# Theoretical calculation 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:

1. For each wind speed, get the corresponding "real" tip-speed ratio and Cp from the NREL curve.
2. 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?

• ridiculous pitch may mean you are not accounting for the direction of the resulting force. only tangential forces contribute to torque that make a turbine spin. the rest just stresses out your structure.
– Abel
Oct 3 at 15:57
• Thank you for your comment. I do not have access to forces or torque, it’s just Cp and the correlations between Cp and lambda/beta. Those correlations have been used extensively in many papers so I just thought they would work out of the box… Oct 3 at 16:57
• you dont need the torque or forces; just the direction of drag and lift determined from coefficients that Cp was likely derived from for a sanity check. (you can calculate torque too if you assume some atmospheric properties.) ridiculous pitch can also be a units thing... degrees, radians, semicircles, minutes. but it looks too close on the plot for something so major
– Abel
Oct 3 at 17:07
• Thank you for your patience. Maybe I’m not being clear. I’m supposing that the only data I have is Cp, tip speed ratio and the coefficients in the correlations. Since the Cp is reproduced perfectly - and beta is supposed to be in degrees in those formulas - I am at loss as to why those correlations cannot properly be inverted to give a meaningful array for beta. As you can see, the correlations do not invoke any torque/force, just Cp, tip speed ratio and pitch angle. Oct 3 at 17:34