Calculate power consumption that varies every second

Question

We know that power consumption or Energy is

E = Power x Time

for utility billing, we used

E = kWatt x Hour

But I am not sure how to get the total Energy if we will based it per second having different power over a period of time.

0.5 Watt (time check 5:00:00 AM)
1.0 Watt  (time check 5:00:04 AM)
0.7 Watt  (time check 5:00:07 AM)
and so on.

Let say I wanted to calculate the energy from 5:00:00 AM - 5:00:07 AM

Is this the right calculation?

E1 =  0.5 x 4 = 2.0 Watt-sec
E2 =  1.0 x 3 = 3.0 Watt-sec
E3 =  0.7 x 1 = 0.7 Watt-sec

E1 + E2 + E3
2 + 3 + .7 = 5.7 Watt-sec
           = .00158 Watt-Hr

I just want to know if my calculation is right. Thanks

Answer 1

If \$P(t)=v(t)\cdot i(t)\$ is the power, then the energy is

\$ E(t) = \int P(t)\,dt = \int v(t)\cdot i(t)\,dt \$

The method that you apply is an approximation to the exact energy calculation.It depends on the accuracy you want in your results, if it helps or not.

If you look at integrating, it is formed by the product of voltage and current multiplied by \$dt\$. The time differential is an infinitesimal increase.
In your case, the time between samples are not infinitesimal, so you can expect an error on the value of energy calculated. If the interval between measurements is kept uniform, you are using the rectangular approximation to the solution of the integral. You should evaluate whether the results you get with the generated error is acceptable.

Answer 2

It is tricky to say as we do not know how your energy usage is being reported. It could be saying 0.5 watts is being used per second up until 5:04AM. Or it could be 0.5 watts being used per second up until 5:00AM. Or it could be 0.5 watts has been used up until then.

I see you have used the first one so following on from that it would be0.5 x 4 x 60 as the time difference is 4 minutes and your rate is per second. Then the final calculation would be as you have done.

As your energy usage is varying per second this will only provide an estimate of the exact amount you would need to use intergration to calculate more acurately.