Charge Scheduling of an Energy Storage System under Time-of-use Pricing and a Demand Charge

A real-coded genetic algorithm is used to schedule the charging of an energy storage system (ESS), operated in tandem with renewable power by an electricity consumer who is subject to time-of-use pricing and a demand charge. Simulations based on load and generation profiles of typical residential customers show that an ESS scheduled by our algorithm can reduce electricity costs by approximately 17%, compared to a system without an ESS, and by 8% compared to a scheduling algorithm based on net power.Comment: 13 pages, 2 figures, 5 table

Octahedral developing of knot complement II: Ptolemy coordinates and applications

It is known that a knot complement (minus two points) decomposes into ideal octahedra with respect to a given knot diagram. In this paper, we study the Ptolemy variety for such an octahedral decomposition in perspective of Thurston's gluing equation variety. More precisely, we compute explicit Ptolemy coordinates in terms of segment and region variables, the coordinates of the gluing equation variety motivated from the volume conjecture. As a consequence, we present an explicit formula for computing the obstruction to lifting a $(\mathrm{PSL}(2,\mathbb{C}),P)$-representation of the knot group to a $(\mathrm{SL}(2,\mathbb{C}),P)$-representation. We also present a diagrammatic algorithm to compute a holonomy representation of the knot group.Comment: 32 pages, 21 figue