The variance of a sum of random variables

[phantom power]

The variance of the sum of random variables is additive, provided that the random variables are independent.

Var{Sum(X1+X2+ ... + Xn)} = Sum(Var{X1}+Var{X2}+ ... + Var{Xn}).

The reason that I bring this up, is that I think about this theorem every time I read somebody who believes that building redundant and geographically distributed renewable generation solves the intermittancy problem.

What happens to the variance of your power supply, as you sum up the output of these distributed installations?

[thereismore]

1. Learn some statistics. The error of the mean decreases as the square root

of the number of variables. That means, mean output gets more stable with the number of sources.

[phantom power]

2. This isn't about estimation. It's about the variance of summed random variables.

[caraher]

4. Think of it this way...

Let one die represent one power source. Your total generation at some instant is the sum of all the dice you roll. If you roll, say, two dice, the mean will probably be relatively far from 3.5. On the other hand, if you roll 200 dice, the mean will be very close to 3.5.

Now sure, the variance will be greater when you roll more dice, but so is the sum of all the dice. As a fraction of the sum, the fluctuations get smaller as you add more independent random variables.

[thereismore]

5. That was my point, thank you. nt

[phantom power]

7. In this case, the sum is what we care about.

[caraher]

3. But is the variance per se important?

I thought the main challenge is making sure that, at any given time, the generation capacity is sufficient, and the intermittancy problem is the simple fact that the sun goes down at night and wind speeds vary.

I do realize that the fluctuations are important to the stability of the grid, but that strikes me as a solvable controls issue.

Also, it's important to note that the assumption that the variables are independent is probably not valid. The variation in available solar power from any one site follows a highly predictable pattern over the course of each 24 hour period, a pattern that in turn varies with a period of one year, subject to short-term fluctuations due to cloud cover. And even those fluctuations will be strongly correlated among solar collectors in a given region. Similar patterns would exist for wind power.

[phantom power]

6. Correlation is the key issue.

If variation across sites is uncorrelated, the variances sum. If they are correlated, in the proper way, the variance will either increase less, or remain steady. (Can anybody construct an example of random variables whose sum decreases?)

I think that, pragmatically speaking, correlation is "low." Which means that variances will sum.

It is also true that you could cast the problem as, "build enough capacity that even though the variance is high, the probability that power drops below some chosen level is less than some chosen probability threshold." I would be interested to see how that plays out in terms of installation size.

[skids]

8. The variance of any power source...

...is from 0W to its peak W. Doesn't matter if it's 1000s of windmills or one coal plant.

Now, for sums of completely uncorrelated variables with occasional large "errors" the error rates actually subtract. This is exploited in digital communications.

The key to it all is avoiding correlation, which is why spaced-out wind farm aggregates are more reliable than single locations.

Of course correlation with solar is pretty high :-). But then, solar is best for peak demand and scheduled power needs (shunt loads, but useful shunt loads).

[phantom power]

9. the variance of a random variable is different than its range.

And I'm not sure what you're saying about digital communications. It's possible to compensate for any error rate, but your transmission rate goes down as a consequence.

None of that really bears on the fact that if you sum uncorrelated random variables, the variance of your sum goes up. As caraher points out, your mean also sums, so it's possible to compensate for that increased variance, but at the cost of additional redundancy: put another way, your peak output becomes much higher than it needs to be, so that the probability of dropping below your desired output remains sufficiently low.

[skids]

10. Just simply not an issue without correlation

Your variance may go up, but it's because you have more
short, large spikes, not more long, small drifts. Variance
is extra sensitive to large value spikes due to the square
term.

My initial comments was snide in that the fact that
capacity factors even for baseload power are not good
enough to even be thinking in terms of variance, since
an outage presents you with many uber-large values
concentrated in a spot on the far side of the mean,
that completely throw off the distribution.

Large short spikes don't require much storage capacity to
smooth out. What matters is correlation between the lower
frequency components, e.g. when the sun sets all solar
without some form of storage turns off at more or less the
same time, barring a grid that spans all time zones.

WRT to communications I was referring to the fact that
in optical fiber systems when you daisy chain amplifiers,
your noise rate actually goes down due to clipping and
cutoff. Seems counterintuitive, but its true.

[XemaSab]

16. Seems to me that the real problem

is the economics of redundancy.

If you put one windmill on every square km across the whole US, sure you'd have enough power for everyone, but wouldn't it be cheaper to say, build solar thermal plants for the cost per actual kilowatt? :shrug:

Wind is a flaky form of energy, methinks.

[jpak]

11. Apples and oranges

It's been demonstrated that geographic distribution of wind turbines does increase grid reliability.

Furthermore, solar, wind and hydro systems compliment each other.

Annual trends in wind speeds and duration, river flow and solar radiation are dissimilar - when one trend is at its annual peak, the others often are not (depends on where you are).

Daily and weekly variations in wind, precipitation and solar radiation compliment each other as well: high pressure = sunny low wind wind days, low pressure = cloudy rainy windy days, frontal passages = windy (day and night).

Trends in annual and daily electrical demand often tracks power output from renewable systems (AC load tracks daily and annual insolation in the Sunbelt and New England, wind farm and hydro output is often greatest in the winter in New England and in the Northwest, New England sea breezes often occur on hot summer days, etc.

Output from tidal power plants is also highly predictable: biomass plants can be efficiently operated to balance power output from tidal power arrays.

