Risk-averse design and operation of renewable energy power grids

The need for effective energy harvesting from renewable resources becomes increasingly important, especially in the light of the inevitable depletion of the fossil fuel energy sources. Among renewable energy sources, wind energy represents one of the most attractive alternatives. In this thesis, we construct several stochastic optimization models, including the traditional risk-neutral expectation based model, and risk-averse models based on linear and nonlinear coherent measures of risk, to study the strategic planning and operation of futuristic power grids where the loads are served from renewable energy sources (wind farms) through High Voltage Direct Current lines. Exact solutions algorithms that employ Benders decomposition and polyhedral approximations of nonlinear constraints have been proposed for the formulated linear and nonlinear mixed-integer optimization problems. The conducted numerical experiments illustrate the efficiency of the developed algorithms, as well as effectiveness of risk-averse models in reducing the power grid's exposure to power shortage risks when the energy is produced from renewable sources. We further extend the risk-averse models to demonstrate how energy storage devices may impact the risk profile of power shortages in the renewable energy power grid. Additionally, we consider convex relaxations of optimal power flow problem over radial networks, that allow for solving mixed-integer optimization problems in traditional alternating current distribution networks. Exactness of a specific second-order cone programming relaxation has been discussed. We finally propose an "extended'' optimal power flow problem and prove its second-order cone programming relaxation to be exact theoretically and empirically.