Optimal power consumption for demand response of thermostatically controlled loads

We consider the problem of determining the optimal aggregate power consumption of a population of thermostatically controlled loads such as air conditioners. This is motivated by the need to synthesize the demand response for a load serving entity (LSE) catering a population of such customers. We show how the LSE can opportunistically design the aggregate reference consumption to minimize its energy procurement cost, given day-ahead price, load forecast, and ambient temperature forecast, while respecting each individual load's comfort range constraints. The resulting synthesis problem is intractable when posed as a direct optimization problem after Euler discretization of the dynamics, since it results in a mixed-integer linear programming problem with number of variables typically of the order of millions. In contrast, in this paper, we show that the problem is amenable to continuous-time optimal control techniques. Numerical simulations elucidate how the LSE can use the optimal aggregate power consumption trajectory thus computed, for the purpose of demand response.