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Severe geomagnetic disturbances (GMDs) increase the magnitude of the electric field on the Earth’s surface (E-field) and drive geomagnetically-induced currents (GICs) along the transmission lines in electric grids. These additional currents can pose severe risks, such as current distortions, transformer saturation and increased reactive power losses, each of which can lead to system unreliability. Several mitigation actions (e.g., changing grid topology) exist that can reduce the harmful GIC effects on the grids. Making such decisions can be challenging, however, because the magnitude and direction of the E-field are uncertain and non-stationary. In this paper, we model uncertain E-fields using the distributionally robust optimization (DRO) approach that determines optimal transmission grid operations such that the worst-case expectation of the system cost is minimized. We also capture the effect of GICs on the nonlinear AC power flow equations. For solution approaches, we develop an accelerated column-and-constraint generation (CCG) algorithm by exploiting a special structure of the support set of uncertain parameters representing the E-field. Extensive numerical experiments based on “epri-21” and “uiuc-150” systems, designed for GMD studies, demonstrate (i) the computational performance of the accelerated CCG algorithm, (ii) the superior performance of distributionally robust grid operations that satisfy nonlinear, nonconvex AC power flow equations and GIC constraints, in comparison with standard stochastic programming-based methods during the out-of-sample testing. [ABSTRACT FROM AUTHOR]

We consider a distributed optimal power flow formulated as an optimization problem that maximizes a nondifferentiable concave function. Solving such a problem by the existing distributed algorithms can lead to data privacy issues because the solution information exchanged within the algorithms can be utilized by an adversary to infer the data. To preserve data privacy, in this paper we propose a differentially private projected subgradient (DP-PS) algorithm that includes a solution encryption step. We show that a sequence generated by DP-PS converges in expectation, in probability, and with probability 1. Moreover, we show that the rate of convergence in expectation is affected by a target privacy level of DP-PS chosen by the user. We conduct numerical experiments that demonstrate the convergence and data privacy preservation of DP-PS. [ABSTRACT FROM AUTHOR]

Geomagnetic disturbances (GMDs), a result of space weather, pose a severe risk to electric grids. When GMDs occur, they can cause geomagnetically-induced currents (GICs), which saturate transformers, induce hot-spot heating, and increase reactive power losses in the transmission grid. Furthermore, uncertainty in the magnitude and orientation of the geo-electric field, and insufficient historical data make the problem of mitigating the effects of uncertain GMDs challenging. In this paper, we propose a novel distributionally robust optimization (DRO) approach that models uncertain GMDs and mitigates the effects of GICs on electric grids. This is achieved via a set of mitigation actions (e.g., line switching, locating blocking devices, generator re-dispatch and load shedding), prior to the GMD event, such that the worst-case expectation of the system cost is minimized. To this end, we develop a column-and-constraint generation algorithm that solves a sequence of mixed-integer second-order conic programs to handle the underlying convex support set of the uncertain GMDs. Also, we present a monolithic exact reformulation of our DRO model when the underlying support set can be approximated by a polytope with three extreme points. Numerical experiments on 'epri-21' system show the efficacy of the proposed algorithms and the exact reformulation of our DRO model. [ABSTRACT FROM AUTHOR]

We introduce a strong extended formulation of the convex recoloring problem on a tree, which has an application in analyzing phylogenetic trees. The extended formulation has only a polynomial number of constraints, but dominates the conventional formulation and the exponentially many valid inequalities introduced by Campêlo et al. (Math Progr 156:303-330, 2016). We show that all valid inequalities introduced by Campêlo et al. can be derived from the extended formulation. We also show that the natural restriction of the extended formulation provides a complete inequality description of the polytope of subtrees of a tree. The solution time using the extended formulation is much smaller than that with the conventional formulation. Moreover the extended formulation solves all the problem instances attempted in Campêlo et al. (2016) and larger sized instances at the root node of the branch-and-bound tree without branching. [ABSTRACT FROM AUTHOR]