How do you understand the kWh that the power company charges you for?

Question

I really didn't know how to phrase this question. But here are the details.

First I understand the simple 10 x 100 W for 1 hour is 1 kWh.

But what I want to know is at what time interval does the power company calculate my usage?

The simple answer is the hour, but that is too simple because I'm sure they know when I turned all my lights on for 10 minutes with a total wattage of 2 kW. Does that mean they charge me 0.33 kWh?

If I turn on a 100 W bulb for 10 minutes every hour for 6 hours. What is the power company going to calculate my usage at? Would it be 16.6 watts each hour? Do they take samples every second and then calculate the kWh from that?

I've read different analogies but one I kind of thought of was a tank that gets measured and emptied every hour. If you turn on your 100 W light bulb for 10 minutes then 16.6 watts is added to the tank. If I then boil my kettle for 10 minutes at 3200 W, 533.33 watts are added to the tank. So in that hour I used 0.5 kWh.

Answer 1

A couple of points:

• k for kilo, K for kelvin.
• W for watt.
• Power (W or kW) is an instantaneous measurement.
• Energy is power × time is charged in kWh (= kW × h).

Now to your questions:

• The energy meter integrates the power used with respect to time. If you have the old spinning disc type of energy meter you would see the disc speed up in proportion to the load. A newer digital meter will usually have an LED indicating measurement pulses. This will blink at a faster rate when demand is increased.

Image source: Free Images Live.

I'm sure they know when I turned all my lights on for 10 minutes with a total wattage of 2 kW. Does that mean they charge me 0.33 kWh?

Correct. 2 kW × 1/6 hours = 2/6 = 1/3 kWh.

If I turn on a 100 W bulb for 10 minutes every hour for 6 hours. What is the power company going to calculate my usage at? Would it be 16.6 watts each hour?

0.1 kW × 1/6 hours = 0.0166 kWh. (It's multiplied by time - not per unit time.)

Do they take samples every second and then calculate the kWh from that?

That's probably about right. The mechanical meters are continuous. The digital meters will sample and may measure each cycle of the mains (50 or 60 times a second).

I've read different analogies ...

Try thinking of a tank that has fuel in it and a meter measuring what's consumed. You turn on the light and it consumes 16.6 μl/s. You switch on your kettle and it consumes 3.2 ml/s. The meter just records the fuel used. It doesn't care weather you used it all on the first day of the billing period or if it was a constant rate throughout.

Answer 2

The term kilowatt-hour is a bit misleading in that it doesn't measure over the span of hours. Power meters measure almost continuously, and integrate the power measurements into an overall energy measure.

An old-style electrical meter had a motor that would spin at a rate proportional to the power (not current -- the actual power) being consumed, and a gear train that ran a bunch of dials on the face of the meter; those dials would be read by the power companies once a month.

Today's meters mimic that electronically. I'm not sure how it's done, but if I were doing it today, the cheapest way to do it with good enough accuracy is to sample the voltage and current at several kilohertz, multiply them into instantaneous power, and then sum up that instantaneous power to mimic continuous integration.

If the meter is working correctly then:

• If your house uses 100W all the time, without variation, for a whole month, then at the end of the month you'll get billed for 72 kW-h.
• If your house uses no power at all except for using 3kW for exactly one day out of the month, then at the end of the month you'll get billed for 72 kW-h
• If your house uses no power at all except that at noon on one day some device that consumes 72kW tries to kick in for one hour, then in a typical residential installation a fuse will blow, because that's asking for 300 amps (in North America) which is silly for a residence.

Answer 3

Texas Instruments has a series of reference designs for Electricity meter Products and reference designs which can provide guidance for how microcontroller based electronic meters can be implemented. Where the reference designs include:

1. The hardware to sample voltage and current.
2. Source code for the microcontroller.

Section 4.2.1.1 RMS Voltage and Current of Implementation of One- or Two-Phase Electronic Watt‑Hour Meter Using MSP430i20xx contains the following which indicates the sample rate of the raw mesasurements for one design:

As described in the previous sections, voltage and current samples are obtained from the ΣΔ converters at a sampling rate of 4000 Hz. All of the samples that are taken in 1 second are kept and used to obtain the RMS values for voltage and current for each phase.

