# Optimal Maintenance Schedule for a Wind Power Turbine with Aging Components
## 4 Multiple-Component Model  
In this section, we expand our one-component model to the cases with \(n\geq 2\) components. We now think of a wind turbine consisting of \(n\) components with the \(j\)-th component having a life length \(L^{j}\) distributed according to a Weibull distribution \(\mathrm{W}(\theta^{j},\beta^{j})\), where parameters \((\theta^{j},\beta^{j})\) may differ for different \(j=1,\ldots,n\). We will assume that components may require different replacement costs:  
\[g_{0}=\text{ the shared logistic and downtime costs associated with a CM activity,}\] \[g^{j}=\text{ the component-specific CM cost,}\] \[h_{0}=\text{ the fixed logistic cost plus the downtime cost during a PM activity,}\] \[h^{j}+tm^{j}=\text{ the component-specific PM replacement cost for $j$-th component at age $t$.}\]  
To be able to update the Formula (1) of the minimal time-average maintenance cost, we would need to determine the times of total renewal of the system that are not readily available in the multiple component setting. For an illustration, turn to Figure 2 dealing with the case of \(n=2\) components. Figure 2: Renewal-reward model for computing \(c\) for the two-component model. Even in the absence of PM activities--see line 1--it is clear that the classical renewal-reward theorem is not directly applicable. Our solution to this problem is to treat each replacement event as total renewal events, sometimes by performing opportunistic maintenance and, in some cases, by increasing the cost function to reflect the future additional age-related replacement costs. This idea is illustrated by lines 2-5 in Figure 2.  
Consider a strategy in which the next PM activity is planned at time \(t\) after each replacement. Line 2 depicts a case in which the second component is broken first and before the time \(t\) of the next PM. As shown by line 3, both components are replaced at the failure time \(L_{1}\), and the next PM is planned at time \(L_{1}+t\). In this way, the time \(L_{1}\) can be viewed as the total renewal time of the two-component system. The incurred replacement cost is  
\[g_{0}+g^{2}+h^{1}+L_{1}m^{1}.\]  
Since, according to the line 3, both components break down after the planned time \(t+L_{1}\) for the next PM, both components are replaced in the PM regime so that the incurred replacement cost is  
\[h_{0}+h^{1}+tm^{1}+h^{2}+tm^{2}.\]  
Continuing in this way by replacing both components at each maintenance event, we arrive at a renewal-reward process described in the general setting as follows. Assume that we start at time \(0\) with \(n\) new components and denote by  
\[L=\min(L^{1},\ldots,L^{n})\]  
the time of the first failure. With independence, we have  
\[\mathrm{P}(L>t)=\mathrm{P}(L^{1}>t)\cdots\mathrm{P}(L^{n}>t)=\exp\{-\theta^{1 }\ell^{\beta^{1}}-\ldots-\theta^{n}t^{\beta^{n}}\}.\]  
Thus, if we plan our next PM at time \(t\) after each maintenance replacement, the first renewal time is \(X=X(t)\) with  
\[g^{j}=\text{ the component-specific CM cost,}\]0  
and the corresponding reward value \(\tilde{R}=\tilde{R}(t)\) is computed as  
\[g^{j}=\text{ the component-specific CM cost,}\]1  
where  
\[g^{j}=\text{ the component-specific CM cost,}\]2  
is the subset of components that would be replaced at the first failure. The renewal-reward theorem allows us to express the time-average maintenance cost \(\frac{\mathrm{E}(\tilde{R})}{\mathrm{E}(X)}\) as an explicit function of the planning time \(t\). However, replacing all of the components at each maintenance event irrespective of the ages of the components in use is definitely a suboptimal strategy. If, at a maintenance event, some component \(j\) is in working condition and its current age \(a\) is rather small, then it might be more beneficial to let it continue working. To still be able to use the renewal argument in such a case, we introduce the idea of a virtual replacement.