Take 2: Characterizing Hysteresis

In my last post, one of the caveats about my results in determining cooling effectiveness was that the real-world data shows hysteresis, or memory, which makes determining the effectiveness of cooling difficult if it the calculation is based only on the temperature outside.
I have attempted to take into account some of this memory by replotting the energy use (kWh) vs. Cooling Degree Days (CDD) data, but with the following change: the CDD numbers on the X axis are the sum of the current CDD value added to a factor times the previous month's CDD value. For for a given month on this plot,

Accumulated CDD = (Factor * previous CDD) + CDD for this month

I utilized the Standard Error function in Excel, STEYX(), to assist me in determining the value of "Factor" by trial and error. I chose a value for "Factor" that minimzed the total standard error of the function; i.e. the sum of the errors of Y as they can be predicted by X. In another intriguiging twist, this yielded a value of greater than 1; in fact, a value around 2.6 actually worked best.
What does this mean? This means that last month's weather matters a lot more (2.6 times more) than this month's weather in determining how much heat is going to entering my home. Another fascinating result, suggesting that once again, simple air convection or conduction (represented by this month's CDD) through the exterior of the house has a lot less to do with cooling load than radiation of the hot environment around me (represented by last month's CDD). Another argument for full-house radiant barriers, if anyone is listening! In fact, the low-emissivity ("low-e") coating on modern windows helps perform exactly that radiant barrier function, but on the sides of the house, somewhere that my attic-based radiant barrier doesn't cover, and something I hope to learn more about in the future. So far, replacing windows has been off of my list due to cost constraints (my entire radiant barrier cost a lot less than 1 window), but there are efficiencies to be gained there as well. But I digress...
I actually also tried this model with a second factor multiplied by the CDD from 2 months ago, but trial and error yielded almost no impact: the factor was less than 0.1. This indicates that the main factors are (in order of importance) last month's temperature, then this month's temperature.



You'll notice a few things right off the bat:

1. The "best fit" lines look a little high. The reason: I made them fit starting at 500 accumulated CDD; I considered everything to the left of that as noise. So the slopes should match the hottest (right hand) part of the curve well, even if their level looks high. The slope is what I'm after; we're going to ignore the intercept (height) of the lines.
2. If you compare to the previous 2005 graph, you'll see that the effect of using accumulated CDDs has been to turn the large loop of 2005 data into a double loop, with a crossing in the middle. Intuitively that makes sense: that minimizes the error between a line drawn through the middle of the loop and the loop itself. The standard error for the raw 2005 CDD data was 564; the standard error for accumulated CDD data is 473, a significant improvement, meaning that a line fits this data better (although clearly still not great!)
   
3. If you compare to the previous 2009 graph, you'll see that the effect of using accumulated CDDs has been to turn the small loop of 2009 data into almost an exact line. The fact that it's a nice fit for the line means that the house's response to temperature has become much more linear; the "loopy" nonlinear part only appears far to the right in the hottest areas. The standard error for the raw CDD data for 2009 was 141; for the accumulated data, it drops to 95. Looking at the graph, you can see that the best-fit line matches quite well.

So now that we've got a better model, particularly for the 2009 data, how do the slopes compare? We've got a slope of:
2005: 0.99 kWh / accumulated CDD
2009: 0.27 kWh / accumulated CDD

Taking reciprocals to convert to effectiveness:
2005: 1.01 accumulated CDD cooled per kWh spent
2009: 3.73 accumulated CDD cooled per kWh spent
Effectiveness ratio: 3.7 to 1
Conclusion: when taking the hysteresis of the system into account, our home improvements look even better. Rather than a 70% improvement in effectiveness based on the immediate CDD method, I may well be looking at a 73% improvement based on a more accurate model.
While not a massive change, this analysis makes me feel a bit better in that I've now accounted for the rather obvious hysteresis in the data, and come out with pretty similar numbers.
There remains the lingering question of the nonlinearity all through the 2005 data and at the very far right of the 2009 data. It would appear that when things get hotter than the house can handle (which seems to happen immediately in 2005, but not until about x=2000 accumulated CDDs in 2009), energy use gets bumped up nonlinearly. There are probably good physics reasons for this nonlinearity, but I will leave it to my readers to write in a let me know what they might be and how I might model them - or even better, prevent them!
In the meantime, this diversion into analysis was fun, but we still have a little bit more story to tell about the home energy efficiency projects completed in 2009. Stay tuned!

Analysis: Effectiveness of Ventilation + Barrier

I will continue the analysis of the cooling effectiveness of my home, skipping forward to 2009. The improvements in 2007 (related to ventilation) have been enhanced by the addition of a foil radiant barrier and a 14 SEER air conditioner (to replace the failed 11 SEER unit).
Here is a graph of energy usage vs. Cooling Degree Days (CDD's, see previous posts for explanation) for 2005 and 2009. Since 2009 isn't over yet, I have substituted 2008 data for both November and December, but neither of those months is real significant for cooling costs.

