The paper relates the power engineering problem of the extremal values of effective energy flow to a well established branch of mathematics; namely, the spectral analysis of linear operators. Variation of the effective energy flow in a linear timeinvariant power distribution system is studied in the light of numerical range of the operator representing the system [Eqs. (18, 19)]. A knowledge of this range and its extremal values depends on the evaluation of eigenvalues and eigenvectors of a complex network function. In most cases an exact evaluation is practically unattainable (Section V). Simple and practical estimations of the energy dissipation range are presented. The novel spectral approach to energy problems may hopefully find useful applications in power transmission and distn'bution.
Z. Background
In the past two decades the problem of the economic dispatch of electrical energy has received wide attention from power engineers. A variety of optimization procedures is presently available for the solution to this class of problems.
Among many known procedures, one may refer to the techniques employed for optimizing active power allocation in a power system. Such procedures usually begin with the calculation of the active or the reactive power in terms of quadratic forms of impedance or admittance functions. The optimization of the desired energy function is traditionally achieved by the Lagrange multiplier technique. (See for instance, Dommel-Tinney
(2), Galiana (3), Rechtschaffen (5).
From the circuit theory point of view the problem of the economic dispatch revolves around the study of energy consumption in a linear n-port with a set of compatible specifications, for example assigned voltage, load or phase requirements at ports. We are not presently concerned with the power dispatch problem which is a broad field in its own right.