Optimal Control of Thermostatic Loads for Planning Aggregate Consumption: Characterization of Solution and Explicit Strategies

We consider the problem of planning the aggregate energy consumption for a set of thermostatically controlled loads for demand response, accounting price forecast trajectory and thermal comfort constraints. We address this as a continuous-time optimal control problem, and analytically characterize the structure of its solution in the general case. In the special case when the price forecast is monotone and the loads have equal dynamics, we show that it is possible to determine the solution in an explicit form. Taking this fact into account, we handle the non-monotone price case by considering several subproblems, each corresponding to a time subinterval where the price function is monotone, and then allocating to each subinterval a fraction of the total energy budget. This way, for each time subinterval, the problem reduces to a simple convex optimization problem with a scalar decision variable, for which a descent direction is also known. The price forecasts for the day-ahead energy market typically have no more than four monotone segments, so the resulting optimization problem can be solved efficiently with modest computational resources.