# Optimal Maintenance Schedule for a Wind Power Turbine with Aging Components
## 2 A Single-Component Model  
We start our exposition of the model by turning to the one-component case. Without planned PM activities, the maintenance cost flow is described by line 1 of Figure 1.  
Here, the consecutive failure times (depicted by crosses) form a renewal process with independent inter-arrival times \(L_{1},L_{2},\ldots\) each having the same Weibull distribution \(W(\theta,\beta)\). After each failure, the broken component is replaced by a new one, and the  
Figure 1: Renewal-reward model for computing \(c\) for the one component model. incurred replacement cost is \(g\). According to the classical renewal-reward theorem, the long-term time-average reward (the maintenance cost in the current setting) is \(c=\frac{g}{E(L)}\), where \(E(L)\) is the expected value of the random variable \(L\). Lines 2-5 in Figure 1 introduce the renewal-reward model used to compute the time-average maintenance cost \(q_{t}\) based on the PM planning strategy: plan the next PM replacement at time \(t\) after each replacement. Line 2 describes a scenario where the first failure occurs after the planned PM time and the first replacement is performed at time \(t\), so the failure is avoided. The corresponding PM replacement cost is assumed to be a linear function \(h+tm\) of the component's age at the time of a PM replacement. According to line 3, after the PM replacement was performed at the cost \(h+tm\), the next failure occurred before the next PM planned time \(2t\). As a consequence, the second replacement was performed in a CM regime incurring the cost \(g\); see line 4. Line 4 says that following the second replacement, the next PM is planned at time \(2t+L_{1}\), that is, at time \(t\) after the failure time \(t+L_{1}\). According to the placement of a cross on line 4, we see on line 5 that the third replacement is performed in the PM regime and the next PM time is scheduled at time \(3t+L_{1}\). Recall that a random variable \(L\) has a Weibull distribution \(\mathrm{W}(\theta,\beta)\) if  
\[\mathrm{P}(L>t)=e^{-\theta t^{\beta}},\quad t\geq 0,\]  
where \(\theta>0\) is a scale parameter and \(\beta>1\) is a shape parameter of \(\mathrm{W}(\theta,\beta)\). The mean value of \(L\)  
\[\mu=\theta^{-1/\beta}\Gamma(1+\frac{1}{\beta})\]  
is computed using the gamma function. The age dependence in the continuous time setting is best viewed in terms of the hazard function \(\theta\theta t^{\beta-1}\), telling how fast the failure rate increases with the component's age \(t\). In this paper, we use the discrete-time version of the Weibull distribution  
\[\mathrm{P}(L>t)=e^{-\theta t^{\beta}},\quad t=0,1,2,\ldots,\]  
so that  
\[\mathrm{P}(L=t)=e^{-\theta(t-1)^{\beta}}-e^{-\theta t^{\beta}},\quad t=1,2,\ldots\]  
Using these expressions and the renewal-reward theorem, we arrive at the following explicit formula involving the key parameters of the one-component model \((\theta,\beta,g,h,m)\). The proof of this result (as well as the forthcoming Proposition 2) is given at the end of the paper. Proposition 1: _Think of an infinite planning horizon and a recurrent strategy of planning the next PM at time \(t\) after each replacement event. Then, the time-average maintenance cost is the following function_  
\[q_{t}=\frac{(1-e^{-\theta t^{\beta}})g+e^{-\theta t^{\beta}}(h+mt)}{\sum_{k=1 }^{t}(e^{-\theta(k-1)^{\beta}}-e^{-\theta k^{\beta}})k+e^{-\theta t^{\beta}}t}\]  
_of the planning time \(t\)._  
As a check, one can send \(t\) to infinity and observe that this results in \(q_{t}\to\frac{g}{E(L)}\), the average maintenance cost in the absence of PM activities. Minimizing the function \(q_{t}\) over the possible planning times \(t\) produces a constant  
\[c=\min_{t\geq 1}\frac{(1-e^{-\theta t^{\beta}})g+e^{-\theta t^{\beta}}(h+mt)}{ \sum_{k=1}^{t}(e^{-\theta(k-1)^{\beta}}-e^{-\theta k^{\beta}})k+e^{-\theta t^{ \beta}}t}, \tag{1}\]  
which we will treat as the long-term maintenance cost per unit of time for the component in question.