Risk-Sensitive Learning and Pricing for Demand Response

We consider the setting in which an electric power utility seeks to curtail its peak electricity demand by offering a fixed group of customers a uniform price for reductions in consumption relative to their predetermined baselines. The underlying demand curve, which describes the aggregate reduction in consumption in response to the offered price, is assumed to be affine and subject to unobservable random shocks. Assuming that both the parameters of the demand curve and the distribution of the random shocks are initially unknown to the utility, we investigate the extent to which the utility might dynamically adjust its offered prices to maximize its cumulative risk-sensitive payoff over a finite number of $T$ days. In order to do so effectively, the utility must design its pricing policy to balance the tradeoff between the need to learn the unknown demand model (exploration) and maximize its payoff (exploitation) over time. In this paper, we propose such a pricing policy, which is shown to exhibit an expected payoff loss over $T$ days that is at most $O(\sqrt{T}\log(T))$, relative to an oracle pricing policy that knows the underlying demand model. Moreover, the proposed pricing policy is shown to yield a sequence of prices that converge to the oracle optimal prices in the mean square sense.