Sign of 3-phase instantaneous reactive power and phase sequence

Question

For three-phase systems, in p-q theory instantaneous reactive power is

$$ q = \frac{(v_a-v_b)i_c + (v_b-v_c)i_a + (v_c-v_a)i_b}{\sqrt{3}} $$

(see, e.g., [1] Eq. (5))

If we substitute in expressions for a balanced inductive load, in which the current lags the voltage by 90 degrees: $$ \begin{align} v_a &= V \sin(\omega t)\\ v_b &= V \sin(\omega t - 2\pi/3)\\ v_c &= V \sin(\omega t + 2\pi/3)\\ i_a &= I \sin(\omega t - \pi/2)\\ i_b &= I \sin(\omega t - \pi/2 - 2\pi/3)\\ i_c &= I \sin(\omega t - \pi/2 + 2\pi/3)\\ \end{align} $$

we find (I used Sympy to verify this): $$ q = \frac{3VI}{2} > 0 \quad\text{(if V, I > 0)} $$ which matches up with, say, a phasor-based computation of total reactive power for a pure inductive load.

However, if we reverse the phase sequence by swapping phases b and c, this equation becomes $$ \begin{align} q' &= \frac{(v_a-v_c)i_b + (v_c-v_b)i_a + (v_b-v_a)i_c}{\sqrt{3}}\\ &=\frac{-(v_c-v_a)i_b - (v_b-v_c)i_a - (v_a-v_b)i_c}{\sqrt{3}}\\ &=\frac{-(v_a-v_b)i_c -(v_b-v_c)i_a -(v_c-v_a)i_b}{\sqrt{3}}\\ &=-q\\ \end{align} $$

and so the instantaneous reactive power changes sign.

My question: to relate p-q theory 3-phase instantaneous reactive power to "conventional" reactive power, must we take the phase sequence into account?

[1] "p-q Theory Power Components Calculations", Afonso, Freitas, and Martins. DOI 10.1109/ISIE.2003.1267279, available here

Answer 1

Phase sequence does not have an effect on P or Q when P or Q is averaged over an integral number of cycles. When thus averaged, Q will be zero and P can be positive, negative or zero. P is negative when the source is absorbing power rather than supplying power.

Answer 2

The problem is that pq theory misinterprets the power phenomea. It calls "reactive power" to something not generally related to the physical true meaning of reactive power, i.e., the energy stored in magnetic (inductor) or electric fields (Capacitor). Only in balanced and single phase systems, there is one to one correspondence, but not in non-sinusoidal or multiphase systems.

The true meaning can only be attached to Maxwell equations through the Poynting Vector. So the result is that you find a "strange" behaviour for q.

PQ theory is widely used for current compensation and it does a good job, but it cannot be used for a physical interpretation of power flow.