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A second look at geographic diversity in wind power 4 comments
Mr. John Petersen has written a few articles recently discussing the reality of geographical diversification as a strategy for leveling out the intermittency in wind power.
seekingalpha.com/article/262050arealitycheckforwindpowerinvestors
seekingalpha.com/article/262713windpowerinvestorsgetanotherrealitycheck
and most recently
seekingalpha.com/article/265055avoidingwindpowerstocksgeographicdiversitydebunked
In the latter article wind power production data (binned into 6 hour intervals) from two months (Jan 2010 and July 2010) is compiled from five different installations in
1) The Pacific Northwest
2) New South Wales
3) Ontario
4) Alberta
5) Ireland
and the data is scales as if all installations were of equal average output. The idea being: If one could connect these five installations into one "supergrid"  how well would their individual varying output average each other out?
Mr. Petersen produced two plots (one for each month) and argued that while the the average output was on the order of 3350 MW out of a "nominal" capacity (after scaling) of 15700 MW there are days in both January and July where the combined output drops below even 1000 MW.
The inference is that there is little or no averaging benefit  even to this hypothetical, global grid of wind farms.
The question therefore arises  does wind production not obey the central limit theorem?
mathworld.wolfram.com/CentralLimitTheorem.html
This is equivalent to asking if wind in Canada is positively correlated to wind in Australia  or Ireland  at some instant in time.
To shed light on the matter I calculated standard statistical moments on all the data kindly provided by Mr. Petersen  both for individual wind installations and for the pooled "supergrid" for both months. I also added in wind production data for Denmark (available from energinet.dk: www.energinet.dk/EN/El/Thewholesalemarket/Downloadofmarketdata/Sider/default.aspx )  downscaling it slightly (as per Mr. Petersen's method) to conform with the other installations  so there are now six different and (possibly) independent installations in the data set.
Firstly, I've plotted histograms for all six regions (January 2010 data only) showing the number of 6 hour intervals where power is within some (25% bin size) of the average power for each installation. Also included is a simple (untruncated Gaussian fit using the simple mean and standard deviation and also a crude Poisson fit  in alphabetical order:
The immediate observation is that with the exception of NSW  all the histograms are poorly represented by either a Poisson distribution or a Gauss distribution. In fact that standard deviation in many cases is of the same order of magnitude as average power (See table below). What this means is that power near zero occurs with high frequency (a 1sigma event). For something to be truly rare  it should be at least two and probably three standard deviations from the mean (3sigma event). Provided, that a Gauss distribution is a good approximation of the data the occurrence of a 3sigma event is, p(P< mean3*sigma) = 0.5*erfc(3/sqrt(2)) = 0.0013 = 0.13%
(erfc(x) is the "complementary error function", and sqrt(x) is the square root)
In words: A Gauss distributed power output will only be less than the mean output minus three times the standard deviation 0.13% of the time. This is why the standard deviation is important.
The central limit theorem tells us that
1) as we pool, data from independent, sources (wind power installations in this case) the data should become an increasingly closer to being Gauss distributed as we increase the number of independent sources (n)
and
2) that that standard deviation of a pool of n data sets (id est  the standard deviation in power output of our supergrid) should scale with with the square root of n:
sigma is proportional to sqrt(n). However, since the average power increases linearly with the number of wind plants, n, the relative standard deviation will scale as sqrt(n)/n = 1/sqrt(n).
In other words  the relative standard deviation of our supergrid of six wind power plants should be roughly 1/sqrt(6) = 0.41 times the relative standard deviation of the individual power plants.
Thus prepared, let's look at the numbers! January first:
As expected from the figures above  all individual areas exhibit terrible relative standard deviations (except in this case NSW). On the other hand, the pooled data look much better. Here the relative standard deviation is down to 0.35  far better than even NSW. We also see, that the supergrid at its worst data point still delivers 35% of its overall mean power while NSW only manages 12% and all the others are at 5% or less. This is a major improvement  and we have just six individual wind farms.
So how does the central limit theorem fare?
Well, first we should check how well the pooled data is represented by a Gauss distribution compared to the nonpooled data:
Clearly, this is way better than all but the NSW data (with which it is comparable  they have the same R^2 value). We can also compare all the Gauss fits:
So far, the Central Limit Theorem fares well. How about those standard deviations?
The simple mean of the relative standard deviations of the individual installations above is 0.82. The supergrid, as mentioned, is only 0.35 (id est  doing much better!)  and incidentally 0.82/sqrt(6) = 0.33  pretty darn close to the mark!

What about the July data?
Same story. I didn't generate the plots, but here's the table:
Now the NSW data looks more normal (bad), but this makes it even more clear how big the benefit from pooling just six sources is  relative standard deviation is down to 0.42  whereas no individual source is below 0.68 and most are above 0.8.
Average standard deviation is 0.86 and 0.86/sqrt(6) = 0.35 so once again, the central limit theorem seems to do a decent job. As expected.
Updated versions of Mr. Petersen's plots for all six sources are given below:
Clearly, six sources are by no means enough, but now we can say with some weight, that more do help  and we can quantify by how much.
For the sake of argument, let's say that you want a drop to 50% of mean output to be a 3sigma event. How many (independent) wind parks must you pool?
Well, each park is has a relative standard deviation of 0.84 (average of 0.82 and 0.86  to use all available data) so
(rel. std. dev.)(n) = 0.86/sqrt(n)
and we want (rel. std. dev.) to be 0.5/3 (for 50% power to be a 3sigma event) so we get:
n = (0.86/(0.5/3))^2 = 27.
So, if we connect 2530 independent wind farms we will have at least 50% of the average power at least 99.87% of the time (and at least 66% of the average power 98% of the time). As an added bonus  we will hardly ever have too much electricity either (150% of average power only occurs 0.13% of the time).
Is it possible to find 2530 independent locations for a realistic grid? Perhaps not, but there is certainly a huge benefit to geographic pooling in wind power.

See also Tom Konrads recent discussion of the problem on SA:
seekingalpha.com/article/265511windpowerinvestorsshouldalsobetransmissioninvestors
Disclosure: I am long MY, OTC:APWR.
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