Theoretical calculations of blade pitch angle

I would like to start by apologizing for my ignorance, I am sure I am missing something very obvious here.

I am using data for the NREL 5MW reference turbine, and specifically power curve and power coefficient curve. I know that the relationship between the power coefficient (Cp) and the blade pitch angle is not so obvious, but I thought I could use the general approximation formulas:

Where c1, c2, c3, etc… and x are constants, lambda is the tip-speed ratio and beta the blade pitch angle. Since I have all the values for c1, c2, c3, x, etc…, the tip speed ratio and the power coefficient from the NREL report, the only approach I thought about was:

  • For each wind speed, get the corresponding “real” tip-speed ratio and Cp from the NREL curve.
  • For each wind speed, find beta that minimizes the square difference between the “real” Cp and the one calculated using the equations above

The minimization routine works perfectly, the Cp (from rated wind speed to cut-out) is reproduced to the 8th decimal at all wind speeds after rated (my minimization is labelled as “Mine”):

However, plotting the calculated blade pitch angles against the ones reported by NREL I get this:

You can see that my calculated beta (labelled as “Mine”) diverges almost straight away from the official values, ending up in some ridiculous pitch angle at high wind speed.

Could you please comment on whether the approach appears sensible? Is there maybe a more intelligent (or easier) way to tackle this problem - maybe with other formulas I know nothing about?

Thank you in advance for any suggestions.


Dear Andrea,

I’m not familiar with your equations, but they look like they are empirical fits to a Cp(lambda,beta) surface. Have you calculated the coefficients (C_1 through C_10) by fitting to the Cp(lambda,beta) surface provided by NREL?

Best regards,

Dear Jason,

thank you very much for your answer, I sincerely appreciate it. You are indeed 100% correct that those equations are empirical fit to a Cp(lambda,beta) surface.

Those equations are actually a generalization that encompasses 6 different studies done in the past by Dai et al (2016), Ochieng et al (2014), De Kooning et al (2010), Thongam et al (2009), Heier et al (2009) and Slootweg et al (2003). All those papers employs similar approximations of Cp vs lambda & beta by data fitting very many turbines’ data and coming out with the c1, c2, c3, etc… coefficients. The generalization that I posted can be used to represent all those models (by changing the C1, c2, etc… coefficients depending on the model) and it has been published in, for example, here: “A parametric model for wind turbine power curves incorporating environmental conditions” ( … 8120306613).

Now, it appeared to me those Cp(lambda,beta) models were general enough (albeit not particularly accurate) to be used in any turbine modelling, so I thought I could just try all of them and see if I could get at least a sensible answer to the inverse problem of calculating beta given Cp. My tests today, using all 6 models, of the “best fit” I could find for beta:

As you can see, none of them are even remotely close to the one labelled “NREL” - which is the one I found in the data you posted on the forum a while back.

A bit of context on why I would like to do this. I have a personal approach (derived from the paper I linked above “A parametric model for wind turbine power curves incorporating environmental conditions” but I have greatly extended it) that allows to build super fast (although very approximate and quick and dirty) power curves/Cp curves from just nominal power and rotor diameter as inputs. Using this extended implementation, I can also get tip-speed ratio and rotor speed.

Now, I have recently started looking at BEM models and, although for sure I do not have information on airfoil data for unknown turbines (i.e., a 20 MW theoretical turbine, let’s say), I thought I could start by trying to model the NREL 5MW reference turbine. Given that the airfoil data is available, the only piece of information I’d miss is the blade pitch angle. I know it’s available from the data you posted, but what if I am dealing with an unknown turbine? I could always (for learning purposes, of course), use the same (or similar) airfoils data for an unknow turbine: it will probably not work out of the box, but I could of course experiment with them if I knew that my blade pitch angles were not too far off. Which then brings the question: if you’re working with BEM code (such as ccblade, for example) and you make up a set of airfoils that appear interesting, the only piece of missing information is then beta vs. wind speed (or rotor speed, or TSR): where would you get that info from if you don’t have access to sophisticated tools?

I guess in the end the inverse problem I am trying to solve is maybe ill posed, unless there is some other approximate way to model the blade pitch angle.

Thank you again.


Dear Andrea,

I’m not sure I fully understand your question, but the optimal pitch angles for wind speeds above rated can be found from knowledge of the Cp(lambda,beta) surface by finding the pitch angle that gives you rated power at the given TSR/wind speed. The Cp(lambda,beta) surface can be calculated from simple BEM tools like CCBlade.

If the fit to the Cp(lambda,beta) surface is not accurate, then the approach you are using would result in an inaccurate pitch angles.

Best regards,