3,954 research outputs found
From Electrical Power Flows to Unsplittabe Flows: A QPTAS for OPF with Discrete Demands in Line Distribution Networks
The {\it AC Optimal Power Flow} (OPF) problem is a fundamental problem in
power systems engineering which has been known for decades. It is a notoriously
hard problem due mainly to two reasons: (1) non-convexity of the power flow
constraints and (2) the (possible) existence of discrete power injection
constraints. Recently, sufficient conditions were provided for certain convex
relaxations of OPF to be exact in the continuous case, thus allowing one to
partially address the issue of non-convexity. In this paper we make a first
step towards addressing the combinatorial issue. Namely, by establishing a
connection to the well-known {\it unsplittable flow problem} (UFP), we are able
to generalize known techniques for the latter problem to provide approximation
algorithms for OPF with discrete demands. As an application, we give a
quasi-polynomial time approximation scheme for OPF in line networks under some
mild assumptions and a single generation source. We believe that this
connection can be further leveraged to obtain approximation algorithms for more
general settings, such as multiple generation sources and tree networks
A Novel Decomposition Solution Approach for the Restoration Problem in Distribution Networks
The distribution network restoration problem is by nature a mixed integer and
non-linear optimization problem due to the switching decisions and Optimal
Power Flow (OPF) constraints, respectively. The link between these two parts
involves logical implications modelled through big-M coefficients. The presence
of these coefficients makes the relaxation of the mixed-integer problem using
branch-and-bound method very poor in terms of computation burden. Moreover,
this link inhibits the use of classical Benders algorithm in decomposing the
problem because the resulting cuts will still depend on the big-M coefficients.
In this paper, a novel decomposition approach is proposed for the restoration
problem named Modified Combinatorial Benders (MCB). In this regard, the
reconfiguration problem and the OPF problem are decomposed into master and sub
problems, which are solved through successive iterations. In the case of a
large outage area, the numerical results show that the MCB provides, within a
short time (after a few iterations), a restoration solution with a quality that
is close to the proven optimality when it can be exhibited
Equivalent relaxations of optimal power flow
Several convex relaxations of the optimal power flow (OPF) problem have
recently been developed using both bus injection models and branch flow models.
In this paper, we prove relations among three convex relaxations: a
semidefinite relaxation that computes a full matrix, a chordal relaxation based
on a chordal extension of the network graph, and a second-order cone relaxation
that computes the smallest partial matrix. We prove a bijection between the
feasible sets of the OPF in the bus injection model and the branch flow model,
establishing the equivalence of these two models and their second-order cone
relaxations. Our results imply that, for radial networks, all these relaxations
are equivalent and one should always solve the second-order cone relaxation.
For mesh networks, the semidefinite relaxation is tighter than the second-order
cone relaxation but requires a heavier computational effort, and the chordal
relaxation strikes a good balance. Simulations are used to illustrate these
results.Comment: 12 pages, 7 figure
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