## 2 Materials and Methods
### Overall Methodology  
As long-term weather forecasts become increasingly uncertain, plans should be revised and extended continuously to stay reliable. Therefore, methods from the field of control theory, in particular, the Model Predictive Control (MPC) scheme, offer suitable tools to achieve this continuous adaptation. While other heuristic or metaheuristic optimization schemes can be found in the current literature \begin{table}
\begin{tabular}{l l l} \hline \hline \multicolumn{1}{c}{**Indices:**} & \multicolumn{1}{c}{**Indices for Time Instances (\(k\))**, **Vessels (\(v\))**, and **Operations (\(o\)) with**} \\  & & \(k\), \(v\)\(\in\)\(\mathbb{N}^{+}\) and \(o\in\{1,2,3,4\}\).** \\ \hline \multicolumn{3}{l}{Parameters determined a-priori:} \\ \(N\) & Integer & Length of the prediction horizon as \(N=P*T\) \\ \(T\) & Integer & Sampling step size \\ \(P\) & Integer & Number of planning periods considered by the MILP \\ \(V\) & Integer & Number of installation vessels \\ \(CAP\) & Integer & Maximum loading capacity for vessels \\ \(BAYS\) & Integer & Number of loading bays \\ \(D_{o,k}\) & Integer & Estimated duration of operation \(o\) at time instance \(k\) \\ \(START_{v}\) & Integer & Index of time instance when a vessel \(v\) is available \\ \(INUSE^{bays}_{k}\) & Integer & Denotes the number of busy loading bays for each time instance \(k\) \\ \(C^{o}_{v}\) & Float & Cost for being offshore per hour by vessel \(v\) \\ \(C^{m}_{v}\) & Float & Cost for moving between port and site by vessel \(v\) \\ \(C^{p}_{v}\) & Float & Cost for port operations per hour by vessel \(v\) \\ \(B^{oavt}\) & Float & Benefit for installing a turbine \\ \(B^{early}\) & Float & Benefit for finishing an operation early \\ \hline \multicolumn{3}{l}{Decision variables (start of operation events):} \\ \(Y^{oavt}_{v,k}\) & Binary & Denotes if a vessel \(v\) starts an _installation operation_ at time instance \(k\) \\ \(Y^{load}_{v,k}\) & Binary & Denotes if a vessel \(v\) starts a _loading operation_ at time instance \(k\) \\ \(Y^{PortPort}_{v,k}\) & Binary & Denotes if a vessel \(v\) starts to _move to the port_ at time instance \(k\) \\ \(Y^{Golite}_{v,k}\) & Binary & Denotes if a vessel \(v\) starts to _move to the construction site_ at time instance \(k\) \\ \hline \multicolumn{3}{l}{Support variables (ongoing operations and states):} \\ \(X^{exp}_{v,k}\) & Integer & Amount of currently loaded components \\ \(Y^{loc}_{v,k}\) & Binary & Denotes if vessel \(v\) is in port (0) or offshore (1) \\ \(Y^{busy}_{v,k}\) & Binary & Denotes if vessel \(v\) is currently performing an operation (1) or is idle (0) \\ \(Y^{dBay}_{v,k}\) & Binary & Denotes if vessel \(v\) is currently using a loading bay (1) or not (0) \\ \(X^{fOut}_{v,k}\) & Integer & Denotes at which time instance an installation operation conducted by vessel \(v\) and started at instance \(k\) will finish \\ \(X^{fOp}_{v,k}\) & Integer & Denotes at which time instance a non-installation operation conducted by vessel \(v\) and started at instance \(k\) will finish \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters and Variables used in the extended Mixed-Integer Linear Program. on renewable energies (e.g. [35]), only very few directly include feedback loops that are required to deal with uncertainty. The overall proposed methodology classifies as a predictive-reactive approach. For the reactive component, the methodology relies on a combination of a continuous rescheduling (MPC) with recourse actions. The continuous rescheduling is used to reduce the complexity of the overall problem to smaller prediction horizons. Semi-robust, predictive plans are calculated by estimating disturbances, i.e., dynamic operation times due to changing weather conditions to avoid a high number of recourse actions. In contrast to robust scheduling approaches, the proposed methodology does not assume the worst case but tries to make realistic assumptions based on provided weather forecasts.