## 3 Problem Formulation
\begin{split} P_{c_{i}}^{\text{min}}&\leq P_{c_{i}} \leq P_{c_{i}}^{\text{max}},\\ P_{s_{i}}^{\text{min}}&\leq P_{s_{i}}\leq P_{s_{i}}^{ \text{max}},\\ P_{w_{i}}^{\text{min}}&\leq P_{w_{i}}\leq P_{w_{i}}^{ \text{max}},\\ P_{b_{i}}^{\text{min}}&\leq P_{b_{i}}\leq P_{b_{i}}^{ \text{max}}.\end{split}  \tag{4}\]  
Again, \(P_{c_{i}},P_{s_{i}},P_{w_{i}}\), and \(P_{b_{i}}\) represent power generated by agents (i.e., technology assets), each with a minimum (\(P_{(\cdot)}^{\text{min}}\)) and maximum (\(P_{(\cdot)}^{\text{max}}\)) limit. Conventional generating units have individual minimum and maximum power limitations determined by mechanical and electrical properties [10]. Solar and wind power plants can produce active power from zero up to their maximal limits (i.e., rated power) [54; 57]. Finally, energy storage systems have capacity restrictions depending on battery design, charging or discharging rates, and the inverter used [10; 28; 56]. The power interaction with the grid \(P_{g}\) is considered unconstrained. In the case of energy storage, the stored energy over a time horizon \(E_{b_{i}}(t)\) can be defined as follows:  
\[E_{b_{i}}^{0}+\sum_{t=k}^{k+T_{d}}P_{b_{i}}(t)\cdot h=E_{b_{i}}(t), \tag{5}\]  
where \(E_{b_{i}}^{0}\) is the energy storage initial state, \(T_{d}\) is considered time horizon, \(h\) is (dis)charging interval and \(k\) represents current time step. Further, the battery systems need to consider capacity constraints, which can be formulated as follows:  
\[E_{b_{i}}^{min}\leq E_{b_{i}}(t)\leq E_{b_{i}}^{max}, \tag{6}\]  
where \(E_{b_{i}}^{min}\) is the state of the charge minimal level and \(E_{b_{i}}^{max}\) is the state of the charge maximum level for the \(i\)-th energy storage unit. Equation (5) ensures that the cumulative energy (dis)charged from/to the battery system during the specified period \(T_{d}\) does not exceed the total energy stored in the battery system at that particular time. In the case of a grid-connected system, the overall power system stability is maintained by robust balancing mechanisms operated by distribution or transmission system operators. In the case of island operation, the stability is maintained locally using controllable pro duction, consumption, and storage assets. In this case, renewable volatile sources should not provide more than a maximum of 30-40% of the total required network load \(P_{D}\) to maintain system stability [29; 57]. From now on, the paper focuses on island power systems for clarity. Here, the overall renewable production is defined as:  
\[P_{\text{ren}}=P_{\text{s}}+P_{w}=\sum_{i=1}^{N_{\text{s}}}P_{s_{i}}+\sum_{i=1}^ {N_{\text{tr}}}P_{w_{i}}. \tag{7}\]  
Other energy sources can be collected as:  
\[P_{\text{rest}}=P_{c}+P_{b}=\sum_{i=1}^{N_{\text{c}}}P_{c_{i}}+\sum_{i=1}^{N_{ b}}P_{b_{i}}, \tag{8}\]  
and overall load is covered by these energy sources as:  
\[P_{D}=P_{\text{rest}}+P_{\text{ren}}. \tag{9}\]  
Therefore, design constraints related to operation stability can be defined as:  
\[P_{\text{ren}}\leq X_{\text{level}}\cdot P_{D}, \tag{10}\]  
where \(X_{\text{level}}\) denotes the maximum secure renewable contribution to the total load. For instance, if the maximum is 30%, then \(X_{\text{level}}=0.3\). In order to ensure the operation feasibility of the islanded power system, power demand must lie between the sum of the minima and the maxima over all sources connected to the network [10; 26; 55]. \[\sum_{i=1}^{N_{\text{c}}}P_{c_{i}}^{min}+\sum_{i=1}^{N_{\text{s}}}P_{s_{i}}^{ min}+\sum_{i=1}^{N_{\text{tr}}}P_{w_{i}}^{min}+\sum_{i=1}^{N_{\text{p}}}P_{b_{i}}^{ min}\leq P_{D}\leq\sum_{i=1}^{N_{\text{c}}}P_{c_{i}}^{max}+\sum_{i=1}^{N_{ \text{c}}}P_{c_{i}}^{max}+\sum_{i=1}^{N_{\text{tr}}}P_{w_{i}}^{max}+\sum_{i=1 }^{N_{b}}P_{b_{i}}^{max} \tag{11}\]