Consequences of computing S = V*I

The standard definition of complex power is $S = VI^* = P + jQ$. With this convention, a lagging (inductive) load absorbs positive $Q$. Using $S = V^*I$ gives the same $P$ but reverses the sign of $Q$.

Derivation

Let $V = |V|\angle\theta_v$ and $I = |I|\angle\theta_i$, with $\phi = \theta_v - \theta_i$ (positive for a lagging current).

$$\begin{aligned} VI^* &= |V||I|\angle(\theta_v-\theta_i) = |V||I|\cos\phi + j|V||I|\sin\phi\\ V^*I &= |V||I|\angle(\theta_i-\theta_v) = |V||I|\cos\phi - j|V||I|\sin\phi \end{aligned}$$

So $V^*I = (VI^*)^* = P - jQ$.

Phasor diagrams (inductive load, current lags)

 S = V I*                 S = V* I
      Q>0  S                 V
     ^    /                 /
     |   / phi             / phi  (V ref)
     |  /                 /------> P
     | /                   \
     |/-------> P           \  S
                             v  Q<0
 Inductive load:           Inductive load:
 Q positive (absorbs)      Q appears negative

Consequences of using $S = V^*I$

• The real power $P$ is unchanged, since $\cos\phi$ is even.
• The sign of $Q$ is reversed: an inductor (lagging current) would appear to supply reactive power, and a capacitor to absorb it.
• This contradicts the standard convention (IEEE/IEC) that inductive loads absorb VAr. Generator, motor and load data would be read wrongly.
• The power factor angle and the power triangle are flipped, so lagging and leading would be confused.
• In load flow and compensation studies, capacitor banks would be sized in the wrong direction.

Example: Take $V = 100\angle0^\circ$ V and $I = 10\angle-30^\circ$ A (inductive load).

• $VI^* = 1000\angle30^\circ = 866 + j500$: the load absorbs 500 var, which is correct.
• $V^*I = 1000\angle-30^\circ = 866 - j500$: this wrongly shows the inductive load as a source of 500 var.

$S = VI^*$ is used so that inductive Q is positive.