1. Again, as in the previous post, the hysteresis of the system is quite apparent.
   
2. As expected from personal experience, the 2009 loop extends almost 100 CDDs further to the right than the 2005 loop, indicating the record breaking HOT summer here.
3. The curve for 2009, while showing some positive slope, looks almost unfazed by the increasing heat.

So at a glance, it appears that adding the radiant barrier and more efficient air conditioner to the already improved 2007 system has helped. But how much? Let's perform our linear best fit again and look at the results:


Including the numbers from our last post covering 2007, here are the final results of this analysis method:
2005 Slope: 2.8 kWh/CDD
2007 Slope: 1.1 kWh/CDD
2009 Slope: 0.84 kWh / CDD
So, by 2009, the effectiveness of my cooling system has increased again. Stated as "kWh of energy to cool 1 CDD", the effectiveness has gone up from:
(1/2.8)= 0.35 CDD per kWh expended to
(1/0.84)=1.19 CDD per kWh expended.
Finally: something meaningful to compare. Stated as simply as I can, based on linear best-fit analysis to remove weather dependencies:

The energy cost of cooling my home 1 degree has dropped 70% due to my efforts.

Analyzing Usage and Weather

I located some logged online Cooling Degree Day data nicely packaged by these folks.
To review, one Cooling Degree Day (CDD) recorded at a base of 65 degrees, for example, is a day where the temperature averaged over a day was 66 degrees. If the temperature for that day averaged 85 degrees, the CDD would be 20 for that day. Negative CDDs are ignored - actually, they would be counted as Heating Degree Days, which I am currently not using in my analysis (I may at some point in the future when I look at my natural gas usage, which heats my house).
The first question was which weather station to use; there are many in my area, and their numbers are all different. Not _too_ different, but somewhat. I decided to use the airport data since that data has the fewest gaps, even though the airport is a good half-hour drive away, and is well outside the urban heat island that I live on. One of the issues in dealing with long term archived data is dealing with the gaps. In my case, only a few months were missing, and I filled those in with data from another source.
I have the vague idea that my energy usage is mostly driven by air conditioner usage, supported by the fact well known to many Texans that the electric bills are largest in July and August, and are still not fun in June and September. Of course, I'm also using energy for other things such as lighting, computers, televisions, refrigerator, etc. but those loads should not vary as much seasonally, although, in yet another complication, lighting usage varies seasonally as the days get shorter.
So we have some known issues with our analysis, which we can hope will not matter too much:

1. Location difference between airport (out of town) and house (in town, downwind from downtown in the "heat island")
2. Small gap in the CDD data filled by data from a different source
3. Many things sum to make energy usage; air conditioning is only one, albeit a big one
4. CDD's measurements themselves can be taken different ways. Measuring the temperature every hour, and summing those results over a day, yields a different number than just looking at the (highest - lowest)/2 value that some data providers might use.
5. CDD's at base 65 might not be the wrong "baseline" for my house. Perhaps my air conditioner does not kick on until the daily average is over 70, for example.
6. CDD's do not consider sunlight, which delivers far more heat than convected air, particularly on cooler days. It would not surprise me to see some air conditioners running on a day with 0 CDD's but is sunny, and not run on a day with a few CDD's but is cloudy. However, installing a radiant barrier should have reduced this problem for me; I have far less sunny heat gain than before.
7. CDD's do not consider the non-air environment, except as it affects air temperature. What do I mean by this? Well, a lot of us who live here have seen 100 degree days in May. Although unwelcome, those days never seem so bad as 100 degree days in August. Why? For one thing, there is still moisture in the soil in May. Grass is green and growing, plants are moist and lush, and the ground hasn't been baked for months on end to a nice shade of brown. All of these things will reduce the heat radiating around the area and hitting me and my house, even in high air temperature. In August, on the other hand, the grass is dormant and not evaporating water, cooling the ground. The streets, sidewalks, and bricks in the houses are storing a lot of heat built up over the summer that they didn't have in May. All of that heat is radiated and hits me and my house in August, causing more cooling load, even though very little of it affects the air temperature (particularly the air temperature at the airport, which is out of town).

One way that I hope to discover whether some of these factors matter, and perhaps how much they matter, is to simply graphically look at the data. Do the data make sense? If I graph my monthly consumption in kWh vs. the CDD's for that month, a relationship should emerge if there is one. In a nice, pretty world, it would be a linear relationship, with a slope showing how much energy I need to expend to handle one CDD, but we'll see if that is the case.

Here is a look at the Cooling Degree Days at the airport since 2005. You can clearly see the seasonality reflected, and the amazingly cool summer of 2007 right in the middle of the graph. You can also see how our January's have been getting warmer every year fairly consistently, even while the summers fluctuate, and you can see that the summer of 2009 was all-record-breaking in terms of heat. The graph has been 5-point smoothed (each point has been averaged with the 2 before it and the 2 after it) to make it look nicer.