The Optimal Power Flow Problem

By Stephen Wright

The power grid has long been a rich source of problems in mathematical modeling, linear and nonlinear equations, and optimization. An important instance is the optimal power flow problem, which reveals the optimal output of a collection of generators that meets a given set of demands for power around the grid, while respecting limits on current flows along transmission lines. This problem is solved repeatedly on a typical day to take account of fluctuating demands and operating conditions. Operation of the grid has become more complex with the advent of renewable power sources, especially wind farms. The output of these sources is notoriously unpredictable, and if fluctuations are not accounted for during generation planning, excess current on some transmission lines may cause overheating, “tripping” of the lines, and possibly cascading failures, resulting in widespread blackouts.

Bonneville Power Administration shown in outline under 9% wind penetration, where green dots mark actual wind farms. Image courtesy of SIAM Review, 56(3), 461–495

The authors of this issue’s Research Spotlights paper describe a chance-constrained optimization framework that takes into consideration the uncertainty in output of renewable generation sources. By assuming that the output of these sources is a normally distributed random variable (with known mean and variance), a convex optimization problem can be formulated to ensure that the grid operates within desired specifications, with high probability. To provide for flexible operation, power generated at the nonrenewable plants is adjusted according to the output of the renewable plants, according to a linear control law. The convex problem is difficult to solve using package software, so the authors describe a specialized cutting-plane method and report results on a large test set based on a national-scale grid.

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Chance-Constrained Optimal Power Flow: Risk-Aware Network Control under Uncertainty

Daniel Bienstock, Michael Chertkov, and Sean Harnett

SIAM Review, 56(3), 461-495.

Stephen Wright is the George B. Dantzig Professor of Computer Sciences, the Sheldon B. Lubar Chair of Computer Sciences, and the Amar and Balinder Sohi Professor of Computer Sciences at the University of Wisconsin-Madison. He is the section editor of SIAM Review.