91 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
Convex Relaxation of Optimal Power Flow, Part I: Formulations and Equivalence
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,
15(1):15-27, March 2014. This is an extended version with Appendices VIII and
IX that provide some mathematical preliminaries and proofs of the main
result
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
Inexact Convex Relaxations for AC Optimal Power Flow: Towards AC Feasibility
Convex relaxations of AC optimal power flow (AC-OPF) problems have attracted
significant interest as in several instances they provably yield the global
optimum to the original non-convex problem. If, however, the relaxation is
inexact, the obtained solution is not AC-feasible. The quality of the obtained
solution is essential for several practical applications of AC-OPF, but
detailed analyses are lacking in existing literature. This paper aims to cover
this gap. We provide an in-depth investigation of the solution characteristics
when convex relaxations are inexact, we assess the most promising AC
feasibility recovery methods for large-scale systems, and we propose two new
metrics that lead to a better understanding of the quality of the identified
solutions. We perform a comprehensive assessment on 96 different test cases,
ranging from 14 to 3120 buses, and we show the following: (i) Despite an
optimality gap of less than 1%, several test cases still exhibit substantial
distances to both AC feasibility and local optimality and the newly proposed
metrics characterize these deviations. (ii) Penalization methods fail to
recover an AC-feasible solution in 15 out of 45 cases, and using the proposed
metrics, we show that most failed test instances exhibit substantial distances
to both AC-feasibility and local optimality. For failed test instances with
small distances, we show how our proposed metrics inform a fine-tuning of
penalty weights to obtain AC-feasible solutions. (iii) The computational
benefits of warm-starting non-convex solvers have significant variation, but a
computational speedup exists in over 75% of the cases
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
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
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