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The synchronised phasor measurement units (PMUs) can provide extremely accurate measurements that will improve the state estimation performance and observability. Although full observability based only on PMUs appears very impractical today, it will be the case in a few years when these devices become standard equipment at substations. This paper compares the performances of three state estimation formulations, in terms of the convergence properties and the variance in the estimated states. Phasor measurements, as well as zero injections, are incorporated and either rectangular or polar coordinates can be used to represent the bus voltage and branch current phasors measured by PMUs. Current phasors in polar coordinates may give rise to ill-conditioning problems, which are resolved by conversion into rectangular coordinates format, using the uncertainty propagation theory. The behaviour of the elements in the measurement Jacobian matrix, because of the current measurements, is investigated for possible ill-conditioning of the state estimator coefficient matrix. A non-iterative topological-numerical algorithm is proposed for observability checking and restoration in case of observability loss. The proposed algorithms are demonstrated with the IEEE 14-bus and IEEE 118-bus systems.