It was said: "Our SE converges most of the time, so it works very well."
This is a common understanding in power industry which however may not be true. Normally convergent solution contains Bias if bad measurements and/or system modeling errors are not all effectively detected and removed. Therefore, some weird problems occur from the convergent solution, such as negative loads at some buses, big mismatch at some buses, branches generating real power, etc. The biased solution will have negative impacts on the performance of other functions including contingency analysis, LMP calculation, security assessment, etc.
In summary, convergence of a state estimator does not mean the obtained solution is accurate.
It was said: "SE at our company converges more than 99%, therefore we will not put efforts on 1% convergence improvement."
Let us take the best convergence rate 99.8% as an example. Normally state estimator runs every 5 minutes (it is even shorter for some companies), which means state estimator needs to run 288 times every day. 99.8% implies that there is at least one-time divergence every day in average. One-time divergence per day is not a big deal unless the system is in alert state or is under extreme tense, like what happened in 2003 Northeast blackout which led $7-10 billions economic loss.
In 2011 it was reported in "The Future of the Electric Grid: An Interdisciplinary MIT Study" that “the algorithm (of state estimator) is not perfect, and state estimators have trouble estimating a system state during unusual or emergency conditions – unfortunately, when they are most needed”.
In another word, the 1% divergence rate of state estimator will cause system-scale blackout with a big probability if the system is in an alert state. And it is extremenly hard to fix this 1% divergence because today's state estimators are prone to fail when the system is in an alert state.