Revisiting the maximum power transfer for linear n‐ports with uncoupled loads and applications to power systems

Abstract

Summary This paper revisits three classical results of circuit theory: the Thévenin theorem, the maximum power transfer theorem, and Bode's bilinear theorem, as well as its multilinear generalization due to Lin. Combining these results, it proposes a new measurement‐based approach that provides a practical new version of the ‘maximum’ power transfer theorem for n‐ports terminated with uncoupled loads, in the sense that it allows a functional algebraic description of the entire power hypersurface as a function of the chosen port parameters. This is more useful than merely computing port parameters for maximum power transfer, because the power hypersurface can be used along with constraints on port parameters, such as voltages, to find their values for optimal viable power transfer. An example of maximum power transfer to two chosen load buses is given for the IEEE‐30 bus system, with voltage constraints, in order to show applicability to practical cases. Copyright © 2015 John Wiley & Sons, Ltd.

Single variable expressions for the efficiency of a reciprocal power transfer system

“…In the ideal case, all the input power from the source can be delivered to the load. In configuration (ii), the efficiency approaches exactly 50%, which is in line with the maximum power transfer theorem [7].…”

“…If approaches zero, the mutual coefficients of the impedance matrix Z, that is, r 12 and x 12 , also approach zero. In configuration (ii), the efficiency approaches exactly 50%, which is in line with the maximum power transfer theorem [7]. • In the other limit, when the extended kQ factor becomes very large, the efficiency in configuration (i) approaches 1.…”

“…This is not the case. Indeed, one has to remember that the graph on Figure 3 for configuration (i) is only valid when port #2 is terminated by a load given by (6) and (7). Any other load will result in a lower efficiency than max .…”

“…Taking into account the maximum power transfer theorem [7], the power transfer to the load is maximized when the load Z L = R L + jX L is the complex conjugate of Z th : R L = r 2 11 r 22 − r 11 r 2 12 + r 11 x 2 12 + r 22 x 2 11 − 2r 12 x 12 x 11 r 2 11 + x 2 11 (14) X L = − x 2 11 x 22 − x 11 x 2 12 + x 11 r 2 12 + x 22 r 2 11 − 2x 12 r 12 r 11 x 2 11 + r 2 11 (15) Applying this load to port #2 realizes configuration (ii), which maximizes the power transfer. Notice that R L equals the negative of X L , but with the resistances and reactances swapped (refer to (13)).…”

Available active power of the N‐port AC Thévenin source: A simple behavioral derivation

“…In this author's opinion, the contributions by Desoer 7 and Flanders 10 are the most important ones and also the most demanding ones, because of both the advanced mathematical tools used in the proofs and the complex structure of their investigations. Recently, this very issue motivated Bhaya et al 16 to try giving, within an application-oriented paper, a simpler derivation of the maximum power transfer theorem for linear N-ports with a general load, based on those works. 7,10 Unfortunately, because of some inaccuracies, their goal has not been wholly achieved, and thus it still remains as a matter of interest.…”