413 research outputs found
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
Optimal Power Flow in Stand-alone DC Microgrids
Direct-current microgrids (DC-MGs) can operate in either grid-connected or
stand-alone mode. In particular, stand-alone DC-MG has many distinct
applications. However, the optimal power flow problem of a stand-alone DC-MG is
inherently non-convex. In this paper, the optimal power flow (OPF) problem of
DC-MG is investigated considering convex relaxation based on second-order cone
programming (SOCP). Mild assumptions are proposed to guarantee the exactness of
relaxation, which only require uniform nodal voltage upper bounds and positive
network loss. Furthermore, it is revealed that the exactness of SOCP relaxation
of DC-MGs does not rely on either topology or operating mode of DC-MGs, and an
optimal solution must be unique if it exists. If line constraints are
considered, the exactness of SOCP relaxation may not hold. In this regard, two
heuristic methods are proposed to give approximate solutions. Simulations are
conducted to confirm the theoretic results
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
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
Exact Convex Relaxation of Optimal Power Flow in Radial Networks
The optimal power flow (OPF) problem determines power generation/demand that
minimize a certain objective such as generation cost or power loss. It is
nonconvex. We prove that, for radial networks, after shrinking its feasible set
slightly, the global optimum of OPF can be recovered via a second-order cone
programming (SOCP) relaxation under a condition that can be checked a priori.
The condition holds for the IEEE 13-, 34-, 37-, 123-bus networks and two
real-world networks, and has a physical interpretation.Comment: 32 pages, 10 figures, submitted to IEEE Transaction on Automatic
Control. arXiv admin note: text overlap with arXiv:1208.407
Graphical Models for Optimal Power Flow
Optimal power flow (OPF) is the central optimization problem in electric
power grids. Although solved routinely in the course of power grid operations,
it is known to be strongly NP-hard in general, and weakly NP-hard over tree
networks. In this paper, we formulate the optimal power flow problem over tree
networks as an inference problem over a tree-structured graphical model where
the nodal variables are low-dimensional vectors. We adapt the standard dynamic
programming algorithm for inference over a tree-structured graphical model to
the OPF problem. Combining this with an interval discretization of the nodal
variables, we develop an approximation algorithm for the OPF problem. Further,
we use techniques from constraint programming (CP) to perform interval
computations and adaptive bound propagation to obtain practically efficient
algorithms. Compared to previous algorithms that solve OPF with optimality
guarantees using convex relaxations, our approach is able to work for arbitrary
distribution networks and handle mixed-integer optimization problems. Further,
it can be implemented in a distributed message-passing fashion that is scalable
and is suitable for "smart grid" applications like control of distributed
energy resources. We evaluate our technique numerically on several benchmark
networks and show that practical OPF problems can be solved effectively using
this approach.Comment: To appear in Proceedings of the 22nd International Conference on
Principles and Practice of Constraint Programming (CP 2016
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