Equilibrium price distributions in energy markets with shiftable demand

The paper examines existence of equilibrium price distributions in energy markets with real-time pricing and consumers with time-flexible demands. Previous works have examined consumer optimal policies for shifting time-flexible loads up to a deadline, in response to an exogenous and stochastic price process. It is shown here that under some mild assumptions on the stochastic price process and the information structure in the market, the individual consumer's optimal policy is a threshold policy. The threshold policy indicates that a consumer will consume only when the price falls below a certain threshold which depends on the time left to his deadline and the information on the price process. This behavior by individual consumer leads to an aggregate behavior by a large number of consumers who implement threshold policies in reaction to the price process; although at each instant of time, different consumers may have different thresholds due to different deadlines and different information assumptions. The aggregate response from the consumers determines the state of the wholesale market, and thus, affects the price process. The question that we intend to answer in this paper is that of consistency of assumptions on the price process and the market outcomes. It is assumed that the price at each time interval is determined by the Independent Service Operator (ISO) on the basis of an estimate of the global consumption as a function of price. It is shown that under technical regularity assumptions on the model components, the price distribution has at least one equilibrium point. Thus, there exists at least one price process that is consistent both with the consumption behavior of the aggregate of consumers who individually implement threshold policies, and the marginal cost pricing mechanism of the ISO.