Quantile Optimization in the Presence of Heavy-Tailed Stochastic Processes, and an application to Electricity Markets

Abstract

In this thesis, we study the electricity market to construct stochastic models that helps us make various decisions under uncertainty. First, we propose an hour-ahead prediction model for electricity prices that captures the heavy-tailed behavior that we observe in the hourly spot market in the Ercot (Texas) and the PJM West hub grids. We present a model according to which we separate the price process into a thin-tailed trailing-median process and a heavy-tailed residual process whose probability distribution can be approximated by a Cauchy distribution. We show empirical evidence that supports our model.

Having the electricity price model as a motivating problem, we present a provably convergent algorithm for computing the quantile of a random variable that does not require storing all of the sample realizations. We then present an algorithm for optimizing the quantile of a random function which may be characterized by a heavy-tailed distribution where the expectation is not defined. The algorithm is illustrated in the context of electricity trading in the presence of storage, where electricity prices are known to be heavy-tailed with infinite variance.

Finally, under the assumption that storage usage will become ubiquitous in the future and the electricity market will be stabilized (stationary and not heavy-tailed) even with the use of intermittent supply, we formulate and solve the problem of making advance energy commitments for wind farms in the presence of a storage device with conversion losses, mean-reverting price process, and an auto-regressive energy generation process from wind. We derive an optimal commitment policy under the assumption that wind energy is uniformly distributed. Then, the stationary distribution of the storage level corresponding to the optimal policy is obtained, from which the economic value of the storage as the relative increase in the expected revenue due to the existence of storage is obtained.