This paper deals with the power flow (PF) and power system state estimation (PSSE) problems, which play a central role in the analysis and operation of electric power networks. The objective is to find the complex voltage at each bus of a network based on a given set of noiseless or noisy measurements. In this paper, it is assumed that at least two groups of measurements are available: (i) nodal voltage magnitudes, and (ii) one active flow per line for a subset of lines covering a spanning tree of the network. The PF feasibility problem is first cast as an optimization problem by adding a suitable quadratic objective function. Then, the semidefinite programming (SDP) relaxation technique is used to handle the inherent non-convexity of the PF problem. It is shown that as long as voltage angle differences across the lines of the network are not too large (e.g., less than 90° for lossless networks), the SDP problem finds the correct PF solution. By capitalizing on this result, a penalized convex problem is designed to solve the PSSE problem. In addition to a linear term inherited from the SDP relaxation of the PF problem, a cost based on the weighted least absolute value is incorporated in the objective for fitting noisy measurements. The optimal solution of the penalized convex problem is shown to feature a dominant rank-one component formed by lifting the true state of the system. An upper bound on the estimation error is also derived, which depends on the noise power. It is shown that the estimation error reduces as the number of measurements increases. Numerical results for the 1354-bus European system are reported to corroborate the merits of the proposed convexification framework. The mathematical framework developed in this work can be used to study the PSSE problem with other types of measurements.

This paper deals with the nonconvex power system state estimation (PSSE) problem, which plays a central role in the monitoring and operation of electric power networks. Given a set of noisy measurements, PSSE aims at estimating the vector of complex voltages at all buses of the network. This is a challenging task due to the inherent nonlinearity of power flows (PFs), for which the existing methods lack guaranteed convergence and theoretical analysis. Motivated by these limitations, we propose a novel convexification framework for the PSSE problem using semidefinite programming (SDP) and second-order cone programming (SOCP) relaxations. We first study a related PF problem as the noiseless counterpart, which is cast as a constrained minimization program by adding a suitably designed objective function. We study the performance of the proposed framework in the case where the set of measurements includes: 1) nodal voltage magnitudes and 2) branch active PFs over at least a spanning tree of the network. It is shown that the SDP and SOCP relaxations both recover the true PF solution as long as the voltage angle difference across each line of the network is not too large (e.g., less than 90^{\circ } for lossless networks). By capitalizing on this result, penalized SDP and SOCP problems are designed to solve the PSSE, where a penalty based on the weighted least absolute value is incorporated for fitting noisy measurements with possible bad data. Strong theoretical results are derived to quantify the optimal solution of the penalized SDP problem, which is shown to possess a dominant rank-one component formed by lifting the true voltage vector. An upper bound on the estimation error is also derived as a function of the noise power, which decreases exponentially fast as the number of measurements increases. Numerical results on benchmark systems, including a 9241-bus European system, are reported to corroborate the merits of the proposed convexification framework.