## 2 Problem Description
### Aiming Strategy Problem
#### 2.1.3 Sector Heliostat Field Optimization Problem  
The 2-dimensional diagram of \(M_{9}\) and a flux map, indicating flux density at each heliostat, are shown in Figure 4. Pre-calculated 2-dimensional flux maps are reshaped into vectors \(Pr\) to reduce real-time optimization calculations. Each element \(Pr_{i}\) represents the flux density at the \(i\)th cell, and \(num\) denotes the total cell number on the receiver panel (Figure 2). Pre-calculated individual heliostat flux maps, denoted as \(pr_{i}\), are used to evaluate the optimized performance of the algorithm.  
\[Pr=[pr_{1},pr_{2},\ldots,pr_{num}] \tag{6}\]  
Figure 3: Full heliostat field optimization problem reconstruction diagram.  
Any heliostat \(j\) can aim at any aiming point \(k\), denoted as \(g_{j,k}\). An aiming strategy \(\lambda\) can be represented by 0-7 decision variables when each \(g_{j,k}\) is given. Each heliostat can only choose one aiming point, as shown in Equation (7):  
\[\begin{array}{ll}\lambda=\{g_{1,1},g_{1,2},\ldots,g_{m,n}\},\\ g_{j,k}=0,\ldots,7&\forall j=1..m,\forall k=1..n\\ \sum\limits_{k}h_{j,k}=1,&\forall j=1..m,\end{array} \tag{7}\]  
where \(n\) is the total number of aiming points in one panel, and \(h_{j,k}\) indicates whether the \(j\)th heliostat aims on the \(k\)th aiming height.  
According to most commonly used aiming strategy optimization models [49; 50; 51], Formula (2) can be expanded into the fitness value function of this paper and shown as Formula (8):  
\[\begin{array}{ll}maxF(M_{l})=&\Gamma\Bigg{(}\frac{\sum\limits_{l}Pr_{l}}{Pr_ {\max}}-\gamma_{gen}\frac{\sum\max(Pr_{l}-Pr_{\max},0)}{Pr_{\max}}\Bigg{)}\\ \text{s.t.}&F_{i,j,k}=Ft(\text{cell}_{i},\text{heliostat}_{j},\text{point}_{k })&\forall i,j,k\\ &Pr_{i}=\sum\limits_{j}\sum\limits_{k}F_{i,j,k}ref_{j}g_{j,k},&\forall i.\\ &h_{j,k}=(0,1),&\forall j,k.\\ &\sum\limits_{k}h_{j,k}=1,&\forall j,k.\\ &ref_{j}=(0,1),&\forall j.\\ &M_{l}=(0,32)&\forall l,\end{array} \tag{8}\]  
where \(F_{i,j,k}\) represents the flux density generated at the \(i\)th cell of the receiver when a single heliostat \(j\) aims at the \(k\)th aiming point; \(Ft(\.\ )\) denotes a pre-calculated single heliostat flux map; \(ref_{j}\) is defined as a dynamic cloud circumstance; and \(Pr_{i}\) denotes the flux density generated by the total flux distribution at the \(i\)th section of the receiver. \(Pr_{max}\) denotes the allowable flux density (AFD). The AFD penalty factor, denoted by \(\gamma_{gen}\), is a hyper-parameter that balances the target and constraint and can be adjusted as needed to balance the efficiency and reliability of the receiver. The \(\Gamma\) represents the shrinkage coefficient, which is positively related to the concentration ratio of SPT. In order to more intuitively show the fitness value function changes with the total output flux of the mirror field, the fitness function value is scaled, and \(\Gamma\) is set as 1000 according to the configuration and design in [52; 53]. The fitness function value represents the performance of the real-time  
Figure 4: Heliostat plant (left) and the 2-dimensional illustration of a partial section of a heliostat plant \(M9\) (right).  
aiming strategy optimization method on the total output flux of the heliostat field under the condition of the heliostat field constraint and is used to evaluate the optimization effect of the real-time aiming strategy.