Morteza Ashraphijuo

Promises of Conic Relaxation for Contingency-Constrained Optimal Power Flow Problem

This paper is concerned with the security-constrained optimal power flow (SCOPF) problem, where each contingency corresponds to the outage of an arbitrary number of lines and generators. The problem is studied by means of a convex relaxation, named semidefinite program (SDP). The existence of a rank-1 SDP solution guarantees the recovery of a global solution of SCOPF. We prove that the rank of the SDP solution is upper bounded by the treewidth of the power network plus one, which is perceived to be small in practice. We then propose a decomposition method to reduce the computational complexity of the relaxation. In the case where the relaxation is not exact, we develop a graph-theoretic convex program to identify the problematic lines of the network and incorporate the loss over those lines into the objective as a penalization (regularization) term, leading to a penalized SDP problem. We perform several simulations on large-scale benchmark systems and verify that the global minima are at most 1% away from the feasible solutions obtained from the proposed penalized relaxation.

Promises of conic relaxation for contingency-constrained optimal power flow problem

This paper is concerned with the security-constrained optimal power flow (SCOPF) problem, where each contingency corresponds to the outage of an arbitrary number of lines and generators. The problem is studied by means of a convex relaxation, named semidefinite program (SDP). The existence of a rank-1 SDP solution guarantees the recovery of a global solution of SCOPF. We prove that the rank of the SDP solution is upper bounded by the treewidth of the power network plus one, which is perceived to be small in practice. We then propose a decomposition method to reduce the computational complexity of the relaxation. In the case where the relaxation is not exact, we develop a graph-theoretic convex program to identify the problematic lines of the network and incorporate the loss over those lines into the objective as a penalization (regularization) term, leading to a penalized SDP problem. We perform several simulations on large-scale benchmark systems and verify that the global minima are at most 1% away from the feasible solutions obtained from the proposed penalized relaxation.

Characterization of rank-constrained feasibility problems via a finite number of convex programs

In this paper, the rank-constrained matrix feasibility problem is considered, where an unknown positive semidefinite (PSD) matrix is to be found based on a set of linear specifications. First, we consider a scenario for which the number of given linear specifications is at least equal to the dimension of the corresponding space of rank-constrained matrices. Given a nominal symmetric and PSD matrix, we design a convex program with the property that every arbitrary matrix could be recovered by this convex program based on its specifications if: i) the unknown matrix has the same size and rank as the nominal matrix, and ii) the distance between the nominal and unknown matrices is less than a positive constant number. It is also shown that if the number of specifications is nearly doubled, then it is possible to recover all rank-constrained PSD matrices through a finite number of convex programs. The results of this paper are demonstrated on many randomly generated matrices.

A characterization of sampling patterns for low-tucker-rank tensor completion problem

In this paper, we characterize the deterministic conditions on the locations of the sampled entries, which are equivalent (necessary and sufficient) to finite completability of a tensor given some components of its Tucker rank. In order to derive this characterization, we propose an algebraic geometric analysis on the Tucker manifold, which allows us to incorporate multiple rank components in the proposed analysis in contrast with the conventional geometric approaches on the Grassmannian manifold. Then, using the developed tools for this analysis, we also derive a sufficient condition on the sampling pattern that ensures there exists only one completion for the sampled tensor (unique completability).

Inverse function theorem for polynomial equations using semidefinite programming

This paper is concerned with obtaining the inverse of polynomial functions using semidefinite programming (SDP). Given a polynomial function and a nominal point at which the Jacobian of the function is invertible, the inverse function theorem states that the inverse of the polynomial function exists at a neighborhood of the nominal point. In this work, we show that this inverse function can be found locally using convex optimization. More precisely, we propose infinitely many SDPs, each of which finds the inverse function at a neighborhood of the nominal point. We also design a convex optimization to check the existence of an SDP problem that finds the inverse of the polynomial function at multiple nominal points and a neighborhood around each point. This makes it possible to identify an SDP problem (if any) that finds the inverse function over a large region. As an application, any system of polynomial equations can be solved by means of the proposed SDP problem whenever an approximate solution is available. The method developed in this work is numerically compared with Newtons method and the nuclear-norm technique.

A characterization of sampling patterns for low-rank multi-view data completion problem

In this paper, we consider the problem of completing a sampled matrix U = [U1|U2] given the ranks of U, U1, and U2 which is known as the multi-view data completion problem. We characterize the deterministic conditions on the locations of the sampled entries that is equivalent (necessary and sufficient) to finite completability of the sampled matrix. To this end, in contrast with the existing analysis on Grassmannian manifold for a single-view matrix, i.e., conventional matrix completion, we propose a geometric analysis on the manifold structure for multi-view data to incorporate more than one rank constraint. Then, using the proposed geometric analysis, we propose sufficient conditions on the sampling pattern, under which there exists only one completion (unique completability) given the three rank constraints.

Power system state estimation with a limited number of measurements

This paper is concerned with the power system state estimation (PSSE) problem, which aims to find the unknown operating point of a power network based on a given set of measurements. The measurements of the PSSE problem are allowed to take any arbitrary combination of nodal active powers, nodal reactive powers, nodal voltage magnitudes and line flows. This problem is non-convex and NP-hard in the worst case. We develop a set of convex programs with the property that they all solve the non-convex PSSE problem in the case of noiseless measurements as long as the voltage angles are relatively small. This result is then extended to a general PSSE problem with noisy measurements, and an upper bound on the estimation error is derived. The objective function of each convex program developed in this paper has two terms: one accounting for the non-convexity of the power flow equations and another one for estimating the noise levels. The proposed technique is demonstrated on the 1354-bus European network.

A strong semidefinite programming relaxation of the unit commitment problem

The unit commitment (UC) problem aims to find an optimal schedule of generating units subject to the demand and operating constraints for an electricity grid. The majority of existing algorithms for the UC problem rely on solving a series of convex relaxations by means of branch-and-bound or cutting-planning methods. In this paper, we develop a strengthened semidefinite program (SDP) for the UC problem by first deriving certain valid quadratic constraints and then relaxing them to linear matrix inequalities. These valid inequalities are obtained by the multiplication of the linear constraints of the UC problem such as the flow constraints of two different lines. The performance of the proposed convex relaxation is evaluated on several instances of the UC problem. For most of the instances, globally optimal integer solutions are obtained by solving a single convex problem. Since the proposed technique leads to a large number of valid quadratic inequalities, an iterative procedure is devised to impose a small number of such valid inequalities. For the cases where the strengthened SDP does give a global integer solution, we incorporate other valid inequalities, including a set of Boolean quadric polytope constraints. The proposed relaxations are extensively tested on various IEEE power systems in simulations.