###
Why it's wrong to try relating cumulative consumption
to cumulative degree days

You sometimes see words to this effect in an energy consultant's survey report:
*
"... Cumulative gas consumption is plotted against cumulative degree days
in figure xx. This diagram exhibits various changes in gradient which
could not be explained."
*

Here's the explanation.
In the ideal case a perfectly-behaved heating system would consume energy
**E** in relation to degree days **D** according to a straight-line
relationship:

**E** = **k**_{0} + **k**_{1}**.D**

Where **k**_{0} is the fixed monthly consumption and **k**_{1}
is the consumption per degree day.
At the end of the **n**th month the cumulative degree days will be
**D**_{n} and the cumulative consumption, **E**_{n}, will be given by

**E**_{n} = **n.k**_{0} + **k**_{1}**.D**_{n}

The consultant is interested in the ratio **E**_{n}**/D**_{n}, which he thinks
should remain constant. But:

**E**_{n}**/D**_{n} = (**n.k**_{0} + **k**_{1}**.D**_{n})**/D**_{n}

which can be rearranged as

**E**_{n}**/D**_{n} = **k**_{0} ( **n/D**_{n} ) + **k**_{1}

The right-hand side of this equation is constant only if **k**_{0} is
zero (a building with no fixed component of demand) or if **n/D**_{n} is
constant, *which it never is*, because a different number of degree days are
added to **D**_{n} each time **n** increases by 1.
In winter, when values of **D** are high, the term **n/D**_{n} falls; in
summer, when values of **D** are low, it goes back up.

To be sure, the seasonal variation
becomes relatively insignificant as **n** and **D**_{n} tend towards
infinity, but that isn't quite the point. If even an 'ideal' heating
system is guaranteed to exhibit variation of slope, the method of cumulative
consumption versus cumulative degree days is a waste of time. There is,
however, a method which works.