9,130 research outputs found
A differential analysis of the power flow equations
The AC power flow equations are fundamental in all aspects of power systems
planning and operations. They are routinely solved using Newton-Raphson like
methods. However, there is little theoretical understanding of when these
algorithms are guaranteed to find a solution of the power flow equations or how
long they may take to converge. Further, it is known that in general these
equations have multiple solutions and can exhibit chaotic behavior. In this
paper, we show that the power flow equations can be solved efficiently provided
that the solution lies in a certain set. We introduce a family of convex
domains, characterized by Linear Matrix Inequalities, in the space of voltages
such that there is at most one power flow solution in each of these domains.
Further, if a solution exists in one of these domains, it can be found
efficiently, and if one does not exist, a certificate of non-existence can also
be obtained efficiently. The approach is based on the theory of monotone
operators and related algorithms for solving variational inequalities involving
monotone operators. We validate our approach on IEEE test networks and show
that practical power flow solutions lie within an appropriately chosen convex
domain.Comment: arXiv admin note: text overlap with arXiv:1506.0847
Distributed Random Convex Programming via Constraints Consensus
This paper discusses distributed approaches for the solution of random convex
programs (RCP). RCPs are convex optimization problems with a (usually large)
number N of randomly extracted constraints; they arise in several applicative
areas, especially in the context of decision under uncertainty, see [2],[3]. We
here consider a setup in which instances of the random constraints (the
scenario) are not held by a single centralized processing unit, but are
distributed among different nodes of a network. Each node "sees" only a small
subset of the constraints, and may communicate with neighbors. The objective is
to make all nodes converge to the same solution as the centralized RCP problem.
To this end, we develop two distributed algorithms that are variants of the
constraints consensus algorithm [4],[5]: the active constraints consensus (ACC)
algorithm, and the vertex constraints consensus (VCC) algorithm. We show that
the ACC algorithm computes the overall optimal solution in finite time, and
with almost surely bounded communication at each iteration. The VCC algorithm
is instead tailored for the special case in which the constraint functions are
convex also w.r.t. the uncertain parameters, and it computes the solution in a
number of iterations bounded by the diameter of the communication graph. We
further devise a variant of the VCC algorithm, namely quantized vertex
constraints consensus (qVCC), to cope with the case in which communication
bandwidth among processors is bounded. We discuss several applications of the
proposed distributed techniques, including estimation, classification, and
random model predictive control, and we present a numerical analysis of the
performance of the proposed methods. As a complementary numerical result, we
show that the parallel computation of the scenario solution using ACC algorithm
significantly outperforms its centralized equivalent
On Quasi-Newton Forward--Backward Splitting: Proximal Calculus and Convergence
We introduce a framework for quasi-Newton forward--backward splitting
algorithms (proximal quasi-Newton methods) with a metric induced by diagonal
rank- symmetric positive definite matrices. This special type of
metric allows for a highly efficient evaluation of the proximal mapping. The
key to this efficiency is a general proximal calculus in the new metric. By
using duality, formulas are derived that relate the proximal mapping in a
rank- modified metric to the original metric. We also describe efficient
implementations of the proximity calculation for a large class of functions;
the implementations exploit the piece-wise linear nature of the dual problem.
Then, we apply these results to acceleration of composite convex minimization
problems, which leads to elegant quasi-Newton methods for which we prove
convergence. The algorithm is tested on several numerical examples and compared
to a comprehensive list of alternatives in the literature. Our quasi-Newton
splitting algorithm with the prescribed metric compares favorably against
state-of-the-art. The algorithm has extensive applications including signal
processing, sparse recovery, machine learning and classification to name a few.Comment: arXiv admin note: text overlap with arXiv:1206.115
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