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minorman
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Education: Engineer (Ms. Sci. Eng.) and PhD My investment interests are focused on the global "macro picture" - with particular emphasis on resources and commodities. I am only interested in long-term positions - typically 4-10 years.
  • A second look at geographic diversity in wind power 4 comments
    Apr 25, 2011 2:33 AM | about stocks: GEX, ICLN, PBD, PBW, PUW, PWND, MY, APWR, VWSYF
    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/262050-a-reality-check-for-wind-power-investors
    seekingalpha.com/article/262713-wind-power-investors-get-another-reality-check
     
    and most recently
    seekingalpha.com/article/265055-avoiding-wind-power-stocks-geographic-diversity-debunked

    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/The-wholesale-market/Download-of-market-data/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:
    Production histogram for AlbertaBPA (Pacific Northwest)Denmark (east and west)New South WalesNew South WalesOntarioThe 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 1-sigma event). For something to be truly rare - it should be at least two and probably three standard deviations from the mean (3-sigma event). Provided, that a Gauss distribution is a good approximation of the data the occurrence of a 3-sigma event is, p(P< mean-3*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:
    Derived information from Jan 2010 data
    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 non-pooled data:
    Supergrid histogram
    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:
    Comparison of 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:
    Derived values from July 2010 data
    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:
    Supergrid Jan 2010
    Supergrid July 2010

    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 3-sigma 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 3-sigma event) so we get:
    n = (0.86/(0.5/3))^2 = 27.
    So, if we connect 25-30 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 25-30 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/265511-wind-power-investors-should-also-be-transmission-investors



    Disclosure: I am long MY, OTC:APWR.
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Comments (4)
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  • terravario
    , contributor
    Comments (101) | Send Message
     
    Minorman. Thank you for your indepth analysis of geographical diversity in wind power. What an improvement over what John Peterson produced in his last three articles. We are planning a grid in the Great Lakes that takes into account geographical diversity and our vast resources of hydro and pumped hydro capacity to fill in the valleys of production. I think we need to look at the Eastern Interconnect (including the Great Lakes) and the Quebec Interconnect as the backbone of a renewable energy driven supergrid.
    27 Apr 2011, 10:24 PM Reply Like
  • NanooGeek
    , contributor
    Comments (214) | Send Message
     
    I see no mention of transmission line losses. While they might not negate a geographic diversity scheme, they are another shard of debris blowing at the sail of "wind is free".

     

    There are other costs to wind generation that are rarely discussed. For example, the need to provide power when generators are on standby. According to a 01Jun2011 article on the PacificNorthwest, a commercial wind turbine can require more electricity on standby than an average house: ruralite.org/magazine/.../
    8 Jun 2011, 01:05 AM Reply Like
  • minorman
    , contributor
    Comments (233) | Send Message
     
    Author’s reply » Both are valid points, but transmission line losses are less than 10%/1000 km. Transcontinental supergrids appear impractical, but grids can be *very* big. Particularly with HVDC transmission.

     

    The average house uses perhaps 3 kW, while a wind turbine produces an average of - say 1000 kW*20% (capacity factor) = 200 kW. It may be idle 20% of the time so average standby consumption is 3 kW*20% (idle time) = 0.6 kW.

     

    Overall efficiency loss = 0.6kW/200 kW = insignificant.
    8 Jun 2011, 01:49 AM Reply Like
  • wookey
    , contributor
    Comments (2) | Send Message
     
    Average American houses might use 3kW. Average British houses use about 500W (12kWh/day). Hmm. the article linked-to specifies 10-20kW per family home (what do they do with it all - electric heating?) and, more relevant to the question, 5-20kW per turbine during standby. So standby losses 1/200 to 4/200 or 0.5% to 2%. Which is still small, but perhaps a little more than insignificant at the higher end.
    7 Jul 2011, 03:40 PM Reply Like
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