preventive replacements of the components based on their current age.

Typically, a maintenance model distinguishes between a corrective maintenance (CM) event, when a component should be attended after it breaks down, and a PM event, when one or several older components are renewed before they break down, see the recent survey . An optimal PM scheduling is aimed at reducing the lost production due to the downtime caused by CM events.

There is a multitude of papers devoted to the optimal PM scheduling for multi-component systems; see . The article proposes a joint optimization of the maintenance policy and the inspection interval for a multi-unit series system with economic dependence. It develops an algorithm aiming at a maintenance policy for a multi-component system minimizing the maintenance cost, under the assumption that one unit of the system is subject to condition monitoring, while for the other units only the age information is available. develop a method to quantify the uncertainty of the remaining life length, resulting in an effective condition-based maintenance approach to optimal scheduling.

The article looks at opportunistic maintenance (OM), which is a special kind of a PM activity occurring at the time of a CM replacement: replacing still-functioning components together with the broken one may save some mobilization costs. OM activities are shown to be extremely beneficial for the offshore wind farms, due to the large mobilization costs.

In , optimization models are developed to determine the optimal PM schedules in repairable and maintainable systems. They show that if mobilization costs are the same irrespective of the number of components to be attended, then multiple simultaneous PM activities become cost-effective. However, their optimization models are nonlinear and non-convex, which makes them computationally hard to solve; see Sect. 1.3 in .

The preventive maintenance scheduling problem with interval costs (PMSPIC) model from was the major inspiration for this work. The main feature of the PMSPIC model is the idea of interval cost: given a time interval between two consecutive PM activities, the expected maintenance cost should take into account eventual breakdowns of components during this time interval. The PMSPIC model has a long computational time, which motivated us to build a new optimization model for PM scheduling of a wind turbine.

In this paper, we build on the state of the art with a new algorithm, NextPM. Given the current ages of the key components of the system, NextPM computes the best time to perform the next maintenance activity and determines which components should be replaced at that time. The algorithm can be solved in 1 s and thus has a potential for being used as a key module in a maintenance scheduling app for wind turbines.

The paper is organized as follows. Section  presents a novel optimization model for maintenance scheduling of a multi-component system. In the context of wind farm maintenance, each wind turbine is viewed here as a system comprising multiple components such as the gearbox, power generator, rotor, and main bearing. Whenever one of the components is broken, the whole system stops functioning. After the broken component is replaced by a new one, the system resumes its function. It is assumed that at time 0 all components of the system are new and that the total lifespan of the system is T units of time. The model has a discrete time setting t=0, 1, …, T, where the unit of time can be either a day, a month, or a year, depending on a concrete application; see for a maintenance scheduling with only 1 d ahead. In the same Sect. , the main result of the paper is summarized as Algorithm 1, aiming at an optimal PM schedule for the time period [s, T] with an arbitrary starting time s∈[0,T-1]. Figure  gives a non-technical description of the algorithm.

The key ingredient of Algorithm 1, the NextPM optimization model, is carefully described in Sect. . Section  contains several numerical studies that demonstrate the flexibility of our approach, its accuracy, and computational effectiveness. Finally, Sect.  presents the main conclusions of the paper.

Consider a system composed of n components characterized by different life length distributions. For the component j, it is assumed that its total life length Lj is a random variable having a Weibull distribution with parameters (αj, βj), so that the corresponding survival function is 1PLj>t=e-tαjβj,t≥0,j=1,…,n; see concerning the use of the Weibull distribution for the modeling of multi-component systems. The means and variances of the component life lengths are the following functions of the Weibull parameters: 2μj=αjΓ1+1βj,σj2=αj2Γ1+2βj-μj2,j=1,…,n. Besides the Weibull parameters (αj, βj), j=1, …, n, our optimization model requires the following parameters associated with various maintenance costs: dt is the time-dependent mobilization cost for either a PM or CM activity, t=0, …, T, bj is the CM cost of the component j=1, …, n, and cj is the PM cost of the component j=1, …, n. The full set of the model parameters {d1, …, dT,(α1,β1,b1,c1),…,(αn,βn,bn,cn),λ} includes an extra parameter λ introduced in Sect.  by Eq. ().

