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This time my question about the how power companies monitor usage. We get electric usage bill in [Kilo watt hour]/Energy. What should be looked up on for monitoring the usage the instantaneous energy or power? Or Both?

I trust it is the second one and why is it necessary to monitor both power and energy?

Power is the d/dt of energy.So what change does it make?

And how different will be the result getting from the second method?

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For certain large consumers, electric companies are actually interested in three things: energy, peak power, and the integral of RMS current. For smaller consumers, the latter two quantities are assumed to be close enough to being proportional to energy consumption that billing purely on the basis of energy is sufficient, but that doesn't hold true for some larger consumers.

Electric companies need to build enough equipment to handle the maximum instant power demand; during times when customers aren't using that much power, the equipment won't have to provide energy, but it will still have substantial daily maintenance and depreciation costs. An electric consumer that needs one megawatt of power for one minute per week might compel an electric company to build otherwise-unnecessary equipment, but energy-based billing, even at $0.10/kWh would be less than $10,000/month. A company that sometimes demands a megawatt will be billed monthly based upon that peak demand, even if it only demands that much power intermittently.

Further, billing for current may seem redundant compared with billing for power, but capacitive or reactive loads will draw power from the line during part of each cycle and return power during part of it. Energy-based billing will credit the power returned against the power drawn, which would be fair except that power-company wiring will lose some energy beyond what is is drawn, and will lose some of the energy which is returned. Billing upon energy alone would mean that a consumer which returned 99% of the energy drawn each cycle would only pay 1% as much as a consumer that drew 1% as much energy but returned zero, but the cost of supplying the former customer would be much greater.

It's worthwhile to note that while billing large customers for peak demand and RMS current is a source of revenue, it also serves another more important purpose: it encourages customers to save the utility money. If a customer has a choice between using one megawatt for one minute per month, or 20 kilowatts (1/50 megawatt) for an hour per month, it may be cheaper for the utility to supply the latter customer's needs even though the latter customer uses more energy. Further, in many cases where a consumer might consume excess RMS current (relative to power), it would be possible for the customer to install equipment which would reduce RMS current. Even though such equipment will waste some energy, the reduction of waste in the utility's lines may significantly exceed the amount of energy it consumes. Consequently, even though the equipment increases billable energy usage, utilities would still like to reward those who supply it themselves.

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A perfect power meter would measure

$$ E = \int_0^T{V(t)I(t)dt} $$

where \$ T \$ is your billing period, \$ E \$ is the energy (units can be converted easily) and \$ V \$ and \$ I \$ are the instantaneous voltage and current of the load. That is, they do not measure "power" on its own; they measure voltage and current as separate electrical quantities.

Of course, the real world is not continuous, so in practice it will compute they would compute something like

$$ E = \sum_{n=0}^N{V[n]I[n]\Delta T} $$

with a sampling period \$ \Delta T\$. In practice, they will probably do many rapid measurements at faster than power line frequency, and then sum those results.

Power matters to utility companies because their costs increase as the demand on the grid increases. At some point, the utility may need to fire up an extra generator or purchase power from another jurisdiction to meet local demand. Every utility has different formulas to translate instantaneous power demand to cost. That is why they are concerned with both your total energy usage in a billing period, and your power usage at various times. Strictly speaking, both quantities are measurements of energy over different windows of time.

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