  Electrical power systems are one of the most important infrastructures that
support our society. However, their vulnerabilities have raised great concern
recently due to several large-scale blackouts around the world. In this paper,
we investigate the robustness of power systems against cascading failures
initiated by a random attack. This is done under a simple yet useful model
based on global and equal redistribution of load upon failures. We provide a
complete understanding of system robustness by i) deriving an expression for
the final system size as a function of the size of initial attacks; ii)
deriving the critical attack size after which system breaks down completely;
iii) showing that complete system breakdown takes place through a first-order
(i.e., discontinuous) transition in terms of the attack size; and iv)
establishing the optimal load-capacity distribution that maximizes robustness.
In particular, we show that robustness is maximized when the difference between
the capacity and initial load is the same for all lines; i.e., when all lines
have the same redundant space regardless of their initial load. This is in
contrast with the intuitive and commonly used setting where capacity of a line
is a fixed factor of its initial load.
