Transmission System Operator Optimality Conditions

\[\begin{aligned} \left[\begin{aligned} r_{y} d_{m} \left(\begin{aligned} - \pi^{A}_{acmy} + c^{A}_{acy} \end{aligned} \right) + \lambda^{A}_{acmy} \end{aligned}\right] \geq 0 & \perp q^{A}_{acmy} \geq 0 & \begin{aligned} \forall (a,c) \in \mathcal{AC} , \\ m \in \mathcal{M}, y \in \mathcal{Y} \end{aligned} \\ % \left[\begin{aligned} \frac{r_{y}}{ \| \Delta \|_{y}} \frac{1}{2} c^{\Delta A}_{acy} \\ - \sum_{ y' \in \mathcal{Y} | (y < y' \leq y + L^{A}_{c})} \sum_{m \in \mathcal{M}} \lambda^{A}_{acmy'} \\ + \delta^{A}_{acy} - \delta^{A}_{a^{-1}(a)cy} \end{aligned} \right] \geq 0 & \perp \Delta^{A}_{acy} \geq 0 & \begin{aligned} \forall (a,c) \in \mathcal{AC}, \\ y \in \mathcal{Y} \end{aligned} \\ % \left[\begin{aligned} \frac{r_{y}}{ \| \Delta \|_{y}} \frac{1}{2} c^{\Delta^{RA}}_{arcy} \\ - \sum_{ y' \in \mathcal{Y} | (y < y' \leq y +L^{A}_{c})} \sum_{m \in \mathcal{M}} f^{RA}_{rc} \lambda^{A}_{acmy'} \\ + \sum_{ y' \in \mathcal{Y} | y'> y} \sum_{m \in \mathcal{M}} \lambda^{A}_{army'} \\ \delta^{RA}_{arcy} - \delta^{RA}_{a^{-1}(a)rcy} \end{aligned} \right] \geq 0 & \perp \Delta^{RA}_{arcy} \geq 0 & \begin{aligned} \forall a \in \mathcal{A}, (r,c) \in \mathcal{RA}, \\ (a,r) \in \mathcal{AC}, \\y \in \mathcal{Y} \end{aligned} \\ % \left[ \begin{aligned} \Lambda^{A}_{acy} \\ + \sum_{ y' \in \mathcal{Y} | y-L^{A}_{c} \leq y'< y} \Delta^{A}_{acy}\\ + \sum_{r | (r,c) \in \mathcal{RA}} \sum_{ y' \in \mathcal{Y} | y-L^{A}_{c} \leq y'< y} f^{RA}_{rc} \Delta^{RA}_{arcy}\\ - \sum_{r | (c,r) \in \mathcal{RA}} \sum_{ y' \in \mathcal{Y} | y'< y} \Delta^{RA}_{acry} \\ - q^{A}_{acmy} \end{aligned} \right] \geq 0 & \perp \lambda^{A}_{acmy} \geq 0 & \begin{aligned} \forall (a,c) \in \mathcal{AC} , \\ m \in \mathcal{M}, y \in \mathcal{Y} \end{aligned} \\ % \left[ \begin{aligned} \Delta^{A}_{acy} - \Delta^{A}_{a^{-1}(a)cy} \end{aligned} \right] = 0 & \perp \delta^{A}_{acy} & \begin{aligned} \forall (a,c) \in \mathcal{AC} , \\ y \in \mathcal{Y} \end{aligned}\\ % \left[ \begin{aligned} \Delta^{RA}_{arcy} - \Delta^{RA}_{a^{-1}(a)rcy} \end{aligned} \right] = 0 & \perp \delta^{RA}_{arcy} & \begin{aligned} \forall a \in \mathcal{A}, (r,c) \in \mathcal{RA}, \\ (a,r) \in \mathcal{AC}, \\y \in \mathcal{Y} \end{aligned}\\ \end{aligned}\]