Hilbert space techniques for reactive power compensation in polyphase systems with unequal line resistances

The paper considers the problem of reactive power compensation in polyphase systems with unequal line resistances. It is suggested that a weighted Hilbert space offers a natural framework for loss minimization problems, as they can be solved using projections on appropriate subspaces. The paper discusses the concepts of weighted real (active), reactive and apparent power, and introduces orthogonal current and power decompositions. The particular case of symmetric lines is addressed in more detail, as it sheds additional light on definitions used in IEEE standard 1459. A simple resistive example is used to illustrate the special case of sinusoidal voltage and current.