Daniel K. Molzahn

Sparsity-Exploiting Moment-Based Relaxations of the Optimal Power Flow Problem

Convex relaxations of non-convex optimal power flow (OPF) problems have recently attracted significant interest. While existing relaxations globally solve many OPF problems, there are practical problems for which existing relaxations fail to yield physically meaningful solutions. This paper applies moment relaxations to solve many of these OPF problems. The moment relaxations are developed from the Lasserre hierarchy for solving generalized moment problems. Increasing the relaxation order in this hierarchy results in “tighter” relaxations at the computational cost of larger semidefinite programs. Low-order moment relaxations are capable of globally solving many small OPF problems for which existing relaxations fail. By exploiting sparsity and only applying the higher-order relaxation to specific buses, global solutions to larger problems are computationally tractable through the use of an iterative algorithm informed by a heuristic for choosing where to apply the higher-order constraints. With standard semidefinite programming solvers, the algorithm globally solves many test systems with up to 300 buses for which the existing semidefinite relaxation fails to yield globally optimal solutions.

A Sufficient Condition for Global Optimality of Solutions to the Optimal Power Flow Problem

Recent applications of a semidefinite programming (SDP) relaxation to the optimal power flow (OPF) problem offers a polynomial time method to compute a global optimum for a large subclass of OPF problems. In contrast, prior OPF solution methods in the literature guarantee only local optimality for the solution produced. However, solvers employing SDP relaxation remain significantly slower than mature OPF solution codes. This letter seeks to combine the advantages of the two methods. In particular, we develop an SDP-inspired sufficient condition test for global optimality of a candidate OPF solution. This test may then be easily applied to a candidate solution generated by a traditional, only-guaranteed-locally-optimal OPF solver.

Advanced optimization methods for power systems

Power system planning and operation offers multitudinous opportunities for optimization methods. In practice, these problems are generally large-scale, non-linear, subject to uncertainties, and combine both continuous and discrete variables. In the recent years, a number of complementary theoretical advances in addressing such problems have been obtained in the field of applied mathematics. The paper introduces a selection of these advances in the fields of non-convex optimization, in mixed-integer programming, and in optimization under uncertainty. The practical relevance of these developments for power systems planning and operation are discussed, and the opportunities for combining them, together with high-performance computing and big data infrastructures, as well as novel machine learning and randomized algorithms, are highlighted.

A Survey of Distributed Optimization and Control Algorithms for Electric Power Systems

Historically, centrally computed algorithms have been the primary means of power system optimization and control. With increasing penetrations of distributed energy resources requiring optimization and control of power systems with many controllable devices, distributed algorithms have been the subject of significant research interest. This paper surveys the literature of distributed algorithms with applications to optimization and control of power systems. In particular, this paper reviews distributed algorithms for offline solution of optimal power flow (OPF) problems as well as online algorithms for real-time solution of OPF, optimal frequency control, optimal voltage control, and optimal wide-area control problems.

Recent advances in computational methods for the power flow equations

The power flow equations are at the core of most of the computations for designing and operating electric grids. This system of multivariate nonlinear equations relate the power injections and voltages in an electric power system. A plethora of methods have been devised to solve these equations, from Newton-based methods to homotopy continuation and other optimization-based methods. Although many of these methods often efficiently find a high-voltage, stable solution, challenges remain for finding low-voltage solutions, which play significant roles in certain stability-related computations. While we do not claim to have exhausted the existing literature on all related methods, this tutorial paper introduces some of the recent advances in power flow solution methods to the wider power systems community as well as bringing attention from the computational mathematics and optimization communities to power systems problems. After briefly reviewing some of the traditional computational methods used to solve the power flow equations, we focus on three emerging methods: the numerical polynomial homotopy continuation method, Gröbner basis techniques, and moment/sum-of-squares relaxations using semidefinite programming. In passing, we also emphasize the importance of an upper bound on the number of solutions of the power flow equations and review the current status of research in this direction.

Solution of optimal power flow problems using moment relaxations augmented with objective function penalization

The optimal power flow (OPF) problem minimizes the operating cost of an electric power system. Applications of convex relaxation techniques to the non-convex OPF problem have been of recent interest, including work using the Lasserre hierarchy of “moment” relaxations to globally solve many OPF problems. By preprocessing the network model to eliminate low-impedance lines, this paper demonstrates the capability of the moment relaxations to globally solve large OPF problems that minimize active power losses for portions of several European power systems. Large problems with more general objective functions have thus far been computationally intractable for current formulations of the moment relaxations. To overcome this limitation, this paper proposes the combination of an objective function penalization with the moment relaxations. This combination yields feasible points with objective function values that are close to the global optimum of several large OPF problems. Compared to an existing penalization method, the combination of penalization and the moment relaxations eliminates the need to specify one of the penalty parameters and solves a broader class of problems.

