Dear,
I would understand the limitations of using a fixed and measured main Voltage value for calculating the Real Power (without use of Voltage transformer) whit the emon arduino library.

I understand that the main voltage varies under a maximum and minimum value as stated in the utility contract, so the P calculation is affected by not considering this variations in the VxIxcos(phi) formula.
What is not clear to me is that having this fixed value of V could affect the Real Power calculation against the inductive or reactive nature of the load, and so not taking into account the “real” shift between current and voltage. In other word, fixing the V, does the emon library take in account the cos(phi) during the multiplication of the istantaneous values?

You cannot measure real power without having a sample of the voltage and a sample of the current. It is simply not possible.

What emonLib does, when voltage and current are both available, is it samples voltage at an instant in time, then it samples the current, and multiplies the two to calculate the power at that instant. It repeats this about 2500 times per second, and calculates the average. This is real power. It knows nothing about phase and cos(φ) - in fact cos(φ) exists only in the classroom and in books, because it only works when you have a pure sine wave shape to both current and voltage. When either or both are not pure sine waves, which is usually what happens in the real world, cos(φ) means nothing.

When you do not have a voltage to measure, or even when you have a voltage but you cannot compare it millisecond by millisecond against the current, all you can have is apparent power. This is the number you calculate when you measure the voltage with your meter to get the root mean square (rms) average voltage, and then you do the same with the current, and multiply the two rms values together.

When you do not even know the voltage from one minute to the next, all you can have is a guess at what the apparent power might be. The number you get will be as accurate, or as inaccurate, as your guess for what the true value of the voltage is.

Therefore:
emonLib gives an accurate real power only when it measures voltage & current at the same time.
emonLib gives an estimate of apparent power if it measures current but you tell it an estimate of the voltage.

Thanks a lot for your accurate answer Robert. You clarified completely my doubt.

So the way to calculate the Reactive power (sen(phi) in our books ) is to consider only the sample v(t)*i(t) in which the computed instantaneous value is negative?

And what in case I would discriminate Inductive Reactive Power and Capacitive Reactive Power?

is to delay the voltage (usually) wave by 90°, and do exactly the same calculation as you did for the real power.

What I think you’re really asking is how do you know whether you have a leading or lagging power factor?
Remember the definition of power factor: it is the ratio of real power to apparent power, and emonLib calculates it exactly like that. emonLib will not, cannot, tell you whether it is leading or lagging, because it does not look for the quadrature component. This is what delaying the voltage by 90° will do for you.

Again, power factor = cos (φ) is a result of the definition if, and only if, both voltage and current are perfect sine waves. You can have a power factor for any shape of wave, you cannot have a phase angle unless both waves are identical in shape. When one of them - usually the current - is something like this (red trace)