Generator diversity *and* geographic distribution counts...

See what I'm getting at????

[OKIsItJustMe]

12. Let me simplify things for you a bit

The reason why distributed wind generation can provide baseload power is because, given a large enough geographical area, the wind is essentially always blowing somewhere.

This may help. (Follow the link for the whole paper)

http://www.stanford.edu/group/efmh/winds/aj07_jamc.pdf

VOLUME 46 — JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY — NOVEMBER 2007

Supplying Baseload Power and Reducing Transmission Requirements by Interconnecting Wind Farms

CRISTINA L. ARCHER AND MARK Z. JACOBSON Department of Civil and Environmental Engineering, Stanford University, Stanford, California

(Manuscript received 6 July 2006, in final form 6 February 2007)

ABSTRACT

Wind is the world’s fastest growing electric energy source. Because it is intermittent, though, wind is not used to supply baseload electric power today. Interconnecting wind farms through the transmission grid is a simple and effective way of reducing deliverable wind power swings caused by wind intermittency. As more farms are interconnected in an array, wind speed correlation among sites decreases and so does the probability that all sites experience the same wind regime at the same time. The array consequently behaves more and more similarly to a single farm with steady wind speed and thus steady deliverable wind power. In this study, benefits of interconnecting wind farms were evaluated for 19 sites, located in the midwestern United States, with annual average wind speeds at 80 m above ground, the hub height of modern wind turbines, greater than 6.9 m s ^-1 (class 3 or greater). It was found that an average of 33% and a maximum of 47% of yearly averaged wind power from interconnected farms can be used as reliable, baseload electric power. Equally significant, interconnecting multiple wind farms to a common point and then connecting that point to a far-away city can allow the long-distance portion of transmission capacity to be reduced, for example, by 20% with only a 1.6% loss of energy. Although most parameters, such as intermittency, improved less than linearly as the number of interconnected sites increased, no saturation of the benefits was found. Thus, the benefits of interconnection continue to increase with more and more interconnected sites.

...

In conclusion, this study implies that if interconnected wind is used on a large scale, a third or more of its energy can be used for reliable electric power and the remaining intermittent portion can be used for transportation (i.e., to power batteries or to produce hydrogen), allowing wind to solve energy, climate, and air pollution problems simultaneously.

...

[phantom power]

14. I have a problem with their Figure 4.

I'm going to read further and think about whether its my problem or their problem, but here's the rub:

They claim they're summing the output of the various combinations sites. But mean outputs should sum, and yet their output remains flat as they consider larger combinations of sites. That tells me they aren't combining their sites just by adding up the outputs.

[OKIsItJustMe]

15. On the effect of spatial dispersion of wind power plants on the wind energy capacity credit in Greec

http://www.iop.org/EJ/abstract/1748-9326/3/1/015003

On the effect of spatial dispersion of wind power plants on the wind energy capacity credit in Greece

George Caralis et al 2008 Environ. Res. Lett. 3 015003 (13pp) doi:10.1088/1748-9326/3/1/015003

Full Text: PDF (2.31 MB) | HTML | References

George Caralis^1,2, Yiannis Perivolaris¹, Konstantinos Rados³ and Arthouros Zervos^2
1 InFlow, Wind Energy Consultants, Greece
² School of Mechanical Engineering, National Technical University of Athens, Greece
³ Department of Pollution Control Technologies, Technological Educational Institute of West Macedonia, Greece

Part of Focus on Wind Energy

Abstract. Wind energy is now a mature technology and can be considered as a significant contributor in reducing CO₂ emissions and protecting the environment. To meet the wind energy national targets, effective implementation of massive wind power installed capacity in the power supply system is required. Additionally, capacity credit is an important issue for an unstable power supply system as in Greece. To achieve high and reliable wind energy penetration levels into the system, the effect of spatial dispersion of wind energy installations within a very wide area (e.g. national level) on the power capacity credit should be accounted for.

In the present paper, a methodology for estimating the effect of spatial dispersion of wind farm installations on the capacity credit is presented and applied for the power supply system of Greece. The method is based on probability theory and makes use of wind forecasting models to represent the wind energy potential over any candidate area for future wind farm installations in the country. Representative wind power development scenarios are studied and evaluated. Results show that the spatial dispersion of wind power plants contributes beneficially to the wind capacity credit.

Received 31 May 2007, accepted for publication 13 September 2007
Published 11 January 2008

[phantom power]

17. It seems to me that they answered my question, and also agreed with me.

If you look at their figure-3, you can see that their data shows that to achieve an 80% reliable 300 kW, they required five times that amount in name-plate capacity: 1500 kW.

This is just a graphical proof of what I suggested: because of the high variance, you have to install quite a lot of extra peak capacity to get a sustainable capacity of any desired amount.

I would also like to comment that their notion of "yearly power produced" is deeply bogus. It's an example of confusion power with energy. There is no such thing as "yearly power." Only "yearly energy" has a physical meaning.

However, they compiled a lot of real-world data, which is something I'm unlikely to ever do, so it is a good resource even if their analysis seems rather wrong-headed to me in some ways.

[phantom power]

18. In fact, what this paper tells me is that the multiplier for wind is 5x, not 3x.

[OKIsItJustMe]

13. How's the Spanish wind system doing?

See for yourself, in real time!

http://www.ree.es/ingles/operacion/curvas_demanda.asp