Answer 4

There's a nifty device somewhere on your house/building that keeps track of all the energy going through it: the electric meter. It monitors energy usage and continuously increments a counter for your utility company to read. Typically, they read the meter once every month (more or less, depending on a variety of factors including budget and geography) and bill you for the difference since the last time they read the meter.

If you turned off absolutely everything, the meter would not increment its counter. If you did this for an entire billing cycle, the company would see the same value on the prior reading as the current reading, and you wouldn't be billed for any usage during that time period. (You would, however, be billed for base costs, which you can think of as minimums to cover things like infrastructure, maintenance, and overhead.)

Power (measured in Watts) is voltage multiplied by current. A device, such as your refrigerator, will have varying power requirements based on what it's doing at the time. It might be running the compressor and various other things, and consuming its max power at a given moment. If you measure the power usage at that time, consider that as its instantaneous power usage. When it reaches a low temperature, it shuts off the compressor and might only power a minimum of electronics to monitor the temperature, and be at a very low power, which has a different instantaneous measurement. It is of little use to know these values by themselves, because the duration your appliance stays at these power usage levels is what's important to charging you for energy usage. Thus, the time component ("h" for hours) is tacked onto the kW unit, resulting in "power over time" or, simply put, energy.

I think the heart of your question is in knowing that the power company, i.e. meter, doesn't "sample" your power usage periodically, but rather does so continuously. You may recall a decade or more ago electric meters had physically spinning wheels and dials?

They simply tap off a small portion of the current flowing past the measurement point and increment the dials accordingly. The more power you use, the faster the increment speed.

These days most meters are more efficient, have few or no moving parts, and often transmit the reading electronically/wirelessly to the property management or utility company for billing.

Answer 5

It doesn't matter.

If you travel at 2000mph for 10 minutes you go 330 miles.

If you travel at 100mph for 10 minutes, you go 16.7 miles. If you do it 6 times you go 100 miles. It doesn't matter how long of a rest stop you take in between.

If you shower at 20 gallons/hour for 10 minutes you use 3.3 gallons.

If you shower at 1 gallon/hour for 10 minutes you use 0.167 gallons. If you do it 6 times you use 1 gallon. It doesn't matter how long you take to dry your hair in between.

If you turn on 2kW of lights for 10 minutes, you use 0.33 kWh.

If you turn on a 100W bulb for 10 minutes, you use 0.0167 kWh. If you do it 6 times you use 0.1 kWh. It doesn't matter how long you huddle in the dark in between.

Answer 6

The month

Your concept seems to revolve around "sampling periods". It's not like that. It's more like an analog "flow meter" e.g. when you pump gas. Movement of fluid turns the meter. Likewise movement of current turns the meter "disc". There is no way to "game" the sampling period since there isn't one.

And once a month a human comes and reads the meter. No just kidding - these days they put some electronics in front of that "disc" so they can electronically collect the kWH count once per month.

But they do better than that, actually. They take hourly snapshots - most power companies use 30 minute or 15 minute, even. Why? Several reasons.

• When you buy a service upgrade, they don't upgrade the service drop wire if they can avoid it; they just monitor this data to see how much that drop wire is "heating the sky" and upgrade it when that makes financial sense. They get to do that because they operate under different rulebooks.
• Time-of-day tariffs, obviously, e.g. an hour-by-hour tariff in Chicago.
• Solar metering, if applicable.
• Some tariffs make extensive use of this data. For instance Duke Energy, heavy on hydro and nuclear base load, offers an optional, peak-based tariff that is bonkers. Your monthly bill is $7 x your highest KiloWATT (not kWH) peak hour and that's it.* It's optimized for homes with electric heat. But combine that with some smart panel / Load Sheds to squash all your peaks, and you could have a ridiculously cheap energy bill.

* Well, there's a per-kWH rate but it's under 1 cent, so it's a negligible part of the overall bill. Even if you used battery tech to completely flatten yourself to a continuous load, it'd be half your bill at most. They suspend this sweet deal during summer afternoon peaks, so people who aren't Technology Connections may pay or swelter.

And for reference, a North American 100A-200A service trips the main breaker at 24kW ($168) or 48kW ($336) i.e. that's the highest your peaking charge could possibly go.

