On the optimality of de-synchronized demand response with stochastic renewables and inertial thermal loads

We consider the problem of optimally supporting a collection of thermostatically controlled inertial loads in a smart microgrid environment. One key consideration is to reduce the need for costly nonrenewable reserves, while also meeting the consumer requirements specific to inertial thermal loads at the same time. Several models studied for optimal demand response leads to a response that is synchronized over the collection of loads, which is undesirable since synchronized response entails high demand fluctuations. We propose a model with an additional feature of stochastic variations in end user requirements. We show that employing a cost function that additionally penalizes the error in adhering to end user requirements leads to an optimal demand response that is de-synchronized over the collection of homogeneous loads. That is, identical thermal loads are intentionally staggered so as to hedge against future uncertainty of consumer preferences. We show that such de-synchronization is related to the concavity of the cost-to-go function in the Hamilton-Jacobi-Bellman equation. We provide a simple heuristic to approximate the optimal policy. We also illustrate the results on a simple numerical example.