Wind energy for dummies
Part 1: Power
Introduction
While checking out this map of Vietnam wind energy from Global Wind Atlas (very cool project btw), I realised that I did not really understand the displayed metrics, or more broadly, the science behind wind turbines.
So I decided to document my research and make it a 101 series for beginners like me. In this first post, we will discuss the basic concepts around wind power.
The physics of wind turbines
A wind turbine converts the kinetic energy of the incoming air into a rotational energy acting on the rotor blades, which in turn will set the connected generator going.
As I’m sure you still remember from all your joyful physics courses in high school, the kinetic energy of an object depends on its mass m and its speed v:
We can imagine the air moving toward the rotor in cylindrical form. Its mass m can be expressed as a function of the air density ρ, the rotor area A and the length of the “air cylinder” , which is equal to speed v times t (unit of time):
Finally, the wind power is the kinetic energy per unit time:
We now have a precise idea about the relation between wind speed and its power: when the wind doubles from 5 m/s to 10 m/s, it entails an 8-fold increase in power. If there is one single number to keep in mind from the article, it’s this ratio.
Efficiency
We are now able to quantify the potential power coming from winds. However, the turbine won’t capture all of this power because of a phenomenon called wind deflection.
Basically, if the wind turbine extracts all the kinetic energy, the air won’t have any energy left to leave the turbine, which in turn will block the new air coming into the rotor.
(Not-so-fun fact: if you google “wind deflection”, most of the results are about bullet ballistic.)
One might ask what is the maximum percentage of the potential energy that can be extracted ? It turns out that the magic number is 59% (Betz’ law): a wind turbine will never have an efficiency factor greater than 59%. In reality the number is much lower than that, varies between 20% and 40%.
An important note: efficiency should not be used as metrics to compare wind turbines with other sources of electricity, which is unfortunately very common in public debates. We will dwell into this topic in a future post.
Mean Power Density
In the Global Wind Atlas map, the metric used to represent the power potential of a region is Mean Power Density. How does one compute it from the Power formula above ?
For this section, let’s take a concrete example, considering:
- A wind turbin with a rotor of diameter 25m.
- A density of air of around 1.3 kg/m3.
The first thought is to simply use the mean wind speed for the 10% windiest area to estimate the corresponding mean power.
We have then:
However, for non-Gaussian distribution, the mean can be a misleading statistics. In our case we have to be even extra careful: as the kinetic energy of the wind varies with the cube of the wind speed , a small change in the latter will have significant effect on the estimated mean power.
Therefore let’s take a step back and verify that this is a valid choice.
It turns out that the wind speeds in a specific area usually has the following distribution:
Why the slight skewness in the distribution ? The explanation is quite intuitive: in most regions, there are fewer days where it’s blowing a gale outside compared to those with moderate winds.
We can then compute the power distribution of the above area:
The important observation here is that most of the power come from winds above average speed. Thus the first estimation (119 kW) will likely underestimate the mean power.
The number confirms it, the mean power density of the above distribution is 200 kW, nearly twice our first estimation.
Power curve
Using the same simulation data, we can plot the power curve of that turbine which shows us the expected power at different wind speeds.
This power curve is a highly simplified version of the real one. It suggests that the stronger the wind the better for the wind turbine, and if there is a storm coming to town, we have hit the jackpot.
The reality is of course much more nuanced than that. Each turbine is designed with a cut-in and cut-out wind speed:
- The cut-in speed is when the turbine starts working (typically between 3 –5 m/s).
- The cut-out speed is the upper bound when the turbine stops working to avoid damage on the rotor.
A more realistic power curve should thus looks like the following:
The rated speed is the wind speed at which the turbine reaches its maximum capacity power output (rated power). Above that threshold, the turbine will try to level out the excess incoming energy through a combination of changing blade pitch, power electronics, etc. The goal is to prevent the generator from producing more power than what it (and the electricity grid) are designed to handle.
Conclusion
In this post we have gone through some basic concepts of wind power. We know that this latter depends on air density, the rotor area and most importantly, the wind speed. We also know that there are physical and design factors impacting the efficiency and the power curve of a turbine.
These concepts lay the foundation for future posts where we can start examining the economics and the scalability of this solution in a carbon-neutral world.
Reference
[1] Sustainable energy without the hot air. https://www.withouthotair.com/
[2] Energy fundamentals. https://www.energy-fundamentals.org/
[3] Danish wind industry association. http://xn--drmstrre-64ad.dk/wp-content/wind/miller/windpower%20web/core.htm
[4] FT Exploring. https://www.ftexploring.com/energy/wind-enrgy.html
