# An Integration Model for Flux Density Distribution Formed by a Heliostat
## 4 Analysis and Discussion  
In summary, the average difference, minimum difference, and maximum difference in flux density distribution between the elliptic Gaussian model and experimental data are 2.83%, 2.33%, and 3.64%, and the average difference, minimum difference, and maximum difference in intercept factor are 1.13%, 0.46%, and 1.75%. The error of the model arises from the following aspects:  
* Each point on the heliostat is affected by gravity, temperature, and wind. The slope error will change to different degrees, resulting in the irregular flux density distribution of experimental results and errors with model calculation results. * The solar intensity distribution uses the circular Gaussian function, which is different from the actual solar intensity distribution and will bring some errors. This model considers rays reflected by two planes of the glass mirror, and the focused light spot formed on the receiver plane is divided into three parts, which has been observed experimentally [11]. This is due to the ray reflected by the upper and lower surfaces of the glass. In this paper, when calculating the flux density of the tower heliostat, its influence was also considered, significantly impacting the calculation result. Although the focused spot position of the three reflections differs little, the error transfer effect of the three reflections needs to be calculated separately due to the difference in the number of reflections. The intensity distribution of reflected rays is very different in three parts, resulting in a different flux density distribution and larger light spot dispersion, which is more consistent with the actual situation and introduces fewer calculation errors. Conventional ray-tracing methods consider the single beam and use the circular Gaussian model to describe the transfer effect of the surface error distribution. The comparison results in this paper show that the average difference between SolTrace and experimental data is 3.50%, and the maximum difference is 4.43%. In addition, even if the Monte Carlo method [21] is used to reduce the amount of calculation, it is still necessary to simulate as many reflected rays of the heliostat as possible to obtain high-precision results. Therefore, the speed is relatively slow, which is not suitable for optimizing flux density calculation. When using an i7-4790 CPU 3.6 GHz personal computer, SolTrace needs 7.125 s to calculate the flux density distribution reflected by a single heliostat. In comparison, the elliptic Gaussian model in this paper only needs 2.211 s to calculate. [PERSON] et al. [3] used a circular Gaussian function to describe the flux density formed by a heliostat on the image plane. When calculating the intercept factor based on this method, the error can reach up to 9.3%. The convolution integration method is another common method. A circular Gaussian distribution was first used to describe the surface error of a heliostat, but the difference is relatively large compared with experimental data. In the 1980s, [PERSON] [7] proposed the method of convolution of the solar brightness distribution and elliptic Gaussian error to calculate and solve the flux density distribution, which requires an interpolation calculation of the data table to be established in advance. In [PERSON] paper, HCOEF and KGEN methods were proposed. HCOEF has a fast calculation speed, and the calculation time is approximately 0.27 s. However, when the slope error of the heliostat is small, or the heliostat is very close to the receiver plane, unacceptable errors will occur, and the peak value of the calculated flux density is four to five times the actual value. However, although the calculation results of the KGEN method can fit the actual situation to a certain extent, the calculation speed is very slow, and the calculation time is 31.3 s.  
The HELIOS [8] system uses the two-dimensional fast Fourier transform method to replace convolution with function multiplication.