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

Abstract

This paper is concerned with the securityconstrained 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.

Distributed algorithms for convexified bad data and topology error detection and identification problems

“…where the state v is extended by adding the line power flow vector f through circuit breakers in the suspicious substation. In equation (3), the term M i f represents the effect of these flows in each measurement z i ; M i is an incident matrix defining the interconnection of suspected circuit breakers [12].…”

“…One can choose p = 2 (for detection) or p = 1 (for identification) in (4) for this purpose, where the parameter p (p ≥ 0) is chosen to achieve the desired performance. The goal is to obtain a joint extended state (v,f ) that best fits the measurement set z according to the measurement model in (3).…”

“…The goal is to improve accuracy with a near global optimum estimate by embedding states into proper high dimensional space. This is because there is a strong relationship between SDP formulation for state estimation and the SDP formulation for optimal power flow, e.g., [2], [3] show that the rank-one condition is satisfied with most power grid topologies. However, as a centralized implementation of the semidefinite programming (SDP)-based approach is computationally prohibitive [1], we propose to implement a distributed algorithm based on the underlying power system graph for scalability.…”

Security-constrained economic dispatch using semidefinite programming

“…Molzahn et al [8] extended Lavaei and Low's SDP formulation to incorporate cases with multiple generators at the same bus and with multiple lines between two buses, enabling a more general model of the power system. Of particular relevance to our work, Madani et al [9] explored an SDP relaxation for the SCOPF problem. The authors relaxed the entire SCOPF problem into a single SDP and proposed a method to reduce its effective size based on tree decomposition and matrix completion.…”

Convex Relaxations of Security Constrained AC Optimal Power Flow Under Uncertainty

“…For formulations related to the non-convex AC-OPF formulation, the interested reader can refer to references listed in [1]. For formulations related to the AC-OPF with convex relaxations, the work in [9] extends the semidefinite relaxation of [8] introducing security constraints and uses a loss penalty term on specific lines to achieve zero relaxation gap.…”