  This paper deals with the non-convex 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, for which existing
methods lack guaranteed convergence and theoretical analysis. Motivating by
these limitations, we propose a novel convexification framework for the PSSE
using semidefinite programming (SDP) and second-order cone programming (SOCP)
relaxations. We first study a related power flow (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: (i) nodal voltage
magnitudes, and (ii) branch active power flows over 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 degrees 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.
