408 research outputs found
Geometry of Power Flows and Optimization in Distribution Networks
We investigate the geometry of injection regions and its relationship to
optimization of power flows in tree networks. The injection region is the set
of all vectors of bus power injections that satisfy the network and operation
constraints. The geometrical object of interest is the set of Pareto-optimal
points of the injection region. If the voltage magnitudes are fixed, the
injection region of a tree network can be written as a linear transformation of
the product of two-bus injection regions, one for each line in the network.
Using this decomposition, we show that under the practical condition that the
angle difference across each line is not too large, the set of Pareto-optimal
points of the injection region remains unchanged by taking the convex hull.
Moreover, the resulting convexified optimal power flow problem can be
efficiently solved via }{ semi-definite programming or second order cone
relaxations. These results improve upon earlier works by removing the
assumptions on active power lower bounds. It is also shown that our practical
angle assumption guarantees two other properties: (i) the uniqueness of the
solution of the power flow problem, and (ii) the non-negativity of the
locational marginal prices. Partial results are presented for the case when the
voltage magnitudes are not fixed but can lie within certain bounds.Comment: To Appear in IEEE Transaction on Power System
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
Branch Flow Model: Relaxations and Convexification—Part II
We propose a branch flow model for the analysis and optimization of mesh as well as radial networks. The model leads to a new approach to solving optimal power flow (OPF) that consists of two relaxation steps. The first step eliminates the voltage and current angles and the second step approximates the resulting problem by a conic program that can be solved efficiently. For radial networks, we prove that both relaxation steps are always exact, provided there are no upper bounds on loads. For mesh networks, the conic relaxation is always exact but the angle relaxation may not be exact, and we provide a simple way to determine if a relaxed solution is globally optimal. We propose convexification of mesh networks using phase shifters so that OPF for the convexified network can always be solved efficiently for an optimal solution. We prove that convexification requires phase shifters only outside a spanning tree of the network and their placement depends only on network topology, not on power flows, generation, loads, or operating constraints. Part I introduces our branch flow model, explains the two relaxation steps, and proves the conditions for exact relaxation. Part II describes convexification of mesh networks, and presents simulation results
Branch Flow Model: Relaxations and Convexification (Parts I, II)
We propose a branch flow model for the anal- ysis and optimization of mesh as
well as radial networks. The model leads to a new approach to solving optimal
power flow (OPF) that consists of two relaxation steps. The first step
eliminates the voltage and current angles and the second step approximates the
resulting problem by a conic program that can be solved efficiently. For radial
networks, we prove that both relaxation steps are always exact, provided there
are no upper bounds on loads. For mesh networks, the conic relaxation is always
exact but the angle relaxation may not be exact, and we provide a simple way to
determine if a relaxed solution is globally optimal. We propose convexification
of mesh networks using phase shifters so that OPF for the convexified network
can always be solved efficiently for an optimal solution. We prove that
convexification requires phase shifters only outside a spanning tree of the
network and their placement depends only on network topology, not on power
flows, generation, loads, or operating constraints. Part I introduces our
branch flow model, explains the two relaxation steps, and proves the conditions
for exact relaxation. Part II describes convexification of mesh networks, and
presents simulation results.Comment: A preliminary and abridged version has appeared in IEEE CDC, December
201
Convex Relaxation of Optimal Power Flow, Part II: Exactness
This tutorial summarizes recent advances in the convex relaxation of the
optimal power flow (OPF) problem, focusing on structural properties rather than
algorithms. Part I presents two power flow models, formulates OPF and their
relaxations in each model, and proves equivalence relations among them. Part II
presents sufficient conditions under which the convex relaxations are exact.Comment: Citation: IEEE Transactions on Control of Network Systems, June 2014.
This is an extended version with Appendex VI that proves the main results in
this tutoria
Exact Convex Relaxation of Optimal Power Flow in Tree Networks
The optimal power flow (OPF) problem seeks to control power generation/demand
to optimize certain objectives such as minimizing the generation cost or power
loss in the network. It is becoming increasingly important for distribution
networks, which are tree networks, due to the emergence of distributed
generation and controllable loads. In this paper, we study the OPF problem in
tree networks. The OPF problem is nonconvex. We prove that after a "small"
modification to the OPF problem, its global optimum can be recovered via a
second-order cone programming (SOCP) relaxation, under a "mild" condition that
can be checked apriori. Empirical studies justify that the modification to OPF
is "small" and that the "mild" condition holds for the IEEE 13-bus distribution
network and two real-world networks with high penetration of distributed
generation.Comment: 22 pages, 7 figure
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