## 2 Problem Statement
### Description of the Constraints  
There are 21 sets of constraints in the Challenge, numbered from CT1 to CT21 in the specification [14]. CT6 and CT12 are relaxed in our study; this is possible as we compute dual bounds relaxing these two sets of constraints. Table 3 describes precisely the constraints.  
Constraints CT1 to CT12 concern the production and fuel constraints in the operational level whereas CT13 to CT21 are scheduling constraints in the strategic level. Constraints CT1 couple the production of the T1 and T2 plants with power demands. Constraints CT2 and CT3 define the bounds for T1 productions. Constraints CT4 to CT5 describe the production possibilities for T2 units depending on the fuel stock with outages during the maintenance periods. Constraints CT7 to CT11 involve fuel constraints for T2 units: bounds on fuel stocks, on fuel refueling, and relations between remaining fuel stocks and production/refueling operations.  
Constraints CT13 impose Time Windows (TW) for the beginning weeks of outages, and it can impose some maintenance dates. With CT13 constraints, the maintenance checks follow the order of set \(k\in\mathcal{K}\) without skipping maintenance checks: if an outage \(k+1\) is processed in the time horizon, it must follow the production cycle \(k\). Constraints CT14 to CT21 may be unified in a common format with generic resources. These constraints express minimal spacing/ maximal overlapping constraints among outages, and minimum spacing constraint between coupling/decoupling dates, resource constraints for maintenance using specific tools or manpower skills, and limitations of simultaneous outages with a maximal number of simultaneous outages or maximal T2 power offline.  
\begin{table}
\begin{tabular}{l} \hline \hline
**CT1:** for all production time step \(t\in\mathcal{T}\) and all scenario \(s\in\mathcal{S}\), the total production of T1 and T2 power plants \\ must equalize the demands \(\mathbf{D}_{s,t}\). \\ \hline
**CT2:** for all production time step \(t\in\mathcal{T}\) and all scenario \(s\in\mathcal{S}\), the production domain of T1 plant \(j\in\mathcal{J}\) \\ describes exactly the continuous domain \([\mathbf{Pmin}_{j,s,t},\mathbf{Pmax}_{j,s,t}]\) \\ \hline
**CT3:** the productions of T2 power plants are null during the \(\mathbf{Da}_{i,k}\) weeks of outage \((i,k)\). \\ \hline
**CT4-CT5:** for each period \(t\in\mathcal{T}\) and each scenario \(s\in\mathcal{S}\), the production domain of T1 plant \(j\in\mathcal{J}\) describes exactly the continuous domain \([0,\mathbf{Pmax}_{i,t}]\) when its fuel level is superior to \(\mathbf{Bo}_{i,k}\). \\ \hline
**CT7:** the refueling level for outage \(k\) of T2 plant \(i\) describes the continuous domain \([\mathbf{Rmin}_{i,k},\mathbf{Rmax}_{i,k}]\) \\ \hline
**CT8:** the initial fuel stock for T2 unit \(i\) is \(\mathbf{Xi}_{i}\), known and common for all scenario \(s\). \\ \hline
**CT9:** the fuel stock variation during a production campaign of a cycle \((i,k)\) between \(t\) and \(t+1\), is proportional to the power produced by \(i\) at time step \(t\), with a proportional factor \(-\mathbf{F}_{t}\). \\ \hline
**CT10:** during an outage, the fuel stock variation is the sum of the refueling level and a certain amount of unspent fuel, calculated with a proportional loss with factor \(\mathbf{Q}_{i,k}<1\) to the residual fuel before refueling. \\ \hline
**CT11:** the fuel level is in \([0,\mathbf{Smax}_{i,k}]\) for cycle \(k\) of T2 unit \(i\). The fuel level must be lower than \(\mathbf{A}_{i,k+1}\) to process outage \(k+1\). \\ \hline
**CT13:** outage \((i,k)\) must begin between weeks \(\mathbf{To}_{i,k}\) and \(\mathbf{Ta}_{i,k}\). The maintenance checks follow the order of set \(k\in\mathcal{K}\) without skipping maintenance: if outage \(k+1\) is processed, it must follow the cycle \(k\). \\ \hline \hline \end{tabular}
\end{table}
Table 3: Description of the constraints, with their references in the specification [14].