22 research outputs found
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
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
Tightening QC Relaxations of AC Optimal Power Flow Problems via Complex Per Unit Normalization
Optimal power flow (OPF) is a key problem in power system operations. OPF
problems that use the nonlinear AC power flow equations to accurately model the
network physics have inherent challenges associated with non-convexity. To
address these challenges, recent research has applied various convex relaxation
approaches to OPF problems. The QC relaxation is a promising approach that
convexifies the trigonometric and product terms in the OPF problem by enclosing
these terms in convex envelopes. The accuracy of the QC relaxation strongly
depends on the tightness of these envelopes. This paper presents two
improvements to these envelopes. The first improvement leverages a polar
representation of the branch admittances in addition to the rectangular
representation used previously. The second improvement is based on a coordinate
transformation via a complex per unit base power normalization that rotates the
power flow equations. The trigonometric envelopes resulting from this rotation
can be tighter than the corresponding envelopes in previous QC relaxation
formulations. Using an empirical analysis with a variety of test cases, this
paper suggests an appropriate value for the angle of the complex base power.
Comparing the results with a state-of-the-art QC formulation reveals the
advantages of the proposed improvements
Multi-Objective Dynamic Economic Dispatch with Demand Side Management of Residential Loads and Electric Vehicles
In this paper, a multi-objective optimization method based on the normal boundary intersection is proposed to solve the dynamic economic dispatch with demand side management of individual residential loads and electric vehicles. The proposed approach specifically addresses consumer comfort through acceptable appliance deferral times and electric vehicle charging requirements. The multi-objectives of minimizing generation costs, emissions, and energy loss in the system are balanced in a Pareto front approach in which a fuzzy decision making method has been implemented to find the best compromise solution based on desired system operating conditions. The normal boundary intersection method is described and validated
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