## 4 Case Study  
This work presents a simulated case study about a full heliostat field consisting of 32 sector heliostat fields and follows a staggered radial arrangement. For most parts of the case study in this paper, we only focus on the computation of one of the 32 panels. In any case, a full heliostat field validation study is presented to demonstrate the proposed approach's effectiveness. The specific parameter description of the heliostat field and central receiver is shown in Table 1. The reflectivity of a heliostat is assumed as equal and determined by the cloud shadowing, which is modeled in the former section. In order to simplify the calculation, the atmosphere attenuation will not be considered in this paper, and the reflectivity of the heliostat will be set to \((0,1)\): \(0\) for those covered by clouds, and \(1\) for those not covered. Considering the trade-off between computational speed and optimization, \(8\) vertical aiming points are specified in the panel [30]. In our study, the number of heliostats is \(696\), and the number of aiming points is \(8\), and therefore, the total number of targeting combinations is \(696^{8}\). Figure 6: The flow chart of IUPSO. The cloud shape is simplified to a circle and creates a dynamic cloud movement model to simulate cloud shadowing per unit sampling period, as shown in Figure 7. For simplicity, The clouds are supposed to move in a straight line at a uniform velocity. The initial position of the cloud with the mid-line path of the Y-axis is denoted by \((a_{ini},b)\), and the initial position of the cloud with the mid-line path of the X-axis is presented by \((a,b_{ini})\). The mathematical model of cloud movement is shown as Equations (23) and (24):  
\[ \begin{cases}X_{clo}=a_{ini}-v_{clo}\text{-}gap_{per}\\ Y_{clo}=b\\ \text{s.t.}\ Cloud_{p}=True\\ \end{cases}  \tag{23}\]  
\[ \begin{cases}X_{clo}=a\\ Y_{clo}=b_{ini}-v_{clo}\text{-}gap_{per}\\ \text{s.t.}\ Cloud_{p}=False,\\ \end{cases}  \tag{24}\]  
where \((X_{clo},Y_{clo})\) denotes the real-time position cloud, \(v_{clo}\) denotes the real-time velocity of the cloud, and \(gap_{per}\) denotes the sampling period of the cloud. \(Cloud_{p}=True\) means the clouds move along the Y-axis of the heliostat field, and the clouds move along the X-axis mid-line when the \(Cloud_{p}\) is \(False\). The DNI received by every heliostat among the heliostat fields is rewritten to the reflectivity used for the calculation in this paper and denoted by \(ref_{j}\) and \(ref_{j}\in[0,1]\). The dynamic heliostat field condition can be modeled as Equation (25):  
\[ref_{j}=\left\{ \begin{array}{ll}1,&\text{if}\sqrt{\left(x_{j}-X_{clo}\right) ^{2}+\left(y_{j}-Y_{clo}\right)^{2}}\leq R\\ 0,&\text{if}\sqrt{\left(x_{j}-X_{clo}\right)^{2}+\left(y_{j}-Y_{clo}\right)^{2 }}>R\\ \end{array} \right.,\forall j. \tag{25}\]  
The case study consists of two critical sections: the heliostat field and the dynamic cloud movement model. In order to facilitate the calculation of the cloud cover on the heliostat field, in this study, the cloud is set as a circle, and the motion state is a uniform linear motion along a straight path. The cloud dynamic movement model consists of five main parameters, which are the altitude angle of the sun, azimuth angle, cloud radius, cloud movement speed, and the sampling period of the heliostat field, and can be seen in Table 2.