The incorporation of stochastic loads and generation into the operation of power grids gives rise to an exposure to stochastic risk. This risk has been addressed in prior work through a variety of mechanisms, such as scenario generation or chance constraints, that can be incorporated into OPF computations. Nevertheless, numerical experiments reveal that the resulting operational decisions can produce power flows with very high variance. In this paper we introduce a variety of convex variants of OPF that explicitly address the interplay of (power flow) variance with cost minimization, and present numerical experiments that highlight our contributions.

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Recent work has studied modifications to DC-OPF computations so as to better account for risk arising as a result of stochastic variation in the output of renewable sources. Typically such modifications rely on mathematical constructs such as chance-constraints that can still yield convex formulations. However, numerical simulations show that the computed policies can translate into power flow patterns with high variance. We introduce a number of convex variants of OPF that trade-off variance and cost minimization, describe practical algorithms for the solution of such problems, and present numerical experiments.

We consider the problem of streaming principal component analysis (PCA) when the observations are noisy and generated in a non-stationary environment. Given T , p-dimensional noisy observations sampled from a non-stationary variant of the spiked covariance model, our goal is to construct the best linear k-dimensional subspace of the terminal observations. We study the effect of non-stationarity by establishing a lower bound on the number of samples and the corresponding recovery error obtained by any algorithm. We establish the convergence behaviour of the noisy power method using a novel proof technique which maybe of independent interest. We conclude that the recovery guarantee of the noisy power method matches the fundamental limit, thereby generalizing existing results on streaming PCA to a non-stationary setting.

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