Storage System Operator

\[\begin{aligned} \max_{\substack{q^{S}_{stcomy}, \\ q^{S_{in}}_{stcomy}, \\ q^{S_{out}}_{stcomy}, \\ \Delta^{S}_{scy}, \\ \Delta^{RS}_{srcy}}} \quad & % \sum_{y \in \mathcal{Y}} r_{y} \left[ \begin{aligned} \sum_{c \in \mathcal{C}} \left( \begin{aligned} \sum_{m \in \mathcal{M}} \sum_{t \in \mathcal{T}} \sum_{o \in \mathcal{O}} d_{m} \left[ \begin{aligned} \left(\pi^{S \rightarrow T}_{tn(s)comy} - c^{S_{out}}_{scy} \right) q^{S_{out}}_{stcomy} \\ - \left( \pi^{T \rightarrow S}_{tn(s)comy} + c^{S_{in}}_{scy} \right) q^{S_{in}}_{stcomy} \end{aligned} \right] \\ - \frac{1}{ \| \Delta \|_{y}} c^{\Delta S}_{scy} \Delta^{S}_{scy} \\ \end{aligned} \right) \\ - \frac{1}{ \| \Delta \|_{y}} \sum_{(r,c) \in \mathcal{RS}} c^{\Delta^{RS}}_{srcy} \Delta^{RS}_{srcy} \end{aligned} \right] \\ \text{s.t.} \quad % & \sum_{o \in \mathcal{O}} \sum_{t \in \mathcal{T}} q^{S}_{stcomy} \leq \left( \begin{aligned} \Lambda^{S}_{scy} \\ + \sum_{ y' \in \mathcal{Y} | y-L^{S}_{c} \leq y'<y} \Delta^{S}_{scy} \\ + \sum_{r | (r,c) \in \mathcal{RS}} \sum_{ y' \in \mathcal{Y} | y-L^{S}_{c} \leq y'< y} f^{RS}_{rc} \Delta^{RS}_{srcy}\\ - \sum_{r | (c,r) \in \mathcal{RS}} \sum_{ y' \in \mathcal{Y} | y'< y} \Delta^{RS}_{scry} \end{aligned} \right) \quad & \begin{aligned}\forall c \in \mathcal{C}, m \in \mathcal{M} \\ y \in \mathcal{Y} \end{aligned} \quad & (\lambda^{S}_{scmy}) \\ & \begin{aligned} \left(1- l^{S}_{cmm^{+}(m)} \right) \cdot \left[\begin{aligned} q^{S}_{stcomy} \\ + d_{m} \left(q^{S_{in}}_{stcomy} - q^{S_{out}}_{stcomy}\right) \end{aligned}\right] \\ \geq q^{S}_{stcom^{+}(m)y} \end{aligned} \quad & \begin{aligned} \forall t \in \mathcal{T}, c \in \mathcal{C}, \\ o \in \mathcal{O}, m \in \mathcal{M} \\ y \in \mathcal{Y} \end{aligned} \quad &(\phi^{S}_{stcomy}) & \\ & \left(\begin{aligned} \sum_{ y' \in \mathcal{Y} | y-L^{S}_{c} \leq y'<y} \Delta^{S}_{scy} \\ + \sum_{r | (r,c) \in \mathcal{RS}} \sum_{ y' \in \mathcal{Y} | y-L^{S}_{c} \leq y'< y} f^{RS}_{rc} \Delta^{RS}_{srcy}\\ - \sum_{r | (c,r) \in \mathcal{RS}} \sum_{ y' \in \mathcal{Y} | y'< y} \Delta^{RS}_{scry} \end{aligned} \right) \leq \Omega^{S}_{scy} \quad & \begin{aligned} \forall c \in \mathcal{C}, \\ y \in \mathcal{Y} \end{aligned} \quad &(\omega^{S}_{scy}) & \\ & q^{S}_{stcomy}, q^{S_{in}}_{stcomy}, q^{S_{out}}_{stcomy}, \Delta^{S}_{scy}, \Delta^{RS}_{srcy} \geq 0 \quad & & \end{aligned}\]