The basic premise of “moments of power” is simple.

Imagine an energy conversion system, such as an electric motor. The input variable is the mechanical power demand $P$, and the output variable is the electrical input power $Y$. $Y$ is a function of $P$:

\[Y=f(P)\]

This function can be approximated using a polynomial such as:

\[Y = Y_0 + Y_1 P + Y_2 P^2 + Y_3 P^3\]

The overall electrical energy demand can be found by integrating $Y$:

\[E = \int Y dt\]

So far, so conventional. This value can be established using a simulation in one of many simulation tools. However, there is also an algebraic solution.

Insert the polynomial into the integral leads to

\[E = \int Y_0 + Y_1 P + Y_2 P^2 + Y_3 Y^3 dt\]

which can be separate into

\[E = Y_0 \int dt + Y_1 \int P dt + Y_2 \int P^2 dt + Y_3 \int P^3 dt\]

Those integrals are related to the moments of the distributino of P, which are familiar statistical measures, leading to:

\[E = T ( Y_0 + Y_1 \mu_1 + Y_2 \mu_2 + Y_3 \mu_3)\]

This way, the electrical energy consumption can be calculated using a simple set of multiplications, without the numerical complexity of a simulation. And more importantly, the algebraic solution gives greater insight into how to reduce the energy consumption. This is helpful for analysing any energy system under realistic usage conditions.