Storage System Operator Optimality Conditions

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