Scheduling for electricity cost in a smart grid

We study an offline scheduling problem arising in demand response management in a smart grid. Consumers send in power requests with a flexible set of timeslots during which their requests can be served. For example, a consumer may request the dishwasher to operate for 1 h during the periods 8am to 11am or 2pm to 4pm. The grid controller, upon receiving power requests, schedules each request within the specified duration. The electricity cost is measured by a convex function of the load in each timeslot. The objective of the problem is to schedule all requests with the minimum total electricity cost. As a first attempt, we consider a special case in which the power requirement and the duration a for which a request needs service are both unit-size. For this problem, we present a polynomial time offline algorithm that gives an optimal solution and shows that the time complexity can be further improved if the given set of timeslots forms a contiguous interval.