Molten salt power tower: Image Error

Hello,

I have a question regarding the “Image Error” of the heliostats within a tower thermal plant:

According with SAM’s manual I understand that “Image Error” represents the total conical error distribution for each heliostat. However, according DELSOL3 manual the error is characterized by standard deviations in two perpendicular directions. Could you explain me what this error represents.

Best regards,
Esteban

Dear Esteban,

SAM applies the image error equally in the x and y directions so that an error e is applied such that sigma_x = e and sigma_y = e.

Best regards,
Paul.

Hello,
I have two questions related with the Image error and I would like to have your opinion.

1) If SAM uses the same error for both axis, should we introduce then the RMS value of both axis? e=sqrt((eaxis1^2+eaxis2^2)/2).

2) To which coordinate system are you referring the error?(please refer to "Design Basis Document Rev.0 prepared by Alexis B. Zavoico, for Sandia National Laboratories (pages 41-45) ( prod.sandia.gov/techlib/access-control.cgi/2001/012100.pdf )"). The typical error can be referred to two coordinate systems (mirror normal and reflected beam). I understand that you are using the mirror-normal coordinate system as in your sample SAM file you are introducing Image Error=0.00153rad.
I hope you can clarify my doubts.

Best regards,
Miguel

Hello Miguel,

As you mention, there are several ways to describe heliostat image error. Because your question is pretty common and we haven't yet addressed this fully, I'll provide a little context along with the answer.

The method we've used in SAM asks for the "mirror normal" error, which is physically related to angular deviations (waviness) in the mirror surface, tracking error, or structural deformation that causes the mirror to deviate from the ideal parabolic shape with the focal point on the receiver. A critical aspect of the model is the assumption that the errors can be modeled probabilistically with a Gaussian distribution of mean zero and with a standard deviation specified by the Image Error term.

Effects such as pointing error that would tend to translate a single heliostat image in a particular direction away from the ideal heliostat-to-receiver vector are actually modeled as an error that increases the overall image size symmetrically. This approach is valid if a large number of heliostats have the same pointing error behavior and have images with randomly-sampled translations away from the ideal vector. Ultimately, randomly sampled translational errors from a Gaussian distribution broaden the overall flux image on the receiver in the same way as simply broadening each individual image. So while some slope errors such as surface waviness are approximately Gaussian and others such as pointing errors are not - in the context of a single heliostat, SAM uses the Image Error term to capture all surface, pointing, and specularity effects together.


I'll also point out that mirror normal errors and specularity error must be treated separately since the former class of errors has twice the effect on the total reflected beam error as the latter. Thus, when calculating an equivalent total reflected beam error, we multiply the mirror normal errors by 2:
e_rb = sqrt( (2 * e_mn)^2 + e_spec^2 )

If you have data on a specularity error and on various mirror normal errors and would like to convert that into an Image Error for SAM, the best approach is to calculate the total reflected beam error as shown above, then convert to a single dimensional pointing error as follows:
eaxis_x = eaxis_y = e_rb / ( 2 * sqrt(2) )


Now to answer your questions:
(1) The underlying DELSOL3 optical model used in SAM allows for specification of image error in X and Y coordinates. To simplify the data requested of the user, SAM assigns the same error for both X and Y directions. The total "mirror normal" error is then calculated using the equation you specify: e_mn = sqrt( eaxis_x^2 + eaxis_y^2 ), where eaxis_x = eaxis_y == "Image Error". The "reflected beam error", or the equivalent standard deviation of the reflected image, is given by: e_rb = sqrt( (2 * eaxis_x)^2 + (2 * eaxis_y)^2 ). For the default value of 0.00153 radians, the reflected beam error standard deviation (whole angle) is 0.00433 rad.

(2) The coordinate system for the Image Error value is with respect to local mirror coordinates, where X is horizontal and Y is vertical for a heliostat facing the horizon. Both X and Y error terms are equal to the Image Error value that you specify.

Best,

Mike Wagner
NREL