This book explains the power grid as a hierarchy made up of the transmission, distribution, and microgrid levels.Interfaces among these levels are explored to show how flexibility in power demand associated with residential batteries can be communicated through the entire grid to facilitate optimal power flow computations within the transmission grid.To realize this approach, the authors combine semi-definite optimal power flow with model-order reduction at the distribution level and with a new heuristic algorithm for stable power flow at the transmission level. To demonstrate its use, a...