How to Calculate Wind Turbine Power Output?

How to calculate wind turbine power output? It’s a simple calculation that highlights the complexities of these plain-looking machines.

I’m going to go over this calculation to show you why wind turbines are so popular. At the same time, we’ll learn about the complexities of these amazing machines.

We’ll uncover which variable is the most impactful in the generation of power too. 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 their basic operation.

For starters, when the wind blows past a wind turbine, the blades begin to spin. Duh!

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 plain-looking 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

The following are the variables in our equations:

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.

In short, there’s a limit on the amount of air that can pass through a wind turbine for max power output. This limit we call the 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, let’s start the calculation.

We’ll calculate the power output from a Vestas V164-8.0 MW. This is one of the world’s largest turbines. And of course, it’s an offshore wind turbine.

It’s an understatement to say these wind turbines are HUGE. I haven’t stood next to these monsters, but I’ve done plenty of design work for the 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 rotor diameter is 164 meters, while the V90 is 90 meters.

vestas v90 wind turbine

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 our calculation.

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. Thus, the following becomes our data summary:

  • 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.

First, 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 an approximate calculation.

Also, you can better see the relationship between all the main variables in play. In return, you get 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. And from our equation, we know this will result in uneven power outputs from wind turbines.

On that note, it’s a challenge to smoothly integrate these machines with power grids too. Because power grids want reliable power generation sources. For this reason, the more accurately we can calculate wind turbine power output, the better.

“How to calculate wind turbine power output?” wrap up

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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4 thoughts on “How to Calculate Wind Turbine Power Output?”

    • Hey Mark; as a quick and dirty calculation:

      Per the article, a nominal generation capacity of 800 MW / 62 turbines = 12.9 MW / turbine

      Thus, a 12.9 MW rated wind turbine will generate 12.9 MWh per hour in peak operating conditions.

      Assuming 15 revolutions/minute (rpm), that’s one revolution every 4 seconds.

      Given there are 3600 seconds in an hour, the turbine will generate 0.11% (or 4/3600) of an hour’s worth of generation in a single revolution. Thus, 14.2 kWh/revolution. Then, using the average U.S. wind capacity factor of 0.35, we can chop this value down to 4.97 kWh/revolution.

      The average daily electricity consumption in Massachusetts is about 20.1 kWh according to a quick Google search. So, we fall well short.

      Reply
  1. What I’m really struggling with in the above analysis is the there is no term for blade area.

    It’s just not possible that say a 2 bladed unit generates the same power as a 4 bladed unit.

    No problem with the limits – it’s just that there is such a huge space between blade tips that trailing tip vortices etc will be long gone such as not to impact the trailing blade.

    After many years trying to optimise marine propellers – I know just how important blade area is – in fact a critIcal variable is DAR or Disk Area Ratio or Blade area / Swept area

    Can someone enlighten me on the trade offs of a scenario where the turbine had say 8 blades mounted on a ring at a significant radius – leave aside the practicality at this stage.

    Interesting to see the very old wind turbines driving pumps in outback Australia with maybe 20 blades mounted only on the outer half diameter.

    I understand the logic of ever larger blades – but – can this provide an opportunity for more blades ?

    Reply
    • Great question and you’re right. The blade area would be included as a variable in a more in-depth analysis.

      Now to preface, the boring answer defaults to costs. The end goal with commercial wind turbines is always to maximize energy while minimizing costs and failures. If you design more blades to increase the power produced, you’ll need to beef up your internal components given the higher failure rate due to the higher RPM in certain case scenarios. Yet, the wind turbine becomes only marginally more efficient. 

      To point out, at lower wind speeds, having more blades will lead to greater rotational speeds. But as wind speeds increase, the blades will stall (they create a wall against the wind), and the turbine speed will slow. So, having more blades you can only operate best in a narrow range of low wind speeds, and poorly outside of the range. But, a lower blade count will allow you to generate more energy since it can operate in a wider range of wind speeds given the distribution of how wind typically blows.

      To dig a bit deeper, the rotor power, P = 2*π*T*n, is proportional to the shaft torque (T) and the rotation frequency (n). The frequency (n) is governed by the tip speed ratio. So, the torque (T) will increase with more blades, because every rotating blade creates “dirty air” (i.e the blade reduces the wind speed for the following blade). Thus, the more blades you have, the greater this “wind shadow” will be. 

      And the generator within the turbine moves let’s say 1,800 RPM to convert the wind’s energy into electricity. So, more blades wouldn’t be conducive, as an electric generator is better with higher speeds, especially when you consider the cost of construction, maintenance, and custom blade designs for a given region (e.g. pitch of the blade). Think about the old mechanically driven windmills used for water pumps as you highlighted in Australia. They need a high starting torque to pump water from underground; thus, why they have so many blades.

      Reply

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