We know the typical equation for instantaneous power (which can be proven): \$p(t)=v(t)i(t)\$. In sinusoidal steady state (let's ignore harmonics for simplicity), \$v(t)=\sqrt{2} V_{\text{rms}} \cos{(\omega t + \phi_v)}\$ and \$i(t)=\sqrt{2} I_{\text{rms}} \cos{(\omega t + \phi_i)}\$, where \$\omega\$ and \$T = 2 \pi / \omega \$ are respectively the angular frequency and period of \$v\$ and \$i\$. From this, letting \$ \theta = \phi_v – \phi_i\$, it can be shown that
\$ \begin{align} p(t) &= \underbrace{\left[ V_{\text{rms}} I_{\text{rms}} \cos{(\phi_v – \phi_i)} \right]}_{\text{DC component}} + \underbrace{\left[ V_{\text{rms}} I_{\text{rms}} \right] \cos{(2 \omega t + \phi_v + \phi_i)}}_{\text{AC component}} \tag*{} \\ &= \underbrace{\left[ V_{\text{rms}} I_{\text{rms}} \cos{\theta} \right]}_{\text{unidirectional}} + \underbrace{\left[ V_{\text{rms}} I_{\text{rms}} \cos{(\phi_v + \phi_i)} \right] \cos{2 \omega t} – \left[ V_{\text{rms}} I_{\text{rms}} \sin{(\phi_v + \phi_i)} \right] \sin{2 \omega t}}_{\text{bidirectional}} \end{align} \$
where \$2 \omega\$ and \$T' = 2 \pi / 2 \omega = T/2\$ are respectively the angular frequency and period of \$p\$. Now, since \$p(t) \overset{\text{def}}{=} dw(t)/dt\$, then the energy transferred in an integer multiple \$n\$ of the period of \$p\$ is
\$ \begin{align} W &= \displaystyle\int_0^{nT'} p(t) \, dt \tag*{} \\ &= \underbrace{P \displaystyle\int_0^{nT'} \, dt}_{\text{net energy}} + \underbrace{V_{\text{rms}} I_{\text{rms}} \cos{(\phi_v + \phi_i)} \displaystyle\int_0^{nT'} \cos{2 \omega t} \, dt – V_{\text{rms}} I_{\text{rms}} \sin{(\phi_v + \phi_i)} \displaystyle\int_0^{nT'} \sin{2 \omega t} \, dt}_{\text{no net energy}} \\ &=
P \cdot n \cdot T' + 0 \end{align} \$
It's clear that there's no net energy transferred due to the reactive component of a load impedance. However, I've seen three documents (this webpage in section 1.7, this PDF in page 2, this PDF,) talk about reactive energy and their corresponding meters. Since, as shown above, there's no net energy transferred due to the reactance of loads, what energy do those meters actually read? If it's "reactive energy", what do they mean by that? Two documents I found define reactive energy mathematically as
\$ \dfrac{1}{T} \displaystyle\int_0^{T} v(t) i \left( t+\dfrac{T}{4} \right) \, dt = Q \tag*{} \$
which is non-sense; they're defining reactive energy as reactive power, which is a different quantity; energy has units of joules, yet the right-hand side of the previous relation has units of joules per second (or watts or VArs; they're dimensionally the same). Why do they define it as such? I read these three questions (1, 2, 3), but they don't really address the ones I've asked.
EDIT: My questions are not directly about reactive power, but about reactive energy.
Best Answer
In the AC power industry, reactive energy is electrical energy that is stored rather than converted to some other form of energy and thus "used" or "consumed." Reactive power is the rate of transfer of reactive energy from one storage component to another.
The diagram below shows the typical transfer of power from the electrical grid to a point of use. The source voltage is supplied to the user and is assumed to be an ideal single-phase AC voltage source. The load can be represented as a resistor in parallel with an inductor. The source voltage is the voltage across both components of the load.
The resistor current is in phase with the source voltage. The instantaneous resistor power waveform is the product of the resistor current multiplied by the source voltage. That minimum points on that curve lie on the X axis. The power is positive at all times, indicating that all of the power is transferred from the source to the resistor. The area under the curve represents the energy received by the resistor and dissipated as heat.
The inductor current lags the source voltage by 90 degrees. The product of the source voltage and the inductor current is a sine wave that has positive and negative values that average zero. Since that does not represent real power, is called "volt-amperes, reactive" or "VARs." There are equal areas above and below the curve indicating energy received from the source and returned to the source. That is the reactive energy.
Shown as an ideal circuit, the average and the net reactive energy transfer is zero. However there is real energy that is constantly going back and forth. In an ideal system, the reactive energy is generated when the load is connected, transfers back and forth as long as the load is connected and is given back to the source when the load is turned off. In actuality, something like 7% of the energy is lost in every transfer between the load and the generator. The utility will put some capacitor storage in local substations or even on transmission poles. Using their rate structure, the utilities encourage large users to furnish their own capacitors.
The total volt-amperes (VA) is the sum of the power (watts) and the reactive volt-amperes (VARs). That is shown as a sine wave that dips below the zero axis.
Circuit Data For Above
Supply voltage: 240 Vrms, 339.4 Vpeak
Resistor current: 200 Arms (282.8 pk)
Inductor current: 150 Arms (212.1 pk)
Supply current: 250 Arms (353.8 pk)
Power: 48 kW (96 pk-pk)
Reactive power: 36 kVAR (72 pk-pk)
Apparent power: 60 kVA (120 pk-pk)
There is no net energy, but that is only because energy is transferred in both directions.
They read the rate of energy transfer back and forth.
See above.
VARs are called VARs to distinguish energy that is transferred back and forth from energy that is "consumed." Energy that is "consumed" has a much higher cost than energy that is just transferred back and forth, but VARs still have a cost.
Utility Metering
The unit of measurement that us used for utility energy billing is the kilowatt-hour. That is the area under the power curve integrated over the billing cycle. For fossil fuel generation, energy measured by a kilowatt-hour meter is equal to the energy content of the input fuel plus the losses incurred in generating, transmitting and distributing the energy. Most of those losses are directly proportional to the energy generated.
Utilities may also measure the kilovar-hours. That is the area under the var curve integrated over the billing cycle without regard to the direction of energy flow. Although the net reactive energy transferred is zero, the losses incurred in transmitting and distribution vars is directly proportional to the total vars transferred. There is also an associated capital cost of generation, transmission, and distribution equipment that is proportional to the the total vars transferred.
Billing formulae and metered quantities used are determined by individual utility companies. The basics are generally similar, but a variety of specific methods are used.
References
The basic information presented is presented similarly in text books that cover AC circuits. Here are some references that are specific to the electric power industry:
Edison Electric Institute,Handbook For Electricity Metering
Michael Bearden, Understanding Power Flow and Naming Conventions In Bi-directional Metering Applications