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Power system state estimation is usually formulated as a weighted least-squares problem and solved iteratively by the normal equations method. The normal equations solution method is well-known to exhibit a tendency to be numerically unstable on some networks. A manifestation of this numerical instability is an ill-conditioned set of linear equations that are to be solved at each iterative step. Recent papers have presented orthogonal factorization methods as remedies for this instability. Experience shows, however, that these methods tend to suffer from excessive fill-in when confronted with networks that have a high measurement redundancy. The folklore of power-system state estimation says that large numbers of bus injection measurements (as opposed to line flow measurements) tend to aggrevate the instability, no explanation for this observation, however, has appeared in the literature. This paper presents an analysis of the sources of ill-conditioning in the power system state estimation problem and offers an alternative solution method, due to Peters and Wilkinson, that overcomes this ill-conditioning without losing matrix sparsity.