A Laplacian-Based Approach for Finding Near Globally Optimal Solutions to OPF Problems

A semidefinite programming (SDP) relaxation globally solves many optimal power flow (OPF) problems. For other OPF problems where the SDP relaxation only provides a lower bound on the objective value rather than the globally optimal decision variables, recent literature has proposed a penalization approach to find feasible points that are often nearly globally optimal. A disadvantage of this penalization approach is the need to specify penalty parameters. This paper presents an alternative approach that algorithmically determines a penalization appropriate for many OPF problems. The proposed approach constrains the generation cost to be close to the lower bound from the SDP relaxation. The objective function is specified using iteratively determined weights for a Laplacian matrix. This approach yields feasible points to the OPF problem that are guaranteed to have objective values near the global optimum due to the constraint on generation cost. The proposed approach is demonstrated on both small OPF problems and a variety of large test cases representing portions of European power systems.

Mixed SDP/SOCP moment relaxations of the optimal power flow problem

Recently, convex “moment” relaxations developed from the Lasserre hierarchy for polynomial optimization problems have been shown capable of globally solving many optimal power flow (OPF) problems. The moment relaxations, which take the form of semidefinite programs (SDP), generalize a previous SDP relaxation of the OPF problem. This paper presents an approach for improving the computational performance of the moment relaxations for many problems. This approach enforces second-order cone programming (SOCP) constraints that establish necessary (but not sufficient) conditions for satisfaction of the SDP constraints arising from the higher-order moment relaxations. The resulting “mixed SDP/SOCP” formulation implements the first-order relaxation using SDP constraints and the higher-order relaxations using SOCP constraints. Numerical results demonstrate that this mixed SDP/SOCP relaxation is capable of solving many problems for which the first-order moment relaxation fails to yield a global solution. For several examples, the mixed SDP/SOCP relaxation improves computational speed by factors from 1.13 to 18.7.

Toward topologically based upper bounds on the number of power flow solutions

The power flow equations, which relate power injections and voltage phasors, are at the heart of many electric power system computations. While Newton-based methods typically find the “high-voltage” solution to the power flow equations, which is of primary interest, there are potentially many “low-voltage” solutions that are useful for certain analyses. This paper addresses the number of solutions to the power flow equations. There exist upper bounds on the number of power flow solutions; however, there is only limited work regarding bounds that are functions of network topology. This paper empirically explores the relationship between the network topology, as characterized by the maximal cliques, and the number of power flow solutions. To facilitate this analysis, we use a numerical polynomial homotopy continuation approach that is guaranteed to find all complex solutions to the power flow equations. The number of solutions obtained from this approach upper bounds the number of real solutions. Testing with many small networks informs the development of upper bounds that are functions of the network topology. Initial results include empirically derived expressions for the maximum number of solutions for certain classes of network topologies.

Convex Relaxations of Optimal Power Flow Problems: An Illustrative Example

Recently, there has been significant interest in convex relaxations of the optimal power flow (OPF) problem. A semidefinite programming (SDP) relaxation globally solves many OPF problems. However, there exist practical problems for which the SDP relaxation fails to yield the global solution. Conditions for the success or failure of the SDP relaxation are valuable for determining whether the relaxation is appropriate for a given OPF problem. To move beyond existing conditions, which only apply to a limited class of problems, a typical conjecture is that failure of the SDP relaxation can be related to physical characteristics of the system. By presenting an example OPF problem with two equivalent formulations, this paper demonstrates that physically based conditions cannot universally explain algorithm behavior. The SDP relaxation fails for one formulation but succeeds in finding the global solution to the other formulation. Since these formulations represent the same system, success (or otherwise) of the SDP relaxation must involve factors beyond just the network physics. The lack of universal physical conditions for success of the SDP relaxation motivates the development of tighter convex relaxations capable of solving a broader class of problems. Tools from polynomial optimization theory provide a means of developing tighter relaxations. This paper uses the example problem to illustrate relaxations from the Lasserre hierarchy for polynomial optimization and a related “mixed semidefinite/second-order cone programming” hierarchy.