  A novel method for distributed estimation of the frequency of power systems
is introduced based on the cooperation between multiple measurement nodes. The
proposed distributed widely linear complex Kalman filter (D-ACKF) and the
distributed widely linear extended complex Kalman filter (D-AECKF) employ a
widely linear state space and augmented complex statistics to deal with
unbalanced system conditions and the generality complex signals, both second
order circular (proper) and second order noncircular (improper). It is shown
that the current, strictly linear, estimators are inadequate for unbalanced
systems, a typical case in smart grids, as they do not account for either the
noncircularity of Clarke's \alpha \beta-voltage in unbalanced conditions or the
correlated nature of nodal disturbances. We illuminate the relationship between
the degree of circularity of Clarke's voltage and system imbalance, and prove
that the proposed widely linear estimators are optimal for such conditions,
while also accounting for the correlated and noncircular nature of real-world
nodal disturbances. {Synthetic and real world} case studies over a range of
power system conditions illustrate the theoretical and practical advantages of
the proposed methodology.
