# Optimal Maintenance Schedule for a Wind Power Turbine with Aging Components
## 1 Introduction  
Wind power technology is one of the most efficient sources of renewable energy available today [1]. A large part of the wind energy cost is due to the cost of maintaining wind power equipment (e.g., 25-30% of the total lifecycle costs for offshore wind farms) [2]. The corrective maintenance (CM) of a turbine component, performed after it breaks down, is usually more expensive than a preventive maintenance (PM) event, as some of the equipment is replaced in a planned manner. However, if PM activities are scheduled too frequently, the maintenance costs become unreasonably high, which entails the necessity of a maintenance schedule minimizing the expected replacement, logistic, and downtime costs. There is a broad body of literature devoted to various optimization models of maintenance scheduling. Here, we name just a few of the relevant papers that influenced our own approach. In the article by [PERSON] and [PERSON] [3], some general PM optimization models are presented. The effect of a PM action has been classified into three categories: failure rate reduction, the decrease in the deterioration speed, and age reduction. The paper concludes that it can be profitable to perform PM. Compared to our setting, this paper considers a one-component system and relies on a different survival function. The article [4] looks at opportunistic maintenance, which is a special kind of preventive maintenance. When one component breaks down, the maintenance personnel attending tothe broken component might as well maintain other aged components to save some logistics costs. This is extremely beneficial for offshore wind farms due to the large set-up costs. In [5], optimization algorithms are developed to determine optimal PM schedules in repairable and maintainable systems. It was demonstrated that higher set-up costs make simultaneous PM activities advantageous. However, the suggested models are nonlinear, which means they are computationally hard to solve. Our paper suggests a new optimization algorithm aiming at a PM plan \((x_{s},y_{s})\)--see Section 6--which proposes the next time for the replacement of one or several components at a minimal maintenance cost \(f_{(s,\mathbf{a})}(\mathbf{y}_{s})\) estimated for the whole planning time period \([s,T]\), subject to certain linear constraints on \((x_{s},y_{s})\). Here, \(T\) is the life length of the wind turbine, \(s\in[0,T)\) is the starting time of the planning period, and \(\mathbf{a}=(a^{1},\ldots,a^{n})\) is the vector of component ages at time \(s\). The age dynamics of our algorithm are based on the discrete Weibull distribution; see [6]. To account for the component deterioration due to aging, we assume that the PM cost increases as a linear function of the component's age at the moment of replacement. Previously, the PM replacement cost was usually treated as an age-independent constant; see for example [7; 8]. Our approach, focusing on age dependence, should be compared to that of [9], where the maintenance cost is assumed to depend on the total damage. Another relevant paper [10] quantifies the maintenance cost of a component using the reliability distribution. Our definition of the objective function requires an application of the classical renewal-reward theorem; see for example [11]. This approach is quite straightforward in the one-component case, as explained in Section 2. In the multiple-component case, however, when the true renewal events (when all components are replaced simultaneously) are rarely encountered, our approach requires some kind of pseudo-renewal events. To this end, in Section 5, we introduce the key idea of the virtual maintenance replacement cost \(b_{(t,a)}\) for a generic component with age \(a\) at time \(t\). Section 7 presents two sensitivity analyses and two case studies treating a four-component model of the wind power turbine. In particular, it was demonstrated that under the additional assumptions of [7], the new model produces similar results to those obtained in [7] but at a higher computational speed.