p-q Theory power components calculations

The "generalized theory of the instantaneous reactive power in three-phase circuits", proposed by Akagi et al., and also known as the p-q theory, is an interesting tool to apply to the control of active power filters, or even to analyze three-phase power system in order to detect problems related to harmonics, reactive power and unbalance. In this paper it will be shown that in three phase electrical systems the instantaneous power waveform presents symmetries if 1/6, 1/3, 1/2 or 1 cycle of the power system fundamental frequency, depending on the system being balanced or not, and having or not even harmonics (interharmonics and subharmonics are not considered in this analysis). These symmetries can be exploited to accelerate the calculations for active filters controllers based on the p-q theory. In the case of the conventional reactive power or zero-sequence compensation, it is shown that the theoretical control system dynamic response delay is zero.