## 4 Mathematical Programming Relaxations
### Relaxing Only Constraints CT6 and CT12  
Relaxing CT6 and CT12 constraints, an MILP formulation can be designed with binary variables only for the outage decisions, similar to [26]. The binaries \(x_{i,k,w}\) in [26] are equal to 1 if and only if the beginning week for cycle \((i,k)\) is exactly \(w\). Similar to [20], we define the binaries \(d_{i,k,w}\) as \"step variables\" with \(d_{i,k,w}=1\) if and only if the outage beginning week for the cycle \(k\) of the unit \(i\) is before the week \(w\). We extend the notations with \(d_{i,k,w}=0\) for \(k>K\),\(d_{i,0,w}=1\) for \(w<0\), \(d_{i,k,w}=0\) for \(w<0\) and \(k>0\). CT13 constraints, imposing that maintenance operations \((i,k)\) begin between weeks \(\textbf{T}_{i,k}\) and \(\textbf{T}_{i,k}\), reduce the definition of variables with \(d_{i,k,w}=0\) for \(w<\textbf{T}_{i,k}\) and \(d_{i,k,w}=1\) for \(w\geqslant\textbf{T}_{i,k}\). The other variables have a continuous domain: refueling quantities \(r_{i,k}\) for each outage \((i,k)\), T2 power productions \(p_{i,k,t}\), fuel stocks at the beginning of campaign \((i,k)\) (resp at the end) \(x_{i,k}^{init}\), \(x_{i,k}^{fin}\), T1 power productions \(p_{j,k,t}^{T1}\), and fuel stock \(x_{i,s}^{end}\) at the end of the optimizing horizon. We note that T2 power productions \(p_{i,k,s,t}^{T2}\) are duplicated for all cycle \(k\) to have a linear model, \(p_{i,k,s,t}^{T2}=0\) if \(t\) is not included in the production cycle \(k\). These variables are gathered in Table 4. \begin{table}
\begin{tabular}{l l} \hline \(d_{i,k,w}\in\{0,1\}\) & Binaries to indicate the beginning week for maintenance checks \(k\) of T2 unit \(i\). \\ \(p_{i,k,s}^{T2}\geqslant 0\) & Power production of T2 unit \(i\) at cycle \(k\) at \(t\) for scenario \(s\), 0 if \(t\) is not in cycle \(k\). \\ \(p_{j,s,t}^{T1}\geqslant 0\) & Power production of T1 unit \(j\) at time step \(t\) for scenario \(s\). \\ \(r_{i,k}\geqslant 0\) & Refueling quantity for outage \(k\) of T2 unit \(i\). \\ \(x_{i,s}^{fin}\geqslant 0\) & Fuel stock of T2 unit \(i\) at the end of the optimizing horizon at scenario \(s\). \\ \(x_{i,k,s}^{fin}\geqslant 0\) & Fuel level at the beginning of production cycle \(k\) of T2 unit \(i\). \\ \(x_{i,k,s}^{fin}\geqslant 0\) & Fuel level before the refueling \(k+1\) of T2 unit \(i\), after production cycle \(k\). \\ \hline \end{tabular}
\end{table} 
Table 4: Definition of the variables for the MILP formulation relaxing only constraints CT6 and CT12. for a cycle \(k\) only if the cycle is finished at the end of the optimizing horizon, i.e., if \(d_{i,k+1,W}=1\), which enforces having disjunctive constraints where case \(d_{i,k+1,W}=0\) implies trivial constraints thanks to (12). Constraints (14) are linearization constraints to enforce \(x^{end}_{i,s}\) to be the fuel stock at the end of the time horizon. \(x^{end}_{i,s}\) is indeed the \(x^{fin}_{i,k,s}\) such that \(d_{i,k,W}=1\) and \(d_{i,k+1,W}=0\), for the disjunctive constraints (14) that write trivial constraints in the other cases thanks to (12), defining \(S_{i}=\max_{k}\mathbf{Smax}_{i,k}\). Constraints (15)-(22) have a common shape as scheduling and resource constraints, as presented in [24]. The common format for constraints (15)-(19) may be seen as clique cuts, as written in [25; 26]. Equivalently, using an additional and unitary fictive resource for each CT14-CT18 constraint that is consumed in the specific spacing/overlapping phases among outages, CT14-CT18 are written (15)-(19) as noticed by [37]. We note that the constraints (15) and (16) are not activated if \(\mathbf{Da}_{i,k}+\mathbf{S}_{c}<0\) for a given \(i,k,c\), so that the generic equation is also valid using \(\max(0,\mathbf{Da}_{i,k}+\mathbf{S}_{c})\) operations. CT19-CT21 are also resource constraints, with a non unitary quantity, which is written similarly in Equations (20)-(22). \(LB\) is a dual bound for the original problem, relaxing only constraints CT6 and CT12, requiring less binary variables than the formulation of [37].