How to Calculate Wind Turbine Power Output?

Wind turbines look simple but are complex machines. To understand, you need to know how to calculate wind turbine power output.

I’m going to go over the calculation.

This will show why so many of these giant machines are sprouting up everywhere around you.

At the same time, we’ll uncover the complexities of these machines.

But also, we’ll find which variable is the most impactful in the amount of power generated. Because it’s not super obvious.

How do wind turbines work?

Before we go over how to calculate wind turbine power output, let’s talk about how they operate a little.

For starters, when the wind blows past a wind turbine, the blades begin to spin. We all know that!

But what happens next?

The blades capture the kinetic energy of the wind.

This kinetic energy turns into mechanical energy or called rotational kinetic energy.

The rotating blades then turn a shaft inside of the wind turbine connected to a gearbox. This then spins a generator generating electricity.

In the below schematic, you can see the conversion of power step by step.

wind turbine schematic
Wind turbine schematic (Photo Credit: Jalonsom)

To dig even deeper, the below schematic shows the guts of a wind turbine. So there’s a lot going on inside of these massive structures.

wind turbine section view schematic
Inside a wind turbine (Photo Credit: U.S. Department of Energy)

Important Note: the amount of power generated by wind turbines depends on the following factors: 

  • Mass of air (density)
  • Speed of air (velocity)
  • Amount of air (volume)
  • The swept area of wind turbine blades
  • Blade design

Wind turbine power output calculation equations and variables

Let’s first go over the variables in our equations we’ll use.

m = mass (kg)
v = wind speed (meters/second)
A = rotor swept area (meter^{2})
r = radius (meters)
KE = kinetic energy
P = power
\rho = density (kg/meter^{3})
\frac{dm}{dt} = mass flow rate (kg/second)

swept area of a wind turbine

With our variables defined, let’s now jump into our equations.

Kinetic energy we define as: KE = \dfrac{1}{2}mv^{2}

While power is KE per unit of time: P = \dfrac{1}{2}\frac{dm}{dt}v^{2}

Next, fluid mechanics tells us the mass flow rate is: \frac{dm}{dt} = \rho \times A \times V

We can see this change in mass over a unit of time in the cylinder schematic below.

Keep in mind, the mass flux is the wind flowing through the turbine blades.

Now, we plug in our mass flow rate equation into our power equation and we get: P = \frac{1}{2} \times \rho \times A \times v^{3}

Before we start plugging in actual values, we need to add one other variable to our equation.

mass flow rate

Betz Limit in wind power extraction

In 1919, Albert Betz, a German physicist, made a discovery on the efficiency of wind turbines.

They can’t convert more than 59.3% of the wind’s kinetic energy into mechanical energy.

So, the power efficiency of wind turbines has a limit. It doesn’t matter how perfectly you design a wind turbine either.

In other words, wind turbines can extract only so much energy from the wind.

This is because of generator inefficiencies, drive train friction, and blade design.

But even more, this inefficiency comes from wind slowing down. This happens as the turbine extracts energy from the wind.

As a result, the wind leaving the turbine flows slower than the wind entering the turbine.

This creates a blockage like in traffic on the highway. As cars slow down in front of you, you’ll eventually need to slow down too. No matter how far back you are from the traffic jam.

Another way to think of it is to imagine what would happen if the turbine captured 100% of the wind power. The wind would simply stop.

And for a turbine to properly work, some wind needs to flow out from the back. Otherwise, the blades won’t spin.

As a result, there’s a limit on the amount of air that can pass through a wind turbine at max power output.

This limit we call Betz Limit.

Power Coefficient, Cp

With that out of the way, let’s take a look at the Power Coefficient, Cp.

This is the ratio of extracted power by the wind turbine to the total available power in the wind source.

C_{p} = \frac{P_{T}}{P_{W}}, where C_{P\:MAX} = 0.59.

Where the Betz Limit is the maximum possible value of C_{p}. Thus, 16/27 or 0.59.

We can now update our power equation to read: P = \frac{1}{2} \times \rho \times A \times v^{3} \times C_{p}

Our equation now more accurately represents the power generated by using Betz Limit.