Suppose that the multi-component system is observed at some time s∈[0,T-1], and the latest maintenance times of components j=1, …, n are known to be tj∈[0,s], so that at the time s, the n components have the effective ages (s-t1, …, s-tn). The NextPM optimization model described in Sect.  has the input (t1, …, tn, s, r), where r∈[s+1,T] is the end of the current planning period. The output of NextPM is a PM plan specifying the optimal time τ∈[s+1,r+1] of the next PM event, as well as the set of components P⊂{1,…,n} which should be maintained at the time τ. In particular, the output τ=r+1 means that no PM activity should be scheduled during the planning period [s+1,r], implying that the set P is empty.

The NextPM model is the key module of the following Algorithm 1 aiming at the long-term PM scheduling until the end time T, at which the whole system is expected to be dismantled; see Fig.  for a flow chart illustrating the major steps of Algorithm 1.

Algorithm 1 relies on a rescheduling procedure, where each NextPM step covering r-s units of the planning time is accompanied by a NextOM module. The latter is a modification of the NextPM step (see Sect. ), which addresses the possibility of a component failure before the planned PM, followed by an OM activity.

This section sets up the optimization model NextPM, which is the key ingredient of Algorithm 1 summarized in Sect. . The optimization model PMSPIC of was a major motivation for NextPM, and we start by comparing these two approaches using Figs.  and , which illustrate two different definitions of the objective functions for two optimization models in question.

The main difference between PMSPIC and NextPM model is that while PMSPIC generates a maintenance plan for the whole lifetime of the wind turbine, the NextPM model produces an optimal schedule only for the next PM activity. To this end, PMSPIC looks into the total maintenance cost, while NextPM aims at minimizing the time average maintenance cost.

The three case studies analyzed in this section treat a wind turbine as a system represented by four components. They are all based on the parameter values taken from the paper (see Table ), where the cost unit is USD 1000 and the time unit is 1 month. The lifetime of the wind turbine is assumed to be 30 years. This implies the parameter value T=360 months. As to other model parameters, it is assumed that

s=0, which implies that all four components initially are as good as new;

Comparing the characteristics of four wind turbine components shown in Table , it is important to observe a strict ordering from the perspective of the associated PM costs. For example, consider components 1 and 2. The rightmost column says that the expected life length of the gearbox is smaller by 18.5 months, which on its own suggests a higher rate of replacements for the component 1. But even the other two parameters, CM cost and PM cost, are ordered in a way b1>b2, c1>c2, which is favorable for more frequent replacements of component 1 compared to component 2.

All computational tests are performed on an Intel 2.40 GHz dual-core Windows PC with 16 GB RAM. The mathematical optimization models are implemented in AMPL IDE (version 3.5); the model components (Eqs.  and ) are calculated by MATLAB (version R2015b), and then the optimization problems are solved using CPLEX (version 12.8).

This article introduces a new NextPM optimization model aiming at PM scheduling for a wind turbine viewed as a multi-component system. Which of the components should undergo PM replacements first is decided based on the information on the component ages. Compared to the PMSPIC model from that generates a maintenance plan for the whole lifetime of the wind turbine, the NextPM model produces an optimal schedule only for the next PM activity. By focusing on a shorter planning horizon and implementing a different model structure, we succeeded in substantially reducing the computational time.

NextPM is tested with three case studies based on the data for four components of the wind turbine taken from . Under the seasonal variation, our results show that PM activities should be always scheduled in the summertime. This is due to the lower mobilization costs during the summer months. When the NextPM model is compared to the pure CM strategy, it is found that around 35 % of the maintenance costs can be saved by applying the NextPM model. The third case study compares the performances of NextPM and PMSPIC algorithms, demonstrating the accuracy of the NextPM model despite being much less complex than PMSPIC.

In this paper our NextPM model is applied to a system of four components belonging to a single wind turbine. However, we claim that our approach can handle the case of, say, 10 turbines with 80 components in total (the computational time required by our algorithm grows linearly with the increased number of components, while PMSPIC's computational time grows exponentially fast).

The notable limitation of our setting is that it neglects such important maintenance activities as inspections and minor and major repairs. By considering full replacements as the only kind of CM and PM activities allowed in the model, we were able to tame the mathematical challenge of the problem in hand. Still, even within this simplified model framework, our computational analysis may bring useful insights of more efficient PM planning, depending on a few key parameters of a concrete wind farm. Our results should be viewed as a first promising step towards a much more sophisticated mathematical optimization model that would take into account available condition monitoring data and even recognize the difference in the failure rates for minor repairs, major repairs, and component replacements.