DC power grids with constant-power loads - Part I: A full characterization of power flow feasibility, long-term voltage stability and their correspondence

In this two-part paper we study the feasibility of the power flow equations of DC power grids with constant power loads. In Part I of this paper we present a rigorous introduction into the problem of power flow feasibility of such power grids, and the problem of selecting a desirable operating point which satisfies the power flow equations. Our main contributions are as follows. We introduce and identify all long-term voltage semi-stable operating points, and show that there exists a one-to-one correspondence between such operating points and the power demands for which the power flow equations are feasible. The unique long-term voltage stable operating point associated to a feasible power demand can be found by solving an initial value problem. In addition, we characterize all feasible power demands as the intersection of closed half-spaces, and give a novel proof for the convexity of the set of feasible power demands. Moreover, we recover a known necessary LMI condition for the feasibility of a vector of power demands and prove that it is also sufficient. We present a similar necessary and sufficient condition for feasibility under small perturbation. Part II of this paper further explores the implications to these results. In particular, we prove that the feasible power demands and high-voltage operating points are in one-to-one correspondence as well, and that the high-voltage operating points and long-term voltage stable operating points coincide. In addition, we show how existing sufficient conditions for the feasibility of the AC power flow equations with constant-power loads can be improved in the DC case.