  In this paper, the multi-type branching process is applied to describe the
statistics and interdependencies of line outages, the load shed, and isolated
buses. The offspring mean matrix of the multi-type branching process is
estimated by the Expectation Maximization (EM) algorithm and can quantify the
extent of outage propagation. The joint distribution of two types of outages is
estimated by the multi-type branching process via the Lagrange-Good inversion.
The proposed model is tested with data generated by the AC OPA cascading
simulations on the IEEE 118-bus system. The largest eigenvalues of the
offspring mean matrix indicate that the system is closer to criticality when
considering the interdependence of different types of outages. Compared with
empirically estimating the joint distribution of the total outages, good
estimate is obtained by using the multitype branching process with a much
smaller number of cascades, thus greatly improving the efficiency. It is shown
that the multitype branching process can effectively predict the distribution
of the load shed and isolated buses and their conditional largest possible
total outages even when there are no data of them.
