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 \(\phi \) between any two complex-valued demands. More precisely, a PTAS is presented for the case \(\phi \in [0,\tfrac{\pi }{2}]\), a bi-criteria FPTAS for \(\phi \in [0,{\pi } \text{- } \delta ]\) for any polynomially small \(\delta \), 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 \(\tfrac{1}{2}\cos \frac{\phi }{2}\), while the running time is \({O}(n\log n)\), which can be implemented efficiently in practice.