Answer 7

Remember all those distance/rate/time problems you learned about in grade school? They were a preview of calculus, specifically of integrals and derivatives.

We have distance = velocity x time. Likewise, we have energy = watts x time. More generally, for distance and energy, instead of saying ‘times’, we’d say is the integral of (rate) over (time).

That is, power is a rate (specifically, watts = joules/s) just like km/h is a rate.

Likewise, kWh is an integral of rate, just like km is an integral of km/h for distance travelled. Their math works the same.

Your speedometer gives you instantaneous rate, while the odometer continuously tallies distance travelled, regardless of fluctuations in rate.

Likewise, a utility meter measures watts and tallies the watt-hour (again, technically joules*60) consumption, doing so continuously regardless of moment-to-moment rate variations. Like the odometer, the meter’s tallying action integrates power (that is, rate) over time.

The speedometer and the utility meter are doing a bit of calculus. Moment-to-moment rate variations don’t matter: the meter sees them and they just get added to the tally. For any given time interval then, we have single number that represents the average rate. The Mean Value Theorem tells us so.

Meter tallying vs. load is easier to visualize with a mechanical spinning-disk meter, which tallies power use by counting the disk revolutions. You can tell when a high-power device is on: the disk spins faster, adding more revs to the tally.

There’s another household spinny-wheel device that measures consumption: your water meter, which also tallies use with a spinning wheel that measures water flow rate.

Gas meter? Same thing, a spinny thing measures gas volume and tallies it.

So I’ve said that the instantaneous power rate doesn’t affect the meter tally. That’s a bit more nuanced now that the newer ‘smart’ meters have arrived.

Smart meters can separate energy usage by day-part. This allows billing at different rates depending on the time of day, a scheme intended to encourage shifting high-power loads (like dryers, ovens, pool pumps and especially A/C units) to off-peak hours. You as a consumer have some control over this and it could save you money.

Cynically, time-of-use billing is a way to extract more revenue from ratepayers, subject to the whims of your utility and forces you cannot control (like weather and politics.)

Answer 8

To borrow the car analogy from @user253751, you are being charged simply for the miles travelled.

If you get in your car and drive at 100mph for 1 hour you go 100 miles. If you drive 10 cars at 10mph for an hour you (cumulatively) drive 100 miles.

The odometer in your car doesn't care how fast you went for how long, if you stopped & started & changed speed etc. etc. it just counts up the miles.

Your electricity meter does exactly the same - the more power flows, the faster it counts. Older analogue ones are continuously counting (there is no time interval), newer ones are sampling at some given interval (likely fractions of a second) which will introduce a very very small amount of averaging from one millisecond to the next.

Answer 9

Your understanding is mostly correct.

The time interval the electricity meter calculates over is called infinitesimal: it's non-zero, but very small. It continuously accumulates (totalizes) power times this tiny interval.

That is: the electricity meter is computing - whether electromechanically or electronically - the time integral of real power. Real power as opposed to imaginary power - and there is such a thing!

Answer 10

Any competently designed power measurement device will include an analog low pass filter which will continuously integrate current and voltage. This you can think of the time interval as being infinitely small (see calculus).

Answer 11

Most power companies charge for other things apart from the basic kWh:

But first thre rate for kWh can vary during the time of day - for home use it just tends to be a day rate and a night rate as the load is lower at night, so they make it cheaper so people move some of their use to the night time.

However, for business or commercial users, the story gets more complicated, there are varying rates for kWh based on the time of day and also for the amount you use - larger users can get discounts - and these discounts can be significant if you agree to drop your load when the power company has too large a demand so they effectively cut you off at peak demand times - which is why you can get a serious discount.

Then there are the charges for reactive power, which can also be significant so many large users have power factor control units fitted to control the power factor: if you think you like maths and electrical theory then it is a fun exercise :)

Then some power companies also do a Triad charge which looks at a combination of factors - we used to make sure we avoided pulling power then but we had several methods of generation so we balance dour loads to avoid that.

If you ask your power supplier for the detailed information of how the charges are calculated they will oblige. Good fun putting that into a spreadsheet - did this to work out the "best" power provider as we had a complicated scenario as not only were we interested in the charges for use, we had to factor in the rates they were going to pay for our surplus we generated...