Complex-demand scheduling problem with application in smart grid

We consider the problem of scheduling complex-valued demands over a discretized time horizon. Given a set of users, each user is associated with a set of demands representing different user’s preferences. A demand is represented by a complex number, a time interval, and a utility value obtained if the demand is satisfied. At each time slot, the magnitude of the total selected demands should not exceed a given capacity. This naturally captures the supply constraints in alternating current (AC) electric systems. In this paper, we consider maximizing the aggregate user utility subject to power supply limits over a time horizon. We present approximation algorithms characterized by the maximum angle φ between any two complex-valued demands. More precisely, a PTAS is presented for the case φ ∈ [0, π/2], a bi-criteria FPTAS for φ ∈ [0, π-δ] for any polynomially small δ, assuming the number of time slots in the discretized time horizon is a constant. Furthermore, if the number of time slots is polynomial, we present a reduction to the real-valued unsplittable flow on a path problem with only a constant approximation ratio. Finally, we present a practical greedy algorithm for the single time slot case with an approximation ratio of 1/2 cos φ/2, while the running time is O(n log n), which can be implemented efficiently in practice.