  This paper develops the theoretical framework and the equations of a new
robust Generalized Maximum-likelihood-type Unscented Kalman Filter (GM-UKF)
that is able to suppress observation and innovation outliers while filtering
out non-Gaussian measurement noise. Because the errors of the real and reactive
power measurements calculated using Phasor Measurement Units (PMUs) follow
long-tailed probability distributions, the conventional UKF provides strongly
biased state estimates since it relies on the weighted least squares estimator.
By contrast, the state estimates and residuals of our GM-UKF are proved to be
roughly Gaussian, allowing the sigma points to reliably approximate the mean
and the covariance matrices of the predicted and corrected state vectors. To
develop our GM-UKF, we first derive a batch-mode regression form by processing
the predictions and observations simultaneously, where the statistical
linearization approach is used. We show that the set of equations so derived
are equivalent to those of the unscented transformation. Then, a robust
GM-estimator that minimizes a convex Huber cost function while using weights
calculated via Projection Statistics (PS's) is proposed. The PS's are applied
to a two-dimensional matrix that consists of serially correlated predicted
state and innovation vectors to detect observation and innovation outliers.
These outliers are suppressed by the GM-estimator using the iteratively
reweighted least squares algorithm. Finally, the asymptotic error covariance
matrix of the GM-UKF state estimates is derived from the total influence
function. In the companion paper, extensive simulation results will be shown to
verify the effectiveness and robustness of the proposed method.
