## 2 System Model
\begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Explanation** \\ \hline \(p_{\text{bess}}^{\text{min}}\) & Minimum power output of the energy storage system \\ \hline \(p_{\text{bess}}^{\text{max}}\) & Maximum power output of the energy storage system \\ \hline \(p_{\text{bess}}(k)\) & Power output of the energy storage system at time \(k\) \\ \hline \(E_{\text{bess}}^{\text{min}}\) & Minimum energy capacity of the energy storage system \\ \hline \(E_{\text{bess}}^{\text{max}}\) & Maximum energy capacity of the energy storage system \\ \hline \(E_{\text{bess}}(k)\) & Energy capacity of the energy storage system at time \(k\) \\ \hline \(\eta_{\text{bess}}\) & Charge/discharge efficiency of the energy storage system \\ \hline \(\epsilon_{\text{bess}}\) & Self-discharge rate of the energy storage system \\ \hline \(\Delta P_{DG}\) & Maximum ramping power of the diesel generator \\ \hline \(P_{DG}(k)\) & Power output of the diesel generator at time \(k\) \\ \hline \(p_{DG}^{\text{min}}\) & Minimum power output of the diesel generator \\ \hline \(p_{DG}^{\text{max}}\) & Maximum power output of the diesel generator \\ \hline \(t_{on}(k)\) & Continuous running time of the generator at time \(k\) \\ \hline \(t_{off}(k)\) & Continuous off time of the generator at time \(k\) \\ \hline \(T^{on}\) & Minimum continuous running time required for the generator \\ \hline \(T^{off}\) & Minimum continuous off time required for the generator \\ \hline \(\beta_{cur}^{\text{min}}\) & Minimum proportion of switchable loads \\ \hline \(\beta_{cur}(k)\) & Proportion of switchable loads at time \(k\) \\ \hline \(T_{\text{start},i}\) & Earliest start time for the \(i\)-th controllable load \\ \hline \(T_{\text{finish},i}\) & Latest stop time for the \(i\)-th controllable load \\ \hline \(T_{sch,i}\) & Total operating time requirement for the \(i\)-th controllable load \\ \hline \(E_{sch,i}\) & Total energy demand for the \(i\)-th controllable load \\ \hline \(\delta_{sch,i}(k)\) & Operating state of the \(i\)-th controllable load at time \(k\) \\ \hline \(p_{sch,i}^{\text{min}}\) & Minimum power output for the \(i\)-th controllable load \\ \hline \(p_{sch,i}^{\text{max}}\) & Maximum power output for the \(i\)-th controllable load \\ \hline \(C_{1}\) & Operational costs of the system \\ \hline \(C_{DG}(k)\) & Operating costs of the diesel generator at time \(k\) \\ \hline \(C_{BESS}(k)\) & Operating costs of the energy storage system at time \(k\) \\ \hline \(C_{cur}(k)\) & Penalty for load shedding at time \(k\) \\ \hline \(C_{grid}(k)\) & Cost of buying and selling electricity at time \(k\) \\ \hline \(C_{fuel}(k)\) & Fuel cost for the diesel generator at time \(k\) \\ \hline \(OM_{DG}\) & Maintenance cost per unit time of generator operation \\ \hline \(\lambda_{DC}^{up}(k)\) & Fixed costs for each startup of the diesel generator \\ \hline \(\lambda_{DG}^{down}(k)\) & Fixed costs for each shutdown of the diesel generator \\ \hline \(a,b\) & Constant coefficients of fuel cost for the diesel generator \\ \hline \(OM_{BESS}\) & Maintenance cost per unit time for the energy storage system \\ \hline \(\epsilon_{bESS}^{switch}\) & Cost associated with charging and discharging losses \\ \hline \(a_{cur,i}(k)\) & Penalty coefficient for the \(i\)-th load shedding \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mathematical Notations.