MEJORAMIENTO DE CAPTACIONES (CERCO PERIMÉTRICO)

An optimisation study is performed on the heliostat reflective surface profile to obtain improved performance characteristics. The ray tracer described in Chapter 5 was used for this study. The optimisation requires both a set of input parameters as well as an objective function to reach and quantify an optimum. These are introduced here.

6.2.1 Defining the input variables

The input variables to the optimisation model are in essence very basic, and the single variable is the surface profile. Still, this very simple problem is complicated in that the surface profile must be described mathematically.

Since the surface profile exists in the Euclidean space, the intuitive so- lution would be to define the surface by a parametric equation with two space parameters, z(x, y)as suggested by Landman and Gauché [144]. The optimum profile at the start of the optimisation is, however, unknown. Since a parametric equation relates variables through its predefined structure, such an equation would limit the surface to the structure of the parametric equation. Instead, a method is needed to describe an unconstrained surface mathematically, which itself requires an infinite number of variables to be introduced into the equation.

It is reiterated here that increasing the number of variables results in an exponential growth in computation time. Such a problem may easily become unsolvable due to this impracticality.

Frankot and Chellappa [147] propose a photoclinometry technique that eliminates the presumption that a surface is defined by an equation, thus providing a solution that minimises the number of variables. Previously, the approach was to alter the surface as the variable vector and then analyse the effect on the optics iteratively. The Frankot and Chellappa algorithm allows the process to be reversed so that the surface can be generated using the surface normals. This is possible by describing the surface as an integrable derivative of the surface itself.

The approach, illustrated in Figure 6.1, allows the analysis of only the surface normals per iteration. After the optimisation process is complete, a surface is generated from the surface normals. At this point a parametric equation can easily be fitted to the surface. The process is illustrated in Figure 6.1. Frankot and Surface Fit+ Surface OptimisationNormal Chellappa

Figure 6.1: Vectors are generated on a planar surface. Each surface normal vector

is then optimised individually accoridng to an objective function. Based on these normals a surface can be generated.

The advantage of this approach is that the surface derivative, which de- fines the surface normals and optical performance of the profile, remains the primary area of focus, leaving the profile as merely a secondary result of

the optics. This approach also allows the opportunity to optimise individ- ual points or areas on the heliostat for certain objective functions through discretization without the loss of smoothness or continuity.

The process no longer requires variables to define the surface; rather surface unit normals, requiring only two variables, can be dealt with indi- vidually at discretized points. The unit normal can be defined simply by the displacement of the x and y components of the unit vector. Because fewer unknown variables have to be defined, the optimisation method becomes less computationally expensive.

Vectors are generated on the reflective xy plane using regular sampling. One concern was that the planar generation of the normals may not be representative. Since the focal length is much larger than the profile elevation, dz, it was assumed that the profile elevation has negligible effect on the optics of an infinitesimal point. To validate this assumption, the surface normals to be optimised were generated on both the reflective plane and a sphere of correct focal length. No discrepancies in the results were observed.

6.2.2 Defining the objective function

The objective function serves the purpose of evaluating the variable vector in question and provides a performance measure in the form of a single number. The choice of objective functions is limitless, and any desired effect can be incorporated into the objective function. However, since the surface is generated discretely, the objective function must be able to evaluate the performance of an infinitesimal point on a heliostat. This performance measure will be minimised through the optimisation process. In addition, the objective function must be computationally efficient and will be evaluated many times during the optimisation process.

Literature on facet canting methods have considered the AIPWI as the most appropriate figure of merit because of the direct tie to plant perfor- mance and economics [137; 148]. AIPWI is defined as the annual amount of solar radiation that intercepts the receiver aperture divided by the annual amount of radiation that arrives at the receiver and is thus solely a per- centage measure of annual spillage losses. AIPWI considers only intercept effects, and although a high AIPWI is a requirement for efficient heliostat performance, other effects such as cosine effects can result in a heliostat with high AIPWI to perform poorly [137].

Using an annual performance measure such as AIPWI as the objective function will require a ray tracer to be incorporated into the objective function evaluation. This approach was used by Buck and Teufel [137] but required the use of ten million (107) rays per iteration. This approach is not practical for an unconstrained profile which will require numerous iterations.

mance by considering the axis of the reflected cone from the infinitesimal point on the heliostat. By computing the radial distance from the ray inter- cept to the target centre, d_i, an objective function could be defined according to the ray distribution on the aperture. Objective functions are discussed with each resulting profile in Section 6.4.

The constraint penalties are also specified to ensure that the normals do not drift from the feasible region, avoiding unnecessary computation and indirectly reducing computation. The initial value of the normal was given as the equivalent normal of a spherical profile. Constraints were specified as an angular deviation from the initial value.