15 research outputs found
Matrix Minor Reformulation and SOCP-based Spatial Branch-and-Cut Method for the AC Optimal Power Flow Problem
Alternating current optimal power flow (AC OPF) is one of the most
fundamental optimization problems in electrical power systems. It can be
formulated as a semidefinite program (SDP) with rank constraints. Solving AC
OPF, that is, obtaining near optimal primal solutions as well as high quality
dual bounds for this non-convex program, presents a major computational
challenge to today's power industry for the real-time operation of large-scale
power grids. In this paper, we propose a new technique for reformulation of the
rank constraints using both principal and non-principal 2-by-2 minors of the
involved Hermitian matrix variable and characterize all such minors into three
types. We show the equivalence of these minor constraints to the physical
constraints of voltage angle differences summing to zero over three- and
four-cycles in the power network. We study second-order conic programming
(SOCP) relaxations of this minor reformulation and propose strong cutting
planes, convex envelopes, and bound tightening techniques to strengthen the
resulting SOCP relaxations. We then propose an SOCP-based spatial
branch-and-cut method to obtain the global optimum of AC OPF. Extensive
computational experiments show that the proposed algorithm significantly
outperforms the state-of-the-art SDP-based OPF solver and on a simple personal
computer is able to obtain on average a 0.71% optimality gap in no more than
720 seconds for the most challenging power system instances in the literature
Improving QC Relaxations of OPF Problems via Voltage Magnitude Difference Constraints and Envelopes for Trilinear Monomials
AC optimal power flow (AC~OPF) is a challenging non-convex optimization
problem that plays a crucial role in power system operation and control.
Recently developed convex relaxation techniques provide new insights regarding
the global optimality of AC~OPF solutions. The quadratic convex (QC) relaxation
is one promising approach that constructs convex envelopes around the
trigonometric and product terms in the polar representation of the power flow
equations. This paper proposes two methods for tightening the QC relaxation.
The first method introduces new variables that represent the voltage magnitude
differences between connected buses. Using "bound tightening" techniques, the
bounds on the voltage magnitude difference variables can be significantly
smaller than the bounds on the voltage magnitudes themselves, so constraints
based on voltage magnitude differences can tighten the relaxation. Second,
rather than a potentially weaker "nested McCormick" formulation, this paper
applies "Meyer and Floudas" envelopes that yield the convex hull of the
trilinear monomials formed by the product of the voltage magnitudes and
trignometric terms in the polar form of the power flow equations. Comparison to
a state-of-the-art QC implementation demonstrates the advantages of these
improvements via smaller optimality gaps.Comment: 8 pages, 1 figur
Strengthening QC relaxations of optimal power flow problems by exploiting various coordinate changes
Motivated by the potential for improvements in electric power system economics, this dissertation studies the AC optimal power flow (AC OPF) problem. An AC OPF problem optimizes a specified objective function subject to constraints imposed by both the non-linear power flow equations and engineering limits. The difficulty of an AC OPF problem is strongly connected to its feasible space\u27s characteristics. This dissertation first investigates causes of nonconvexities in AC OPF problems. Understanding typical causes of nonconvexities is helpful for improving AC OPF solution methodologies.
This dissertation next focuses on solution methods for AC OPF problems that are based on convex relaxations. The quadratic convex (QC) relaxation is one promising approach that constructs convex envelopes around the trigonometric and product terms in the polar representation of the power flow equations. This dissertation proposes several improvements to strengthen QC relaxations of OPF problems. The first group of improvements provides tighter envelopes for the trigonometric functions and product terms in the power flow equations. Methods for obtaining tighter envelopes includes implementing Meyer and Floudas envelopes that yield the convex hull of trilinear monomials. Furthermore, by leveraging a representation of line admittances in polar form, this dissertation proposes tighter envelopes for the trigonometric terms. Another proposed improvement exploits the ability to rotate the base power used in the per unit normalization in order to facilitate the application of tighter trigonometric envelopes.
The second group of improvements propose additional constraints based on new variables that represent voltage magnitude differences between connected buses. Using \u27bound tightening\u27 techniques, the bounds on the voltage magnitude difference variables can be significantly tighter than the bounds on the voltage magnitudes themselves, so constraints based on voltage magnitude differences can improve the QC relaxation --Abstract, page iv
Optimal Power Flow with Step-Voltage Regulators in Multi-Phase Distribution Networks
This paper develops a branch-flow based optimal power flow (OPF) problem for
multi-phase distribution networks that allows for tap selection of wye,
closed-delta, and open-delta step-voltage regulators (SVRs). SVRs are assumed
ideal and their taps are represented by continuous decision variables. To
tackle the non-linearity, the branch-flow semidefinite programming framework of
traditional OPF is expanded to accommodate SVR edges. Three types of
non-convexity are addressed: (a) rank-1 constraints on non-SVR edges, (b)
nonlinear equality constraints on SVR power flows and taps, and (c) trilinear
equalities on SVR voltages and taps. Leveraging a practical phase-separation
assumption on the SVR secondary voltage, novel McCormick relaxations are
provided for (c) and certain rank-1 constraints of (a), while dropping the
rest. A linear relaxation based on conservation of power is used in place of
(b). Numerical simulations on standard distribution test feeders corroborate
the merits of the proposed convex formulation.Comment: This manuscript has been submitted to IEEE Transactions on Power
System
Tightening QC Relaxations of AC Optimal Power Flow through Improved Linear Convex Envelopes
AC optimal power flow (AC OPF) is a fundamental problem in power system
operations. Accurately modeling the network physics via the AC power flow
equations makes AC OPF a challenging nonconvex problem. To search for global
optima, recent research has developed a variety of convex relaxations that
bound the optimal objective values of AC OPF problems. The well-known QC
relaxation convexifies the AC OPF problem by enclosing the non-convex terms
(trigonometric functions and products) within convex envelopes. The accuracy of
this method strongly depends on the tightness of these envelopes. This paper
proposes two improvements for tightening QC relaxations of OPF problems. We
first consider a particular nonlinear function whose projections are the
nonlinear expressions appearing in the polar representation of the power flow
equations. We construct a convex envelope around this nonlinear function that
takes the form of a polytope and then use projections of this envelope to
obtain convex expressions for the nonlinear terms. Second, we use certain
characteristics of the sine and cosine expressions along with the changes in
their curvature to tighten this convex envelope. We also propose a coordinate
transformation that rotates the power flow equations by an angle specific to
each bus in order to obtain a tighter envelope. We demonstrate these
improvements relative to a state-of-the-art QC relaxation implementation using
the PGLib-OPF test cases. The results show improved optimality gaps in 68% of
these cases