Wind Dispersion and the Renewable Hubbert Curve

Most critics of wind energy points to the unpredictability of sustained wind speeds as a potential liability in widespread use of wind turbines. Everyone can intuitively understand the logic behind this statement as they have personally experienced the variability in day-to-day wind statistics.

However comfortably we coexist with the concept of "windiness", people don't seem to understand the mathematical simplicity behind the wind speed variability. Actually the complexity of the earth's climate and environment contributes to this simplicity (see my TOD posts Information and Crude Complexity and Dispersion, Diversity, and Resilience). I suggest that it also could prove useful to understand the dispersion of airborne particulates, such as what occurred in the aftermath of the Icelandic volcano.

Let me go through the derivation of wind dispersion in a few easy steps. I start with the premise that every location on Earth has a mean or average wind speed. This speed has a prevailing direction but assume that it can blow in any direction.

Next we safely assume that the kinetic energy contained in the aggregate speed goes as the square of its velocity.

E ~ v ²

This comes about from the Newtonian kinetic energy law ½mv² and it shows up empirically as the aeronautical drag law (i.e. wind resistance) which also goes as the square of the speed. (Note that we can consider E as an energy or modified slightly as a power, since the energy is sustained over time)

As usual, I apply the Principle of Maximum Entropy to the possible states of energy that exist and come up with a probability density function (PDF) that has no constraints other than a mean value (with negative speeds forbidden).

p(E) = k* exp(-kE)

where k is a constant and 1/k defines the mean energy. This describes a declining probability profile, with low energies much more probable than high energies. To convert to a wind dispersion PDF we substitute velocity for energy and simplify:

p(v)dv = p(E)dE

p(v) = p(E) * dE/dv = 2cv * exp(-cv ²)

This gives the empirically observed wind speed distribution, showing a peak away from zero wind speeds and a rapid decline of frequency at higher velocity. I would consider this as another variation of an entropic velocity distribution. Many scientists refer to it as a Rayleigh or Weibull distribution. The excellent agreement in the figure below between the model and empirical data occurs frequently.

Wiki Wind Power
The Weibull model closely mirrors the actual distribution of hourly wind speeds at many locations. The Weibull factor is often close to 2 and therefore a Rayleigh distribution can be used as a less accurate, but simpler model.
Distribution of wind speed (red) and energy (blue) for all of 2002 at the Lee Ranch facility in Colorado. The histogram shows measured data, while the curve is the Rayleigh model distribution for the same average wind speed.

The Wiki explanation pulls its punches by using the heuristic family of Weibull curves. The Rayleigh comes out as the simpler model because it derives from first principles and any deviation from the quadratic exponent seems a bit silly. In the following curve, 1.95 is for all practical purposes the same as 2.


Contrary to other distributions, the wind PDF does not qualify as a fat-tail distribution. This becomes obvious if you consider that the power-law only comes about from the reciprocal measure of time, and since we measure speed directly, we invoke the entropic velocity profile directly as well.

So the interesting measure relates to the indirect way that we perceive the variations in wind speed. Only over the span of time do we detect the unpredictability and disorder in speed -- whether by gustiness or long periods of calm.

We can then pose all sorts of questions based on the entropic wind speed law. For example, how long would we have to wait to generate an accumulated amount of energy?

We can answer this analytically by simply equating the steady-state wind speed to a power and then integrating over all possibilities of the distribution that meet the minimum accumulated energy condition over a period of time.

The naive answer is trivial with the time PDF turning into the following fat-tail power-law:

p(t | E>cv ²) = (cv ²) * exp(-cv ²/ t ) / t ²

This equation corresponds to the following graphed probability density function (where we set an arbitrary E of cv ² = 25). It basically illustrates the likelihood of accumulating a specific energy goal within a given Time.


Because of the scarcity of high wind speeds and the finite time it takes to accumulate energy, one observes a short ramp-up to the peak. Typical wind-speeds round out the peak and the relatively common slow speeds contribute to the long fat-tail.

But since power has an extra velocity factor (Power = Drag*velocity, see Betz' law), it takes longer to integrate the low power values and the exponent changes from a quadratic value of 2 to a value of 5/3, via a chain rule.

p(t ) = (2*d/3) * exp(-d / t ^2/3 ) / t ^5/3

Thanks to E.Swanson at TOD for pointing this out. As you can see in the graph below, the fat-tail becomes fatter and the ramp-up a bit sooner for roughly the same peak value.
That gives us the power-law and a shape that looks surprisingly close to the time depletion curve for an oil reservoir. In fact, since probabilities have such universal properties, the curvature of this profile has the same fundamental basis as the Hubbert oil depletion profile. The huge distinction lies in the fact that wind energy provides a renewable source of energy, whereas oil depletion results in a dead-end.

So the hopelessness of the Hubbert curve when applied to Peak Oil turns to a sense of optimism when you realize that wind power generates a case of a Renewable Hubbert Curve. In other words, anytime you spin-up the wind-turbine you can always obtain a mini Hubbert cycle, if you have patience, just like you need patience with a train schedule.

Amazing the power of a simple consistent model; both dispersive discovery and the entropic wind dispersion model use the same set of ideas from probability. I find it unfortunate that not enough analysts can see through the complexity and discover the underlying elegance and intuitive power of simple entropy arguments.

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Found recently:

http://www.theoildrum.com/node/6386#comment-613465
Once upon a time, long ago and not all that far away, I got an F in math at a famous engineering school because I simply wrote down the answer to a problem on the final instead of going thru all the tedious procedures. I complained to the TA that the answer was obvious and I didn't need the procedure. He said "Ya didn't show that you had learned the procedure, and so the F stays".

I then went down that long soulless hallway to the office of the SUMG (short ugly math genius) who took one glance at the problem and sneered "don't bother me with such trivia, the answer is obviously--". So I went back to the TA with this, and he answered "no procedure, no grade."

So---I look at wind turbines and think "sure this thing works, and it has a energy return of maybe 20 to 40 as is, and there are obvious ways to make it better. That's good enough, so let's go for it".

Right on.