Realizability inequalities for security constrained load flow variables

Abstract

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, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y \in R^{m}</tex> , 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{m}, B_{y}</tex> . Since y is subject to the network equations imposed by Kirchhoff's and Ohm's laws, it satisfies a relation of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y = G(x)</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x \in R^{n} (n \leq m)</tex> is the vector of real components of the complex bus voltages. The vector <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y</tex> is not arbitrary within <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B_{y}</tex> , but must belong to the range space of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G(x)</tex> for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x \in R^{n}</tex> which we call <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{y}</tex> . A solution to a load flow problem with inequality constraints, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y</tex> , is therefore feasible if and only if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y \in R_{y} \cap B_{y}</tex> . We next exploit a special characteristic of power systems, that is, the fact that the typical system operates "near" the flat voltage profile, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x_{f}</tex> , (all bus voltages equal to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/0</tex> in per unit), and demonstrate that near <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x_{f}, R_{y}</tex> has infinitely many supporting hyperplanes, i.e., planes with the property that for all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y \in R_{y}, \beta^{T}_{y} \geq 0</tex> . These conditions, we call realizability inequalities (RI). The combination of selected RI's with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B_{y}</tex> 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.