Important Note: wind turbines cannot operate at this C_{p\:MAX}. This is because design requirements for reliability and durability reduce the C_{p}

Also, wind turbines would need to have the perfect wind conditions. Wind conditions that match the specs of the turbine to maximize power output.  

Thus, in the real world, the C_{P} value falls between the range of 0.25 to 0.45. 

How to calculate wind turbine power output?

Finally, we can start our calculation.

We’re going to calculate the power output from a Vestas V164-8.0 MW. This is one of the world’s largest turbines.

Of course, it’s an offshore wind turbine.

Now, these wind turbines are HUGE. I haven’t stood next to these monsters, but I’ve done plenty of design work for V90-3.0 MW.

Below is a picture I snapped at a project site. Look how small the guy looks standing next to the wind turbine base.

Keep in mind, the V164 has a rotor diameter of 164 meters, while the V90 is 90 meters.

vestas v90 wind turbine

To do our calculation, we need more data on the V164. Below is a table of data from the Journal of Physics.

The data is for LEANWIND 8 MW. It’s based on published data that relates to the V164 unit. So, the data will be good enough for us to use.

Wind speed (m/s)Power (kW)Cp (Power Coefficient)Thrust (kN)Ct (Thrust Coefficient)
41100.131900.92
56000.372730.85
611400.413810.82
719000.435050.8
829000.446480.78
941550.448000.76
1056300.449450.73
1171500.4210520.67
1278650.359720.52
1379700.288470.39
1480000.237650.3
1580000.187000.24
1680000.156440.19
1780000.136040.16

From the table, we’ll use a wind speed of 14 meters/second for max power output.

The following is a summary of our data:

  • V164 blade length: 80 meters
  • Wind speed: 14 meters/second
  • Air density: 1.23 kg/meter^{3}
  • Power coefficient: 0.23

Our power coefficient is low at 0.23. But that’s okay!

Because looking at our equation, wind speed is the most important factor.  The velocity is to the power of three.

To get started, we calculate the swept area of the turbine blades.

A = \pi r^{2}

The V164 blade length is the radius variable in our equation.

A = \pi\times 80^{2} = 20,106.2 \: meter^{2}

Next, we calculate the generated power from the wind turning the turbine blades.

P = \frac{1}{2} \times \rho \times A \times v^{3} \times C_{p}

P = \frac{1}{2} \times 1.23 \times 20,106.2 \times 14^{3} \times 0.23 = 7.80 MW

There we have it. The amount of power generated is exceptional!

Keep in mind, the rated capacity for the V164 is 8 MW.

This is the amount of power the turbine can produce at optimal wind speeds. In other words, the max power output.

Our calculated value is right near 8 MW.

Important Note: our calculation is only an estimate of the power output. For a more accurate calculation, we’d need to consider other design variables. For example, the following:

  • Wind direction
  • The intrusion of water vapor
  • Thermal expansion
  • Equipment corrosion and wear and tear

Importance of understanding how to calculate wind turbine power output

The turbine designer determines the wind turbine power output. But our math exercise still does a good job of giving you a general overview.

You can better see the relationship between all the main variables in play.

This allows you to see how complex these machines actually are. Slight changes in wind speeds cause big swings in power output.

This is especially important for the energy market. Because we sell energy before it’s generated.

So, predicting the power output from a wind farm is important for various businesses. Even more so, because the weather is so unpredictable.

Predicting the weather forecast a day in advance isn’t completely accurate.

One day there’s powerful gusts, and then the next day there’s only a gentle breeze.

From our equation, we know this will result in uneven power output from wind turbines.

What’s more, it’s a challenge to smoothly integrate these machines with power grids. As power grids need reliable power generation sources.

For this reason, the more accurately we can calculate wind turbine power output, the better.

Conclusion

These massive white structures look simple as they stand tall in the distance.

But their design and integration into power grids are far from simple. A lot goes into all aspects of their design.

Hence the importance of learning how to calculate wind turbine power output. I find this to be one of the best ways to appreciate these marvels of engineering.

What are your thoughts on the power output of wind turbines? Where do you see the direction of wind power moving into the future?

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