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 numerical case study based on modified IEEE 9-bus and 33-bus systems for the transmission and distribution grid, respectively, is included. This book shows how exploiting the flexibility on the residential level improves the performance of the power flow with the transmission grid.

This book contains articles presented at the 9th Workshop on Differential-Algebraic Equations held in Paderborn, Germany, from 17-20 March 2019. The workshop brought together more than 40 mathematicians and engineers from various fields, such as numerical and functional analysis, control theory, mechanics and electromagnetic field theory. The participants focussed on the theoretical and numerical treatment of "descriptor" systems, i.e., differential-algebraic equations (DAEs). The book contains 14 contributions and is organized into four parts: mathematical analysis, numerics and model order reduction, control as well as applications. It is a useful resource for applied mathematicians with interest in recent developments in the field of differential algebraic equations but also for engineers, in particular those interested in modelling of constraint mechanical systems, thermal networks or electric circuits.