409 research outputs found
A Mixed-Integer SDP Solution Approach to Distributionally Robust Unit Commitment with Second Order Moment Constraints
A power system unit commitment (UC) problem considering uncertainties of
renewable energy sources is investigated in this paper, through a
distributionally robust optimization approach. We assume that the first and
second order moments of stochastic parameters can be inferred from historical
data, and then employed to model the set of probability distributions. The
resulting problem is a two-stage distributionally robust unit commitment with
second order moment constraints, and we show that it can be recast as a
mixed-integer semidefinite programming (MI-SDP) with finite constraints. The
solution algorithm of the problem comprises solving a series of relaxed MI-SDPs
and a subroutine of feasibility checking and vertex generation. Based on the
verification of strong duality of the semidefinite programming (SDP) problems,
we propose a cutting plane algorithm for solving the MI-SDPs; we also introduce
a SDP relaxation for the feasibility checking problem, which is an intractable
biconvex optimization. Experimental results on a IEEE 6-bus system are
presented, showing that without any tunings of parameters, the real-time
operation cost of distributionally robust UC method outperforms those of
deterministic UC and two-stage robust UC methods in general, and our method
also enjoys higher reliability of dispatch operation
Inexactness of the Hydro-Thermal Coordination Semidefinite Relaxation
Hydro-thermal coordination is the problem of determining the optimal economic
dispatch of hydro and thermal power plants over time. The physics of
hydroelectricity generation is commonly simplified in the literature to account
for its fundamentally nonlinear nature. Advances in convex relaxation theory
have allowed the advent of Shor's semidefinite programming (SDP) relaxations of
quadratic models of the problem. This paper shows how a recently published SDP
relaxation is only exact if a very strict condition regarding turbine
efficiency is observed, failing otherwise. It further proposes the use of a set
of convex envelopes as a strategy to successfully obtain a stricter lower bound
of the optimal solution. This strategy is combined with a standard iterative
convex-concave procedure to recover a stationary point of the original
non-convex problem.Comment: Submitted to IEEE PES General Meeting 201
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
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