# Optimal Maintenance Schedule for a Wind Power Turbine with Aging Components
## 3 Optimal Maintenance Algorithm in the One-Component Case  
In this section, we propose an optimization algorithm for PM scheduling during a discrete-time interval  
\[[s,s+1,\ldots,T-1,T]\]  
assuming that at time \(s\) the component in use is of age \(a\). If \(a=0\), we say that at the beginning of the planning period, the component was as good as new. This will allow us to find an optimal time \(t_{(s,a)}\) for the next PM by minimizing the total maintenance cost during the whole planning period.  
Definition 1: For the one-component model with a planning period \([s,T]\), we call a _PM plan_ any vector  
\[\mathbf{x}_{s}=(x_{s+1},\ldots,x_{T+1})\]  
with binary components satisfying a linear constraint  
\[\sum_{t=s+1}^{T+1}x_{t}=1,\quad x_{s+1}\in\{0,1\},\ldots,x_{T+1}\in\{0,1\}. \tag{2}\]  
For the given planning period \([s,T]\), we define the total maintenance cost \(Q_{(s,a)}(t,u)\) as a function of the planning time \(t\) for the next PM and the failure time \(u\) of the component in use. For \(t\in[s+1,T]\), we put  
\[Q_{(s,a)}(t,u)=\left\{\begin{array}{ll}g+(T-u)c,&\text{if }u\leq t\\ h+(t-s+a)m+(T-t)c,&\text{if }u>t\end{array}\right. \tag{3}\]  
or using indicator functions,  
\[Q_{(s,a)}(t,u)=[g+(T-u)c]1_{\{u\leq t\}}+[h+(t-s+a)m+(T-t)c]1_{\{u>t\}}.\]  
We also put  
\[Q_{(s,a)}(T+1,u)=[g+(T-u)c]1_{\{u\leq T\}}.\]  
This formula recognizes two possible outcomes:  
If \(\{u\leq t\}\), then the breakdown happens before the planned PM time, and the expected total maintenance cost is estimated to be \(g+(T-u)c\), with \(c\) given by (1),  
If \(\{u\geq t+1\}\), so that there is no breakdown before the planned PM time, then the expected total maintenance cost is estimated to be  
\[h+(t-s+a)m+(T-t)c.\]  
If at the starting time \(s\) of the planning period, the component in use has age \(a\geq 1\), and we will use a special notation \(s+L_{a}\) for the first failure time, where  
\[L_{a}\overset{d}{=}\{L-a|L>a\},\]  
is defined in terms of the full life length \(L\) of a genetic component. If \(L\) has a discrete \(\mathrm{W}(\theta,\beta)\) distribution, then  
\[\mathrm{P}(L_{a}>t)=\exp\Bigl{\{}\theta\bigl{(}a^{\beta}-(a+t)^{\beta}\bigr{)} \Bigr{\}},\quad t\geq 0.\]  
Putting \(u=s+L_{a}\) into the formula for \(Q_{(s,a)}(t,u)\), we arrive at a random variable  
\[F_{(s,a)}(\mathbf{x}_{s})=\sum_{t=s+1}^{T+1}Q_{(s,a)}(t,s+L_{a})x_{t}\]that gives us the total maintenance cost of the PM plan \(x_{s}\). Averaging over \(L_{a}\), we obtain the objective function  
\[f_{(s,a)}(x_{s})=\mathrm{E}(F_{(s,a)}(x_{s}))\]  
as the expected maintenance cost of the PM plans \(x_{s}\). Now we are ready to define the optimal maintenance plan as the solution to the following optimization problem:  
minimize \[f_{(s,a)}(x_{s})\] subject to  linear constraints (2).  
Let \(t_{(s,a)}\) be the PM time proposed by the solution of the minimization problem above, and notice that  
\[t_{(0,0)}=\mathrm{argmin}\ (q_{t}).\]  
The following proposition states a consistency property for the set of optimal times \(t_{(s,a)}\), providing intuitive support for the suggested approach.  
Proposition 2: _Suppose for some positive \(\delta\),_  
\[t_{(s,a)}>s+\delta.\]  
_If \(t_{(s,a)}\leq T\), then_  
\[t_{(s+\delta,a+\delta)}=t_{(s,a)}.\]