  We propose a data-based method to solve a multi-stage stochastic optimal
power flow (OPF) problem based on limited information about forecast error
distributions. The framework explicitly combines multi-stage feedback policies
with any forecasting method and historical forecast error data. The objective
is to determine power scheduling policies for controllable devices in a power
network to balance operational cost and conditional value-at-risk (CVaR) of
device and network constraint violations. These decisions include both nominal
power schedules and reserve policies, which specify planned reactions to
forecast errors in order to accommodate fluctuating renewable energy sources.
Instead of assuming the uncertainties across the networks follow prescribed
probability distributions, we consider ambiguity sets of distributions centered
around a finite training dataset. By utilizing the Wasserstein metric to
quantify differences between the empirical data-based distribution and the real
unknown data-generating distribution, we formulate a multi-stage
distributionally robust OPF problem to compute optimal control policies that
are robust to both forecast errors and sampling errors inherent in the dataset.
Two specific data-based distributionally robust stochastic OPF problems are
proposed for distribution networks and transmission systems.
