Ramtin Madani

Convex Relaxation for Optimal Power Flow Problem: Mesh Networks

This paper is concerned with a fundamental resource allocation problem for electrical power networks. This problem, named optimal power flow (OPF), is nonconvex due to the nonlinearities imposed by the laws of physics, and has been studied since 1962. We have recently shown that a convex relaxation based on semidefinite programming (SDP) is able to find a global solution of OPF for IEEE benchmark systems, and moreover this technique is guaranteed to work over acyclic (distribution) networks. The present work studies the potential of the SDP relaxation for OPF over cyclic (transmission) networks. Given an arbitrary weakly-cyclic network with cycles of size 3, it is shown that the injection region is convex in the lossless case and that the Pareto front of the injection region is convex in the lossy case. It is also proved that the SDP relaxation of OPF is exact for this type of network. Moreover, it is shown that if the SDP relaxation is not exact for a general mesh network, it would still have a low-rank solution whose rank depends on the structure of the network. Finally, a heuristic method is proposed to recover a rank-1 solution for the SDP relaxation whenever the relaxation is not exact.

Convex Relaxation for Optimal Distributed Control Problems

This paper is concerned with the optimal distributed control (ODC) problem. The objective is to design a fixed-order distributed controller with a pre-specified structure for a discrete-time system. It is shown that this NP-hard problem has a quadratic formulation, which can be relaxed to a semidefinite program (SDP). If the SDP relaxation has a rank-1 solution, a globally optimal distributed controller can be recovered from this solution. By utilizing the notion of treewidth, it is proved that the nonlinearity of the ODC problem appears in such a sparse way that its SDP relaxation has a matrix solution with rank at most 3. A near-optimal controller together with a bound on its optimality degree may be obtained by approximating the low-rank SDP solution with a rank-1 matrix. This convexification technique can be applied to both time-domain and Lyapunov-domain formulations of the ODC problem. The efficacy of this method is demonstrated in numerical examples.

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.

Convex relaxation for optimal power flow problem: Mesh networks

This paper is concerned with the optimal power flow (OPF) problem. We have recently shown that a convex relaxation based on semidefinite programming (SDP) is able to find a global solution of OPF for IEEE benchmark systems, and moreover this technique is guaranteed to work over acyclic (distribution) networks. The present work studies the potential of the SDP relaxation for OPF over mesh (transmission) networks. First, we consider a simple class of cyclic systems, namely weakly-cyclic networks with cycles of size 3. We show that the success of the SDP relaxation depends on how the line capacities are modeled mathematically. More precisely, the SDP relaxation is proven to succeed if the capacity of each line is modeled in terms of bus voltage difference, as opposed to line active power, apparent power or angle difference. This result elucidates the role of the problem formulation. Our second contribution is to relate the rank of the minimum-rank solution of the SDP relaxation to the network topology. The goal is to understand how the computational complexity of OPF is related to the underlying topology of the power network. To this end, an upper bound is derived on the rank of the SDP solution, which is expected to be small in practice. A penalization method is then applied to the SDP relaxation to enforce the rank of its solution to become 1, leading to a near-optimal solution for OPF with a guaranteed optimality degree. The remarkable performance of this technique is demonstrated on IEEE systems with more than 7000 different cost functions.

Efficient convex relaxation for stochastic optimal distributed control problem

This paper is concerned with the design of an efficient convex relaxation for the notorious problem of stochastic optimal distributed control (SODC). The objective is to find an optimal structured controller for a dynamical system subject to input disturbance and measurement noise. With no loss of generality, this paper focuses on the design of a static controller for a discrete-time system. First, it is shown that there is a semidefinite programming (SDP) relaxation for this problem with the property that its SDP matrix solution is guaranteed to have rank at most 3. This result is due to the extreme sparsity of the SODC problem. Since this SDP relaxation is computationally expensive, an efficient two-stage algorithm is proposed. A computationally-cheap SDP relaxation is solved in the first stage. The solution is then fed into a second SDP problem to recover a near-global controller with an enforced sparsity pattern. The proposed technique is always exact for the classical H2 optimal control problem (i.e., in the centralized case). The efficacy of our technique is demonstrated on the IEEE 39-bus New England power network, a mass-spring system, and highly-unstable random systems, for which near-optimal stabilizing controllers with global optimality degrees above 90% are designed under a wide range of noise levels.

Low-rank solutions of matrix inequalities with applications to polynomial optimization and matrix completion problems

This paper is concerned with the problem of finding a low-rank solution of an arbitrary sparse linear matrix inequality (LMI). To this end, we map the sparsity of the LMI problem into a graph. We develop a theory relating the rank of the minimum-rank solution of the LMI problem to the sparsity of its underlying graph. Furthermore, we propose two graph-theoretic convex programs to obtain a low-rank solution. The first convex optimization needs a tree decomposition of the sparsity graph. The second one does not rely on any computationally-expensive graph analysis and is always polynomial-time solvable. The results of this work can be readily applied to three separate problems of minimum-rank matrix completion, conic relaxation for polynomial optimization, and affine rank minimization. The results are finally illustrated on two applications of optimal distributed control and nonlinear optimization for electrical networks.

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.

Optimized Compact-Support Interpolation Kernels

In this paper, we investigate the problem of designing compact-support interpolation kernels for a given class of signals. By using calculus of variations, we simplify the optimization problem from an nonlinear infinite dimensional problem to a linear finite dimensional case, and then find the optimum compact-support function that best approximates a given filter in the least square sense (ℓ2 norm). The benefit of compact-support interpolants is the low computational complexity in the interpolation process while the optimum compact-support interpolant guarantees the highest achievable signal-to-noise ratio (SNR). Our simulation results confirm the superior performance of the proposed kernel compared to other conventional compact-support interpolants such as cubic spline.

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.