How do residential analog and smart meters measure power?
Analogue meters measure the average value of the instantaneous values of current and voltage multiplied together: -


power = Average(v x i)
energy = integral(v x i)




Power factor is irrelevant because there is only one way to measure power and that is how I have stated it.

Trying to "use" power factor, RMS voltage and RMS current in order to calculate power is a fruitless task given the number of harmonics present in many current waveforms.

A moving coil power measurement device (aka the dynamometer): -



These days microprocessors are used and they sample voltage and current waveforms and multiply the numbers. Surprisingly high sample rates are used to avoid the harmonics in the current waveform aliasing the ADC.

If power factor is needed to be displayed it is done by taking real power and dividing numerically by the product of RMS voltage and RMS current. The same samples that are used to calculate power can be used for calculating RMS voltage and RMS current.

A company that I used to work for (Landis and Gyr) pioneered the hall effect power meter - this was basically a solid state version of the two-coil dynamometer shown above.

Just in case there are any doubts that the power measurment method can cope with harmonics of current take a look at this: -



Here we have voltage at 50Hz and a common harmonic (3rd) of current at 150Hz - see that the instantaneous waveform of power has zero average, just as it should do.