2,240 research outputs found
Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities
We prove a nearly optimal bound on the number of stable homotopy types
occurring in a k-parameter semi-algebraic family of sets in , each
defined in terms of m quadratic inequalities. Our bound is exponential in k and
m, but polynomial in . More precisely, we prove the following. Let
be a real closed field and let with . Let be a
semi-algebraic set, defined by a Boolean formula without negations, whose atoms
are of the form, . Let be the projection on the last k co-ordinates. Then, the number of
stable homotopy types amongst the fibers S_{\x} = \pi^{-1}(\x) \cap S is
bounded by Comment: 27 pages, 1 figur
On the Properties of the Compound Nodal Admittance Matrix of Polyphase Power Systems
Most techniques for power system analysis model the grid by exact electrical
circuits. For instance, in power flow study, state estimation, and voltage
stability assessment, the use of admittance parameters (i.e., the nodal
admittance matrix) and hybrid parameters is common. Moreover, network reduction
techniques (e.g., Kron reduction) are often applied to decrease the size of
large grid models (i.e., with hundreds or thousands of state variables),
thereby alleviating the computational burden. However, researchers normally
disregard the fact that the applicability of these methods is not generally
guaranteed. In reality, the nodal admittance must satisfy certain properties in
order for hybrid parameters to exist and Kron reduction to be feasible.
Recently, this problem was solved for the particular cases of monophase and
balanced triphase grids. This paper investigates the general case of unbalanced
polyphase grids. Firstly, conditions determining the rank of the so-called
compound nodal admittance matrix and its diagonal subblocks are deduced from
the characteristics of the electrical components and the network graph.
Secondly, the implications of these findings concerning the feasibility of Kron
reduction and the existence of hybrid parameters are discussed. In this regard,
this paper provides a rigorous theoretical foundation for various applications
in power system analysi
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