Same energy, different overlap: a paper electricity-schedule workshop

The schedule question

A schedule contains information that a total cannot show. Two activities can use the same combined energy whether they overlap or happen one after another.

This original paper exercise makes that difference visible. It is not a test of a real device, a household energy audit or advice to operate electrical equipment differently.

All loads, durations and start times below are invented for arithmetic.

The U.S. Energy Information Administration distinguishes power at a moment, measured in watts, from energy accumulated over time, measured in watt-hours.

A kilowatt is 1,000 watts; one kilowatt sustained for one hour corresponds to one kilowatt-hour. Those unit definitions supply the starting point, not measurements for this workshop.

Two activity strips

On paper, label activity A as a constant two kilowatts for one hour. Label activity B as a constant one kilowatt for two hours.

Treat each label as an exact rule for this imaginary exercise. A therefore uses two multiplied by one, or two kilowatt-hours.

B uses one multiplied by two, also two kilowatt-hours. Both completed activities together use four kilowatt-hours.

These strips have different shapes even though their energy totals match. A's strip is twice as tall and half as long as B's.

Height represents power and width represents duration. Multiplying height by width produces energy.

The visual is a calculation aid, not an instrument display. No inference about a real machine follows from either shape.

Start together

Place both strips at the zero-hour mark. During the first hour, A contributes two kilowatts and B contributes one, giving a combined three kilowatts.

At the one-hour mark A ends, but B continues. The second hour therefore has a combined load of one kilowatt.

Both activities are finished at the two-hour mark.

Calculate the schedule in two pieces. The first piece contributes three kilowatts multiplied by one hour, or three kilowatt-hours.

The second contributes one kilowatt multiplied by one hour, or one kilowatt-hour. The sum is four kilowatt-hours.

The largest combined load in this deliberately constant-piece model is three kilowatts. That maximum is a different answer to a different question from the energy total.

Remove the overlap

Now leave A between hours zero and one, but move B to hours one through three. The first hour has two kilowatts.

The next two hours have one kilowatt each. The energy calculation becomes two times one, plus one times two: still four kilowatt-hours.

The largest combined load is now two kilowatts, and completion occurs at hour three.

Moving B changed the maximum overlap and the finishing time. It did not change either activity's power or duration, so it did not change their combined energy.

Saying that the new drawing saves one kilowatt-hour would confuse a one-kilowatt difference in maximum load with an energy reduction. There is no hour attached to that difference until a particular duration is specified.

Nor does the exercise establish a financial saving. No price, tariff, demand interval, contract or actual consumption record has been supplied.

The defensible comparison is narrower: in these invented schedules, the second arrangement has a lower maximum combined load and a later completion time, while both arrangements use the same energy.

Half-hour overlap

Keep A at hours zero through one. This time begin B at hour one-half and end it at hour two-and-one-half.

Mark every start or finish before calculating. The first half-hour contains only A; the next half-hour contains both activities; the remaining hour-and-a-half contains only B.

There are three pieces, but they do not all last an hour.

The pieces contribute two times one-half, three times one-half, and one times one-and-a-half kilowatt-hours. Those amounts are one, one-and-a-half, and one-and-a-half.

Together they again make four. The largest combined load returns to three kilowatts, although that overlap now lasts only half an hour rather than a full hour.

This comparison explains why recording only the largest number loses part of the story. Schedule one and schedule three have the same maximum and total energy, but different timing.

Their drawings are not identical. To reconstruct either one, another reader needs the interval boundaries as well as the load assigned to each interval.

Change the duration

For a final paper variation, imagine that activity B can be completed in one hour while remaining at one kilowatt. That is a new assumption, not a claim about the original activity or any real equipment.

B would then use one kilowatt-hour instead of two. With A unchanged, the combined total would fall from four to three kilowatt-hours.

That reduction comes from changing the stipulated duration, not merely moving the strip. Whether the shorter fictional activity overlaps A still determines the combined maximum.

If they run together, their combined load is three kilowatts for the hour. If B follows A, the maximum is two kilowatts and completion takes two hours.

Keep the energy change and the scheduling change separate.

Check by another route

Each unchanged four-kilowatt-hour schedule can be checked in two ways. Add energy activity by activity: two for A and two for B.

Then add it interval by interval using the combined load. The answers must agree because each activity's entire strip appears exactly once in the time drawing.

A disagreement signals a missed interval, double counting or a minutes-to-hours mistake.

Keep this a paper exercise. Its constant loads and exact switching times are invented simplifications; they do not reveal actual operating patterns, wiring limits or provider billing rules.

Its useful result is a habit of explanation: state what changed, retain what stayed fixed, and attach the correct unit to each conclusion.