It's unfortunate that in the electrical industry we use the same term, Power Factor, to describe two completely different phenomena that have no relationship other than the use of the name.
It's important to use the correct terms. The vector product of the RMS voltage and the RMS current is the apparent power. The apparent power can be broken down into the vector sum of the real power component and the reactive power component. If the current and voltage waveforms are in phase the apparent power and real power are the same, and the reactive power is zero. These are all vector quantities with a magnitude and direction.
With harmonic distortion PF there is no reactive power component. Even if the current waveform is very distorted the power is still 100% real. The calculation of harmonic distortion power factor is not a vector and unlike displacement PF cannot be negative. It's a scalar quantity between 0 and 1 that is inversely proportional to the total harmonic distortion.
The utility might not like having to provide the harmonic components in the current and make a customer put in harmonic filters but it's a separate issue from supplying or sinking reactive power. It is possible to have harmonic and displacement PF issues at the same time but they are still completely different phenomena.
Overall, I find your analysis (the math) less than compelling and only marginally useful.
Your last paragraph is fine though. Both reactive power and harmonic power end up being a strain on the generating equipment even though they do not necessarily increase the fuel requirement.
1. If you are talking about a "vector product", you need to specify which one: dot product, which is a scalar, or cross product, which is a vector.
The cross product of two parallel vectors is zero, so you do not want to use that one. But it is a vector.
What gives you the cos(theta) term you need is the dot product. But it is scalar, so the idea of it being composed of two vector components is not terribly useful either.
And for apparent power, you actually want the scalar product of the scalar RMS amplitudes. Then you draw that as a vector of that magnitude with the appropriate decomposition into real and reactive components.
As concluded in another mega-thread the reactive power does not numerically correspond to any real (even cyclical with net zero) transfer of energy.
There can be the equivalent of reactive power in distortion power factor in that certain current harmonics may transfer power to and from POCO with a net of zero even if they are in phase with the voltage fundamental.
2. It seems a lot simpler to me to use the analysis I described in passing, namely that the apparent power is the product of the RMS current and RMS voltage. That works for both displacement and for distortion power factor. You have to recognize that the RMS current is not in fact a sine wave, so the relationship of RMS to peak and average will not be what you are used to. But there are meters that will measure both and their product is the apparent power.
The real power (average power in one direction over a single cycle) can be determined ONLY by an integral of the instantaneous product of current and voltage over one or more cycles.
In the case of displacement power factor the real power can be derived from the apparent power in terms of the cosine of the phase difference.
In the case of distortion power factor, its relationship to the apparent power can only be predicted if you know the amplitude and phase of all of the harmonics of both voltage and current. Often you can still approximate the voltage as a pure sine wave, but not always.
