Realizability inequalities for security constrained load flow variables

A number of theoretical results are introduced with potential applications in the area of load flow and optimal load flow with security inequality constraints. This problem is formulated in the space of general load flow variables,
y \in R^{m}
, that is, these quantities which for security reasons should be fixed to specified levels or restricted to lie between upper and lower bounds (real and reactive power injections and flows, voltage magnitudes). These constraints are called here the admissibility constraints and are represented by a hyperbox in
R^{m}, B_{y}
. Since y is subject to the network equations imposed by Kirchhoff's and Ohm's laws, it satisfies a relation of the form
y = G(x)
where
x \in R^{n} (n \leq m)
is the vector of real components of the complex bus voltages. The vector
y
is not arbitrary within
B_{y}
, but must belong to the range space of
G(x)
for
x \in R^{n}
which we call
R_{y}
. A solution to a load flow problem with inequality constraints,
y
, is therefore feasible if and only if
y \in R_{y} \cap B_{y}
. We next exploit a special characteristic of power systems, that is, the fact that the typical system operates "near" the flat voltage profile,
x_{f}
, (all bus voltages equal to
1/0
in per unit), and demonstrate that near
x_{f}, R_{y}
has infinitely many supporting hyperplanes, i.e., planes with the property that for all
y \in R_{y}, \beta^{T}_{y} \geq 0
. These conditions, we call realizability inequalities (RI). The combination of selected RI's with
B_{y}
has the potential to systematically and relatively simply detect infeasibility in the load flow with inequality constraints as well, as to permit a theoretical analysis of this difficult question. In this paper we prove the existence of RI's near the flat voltage profile, and present a simple numerical technique for their computation. A large number of simulation results were made which demonstrate that RI's exist over a broad range of operating conditions.