The pq theory [AKAGI 2007] proposes a precise mathematical modeling for the definition of the powers in GENERIC three-phase electrical systems (that may contain harmonics and unbalance between phases). This theory allows a differentiated mathematical treatment for such generic systems in which the classical theory based on phasors [CLOSE 1966] can not explain with accuracy, causing errors of interpretation on the real behavior of the flows and exchanges of energy in three-phase systems containing linear and non-linear elements, as the modern systems containing FACTS equipment and HVDC links.

In the system shown in the Figure, the three-phase load R1 ≠ R2 ≠ R3 causes unbalance between the system phases and the SVC causes network harmonics. In this figure the real and imaginary powers are indicated, where "~" refer to oscillating power and  "-" refer to the constant power (average value). The imaginary powers (in red) correspond to the energies exchanged between the 3 phases of the system and they do not contribute in any moment to the energy transfer between the source and the load [AKAGI 2007].

Precisely for this reason, in systems containing HVDC links, it is incorrect the frequent affirmation of engineers that the network "supplies" reactive power to the AC/DC converter, or that the reactive power is "consumed" by the converter. The real three-phase power (in blue) supplied to the resistive load p3φ is equal to the sum of the real power p flowing through the three phases with the zero sequence power p0 flowing through the neutral conductor. It is also observed that the imaginary power q1 which circulates in delta connected SVC capacitors is different from the imaginary power q2 which circulates in the TCR terminals.

In addition to providing this correct interpretation of the powers in unbalanced and distorted three-phase systems, the pq theory has great utility in projects involving the control of active filters, power conditioners and general energy quality applications involving compensation of voltage and current harmonics and compensation of unbalances present in the system.

References: 

[AKAGI 2007] Akagi, H., Watanabe, E. H., Aredes, M., "Instantaneous Power Theory and Applications to Power Conditioning", New Jersey: IEEE Press / Wiley-Interscience, 2007.

[CLOSE 1966] Close, C.M., "Analysis of Linear Circuits", Harcourt College Pub, 1966.