## 3 Meta-Heuristic Algorithm for Aiming Strategy Optimization
### Population Inheritance Mechanism  
A new population inheritance mechanism is introduced in the PSO algorithm to optimize the real-time aiming strategy under dynamic cloud shadows. The Modified PSO algorithm solves optimization problems sequentially and successively by inheriting solutions. The real-time aiming strategy of the previous moment is imported into the current algorithm as the initial solution, the inertia weight \(\varpi^{gen}\) expressed in Equation (11), which, as the number of iterations increases, linear decrement favors first the global search and then the local search. And we inherit the aiming strategy at the moment when the cloud is approaching the heliostat field \(\lambda_{gen-1}\);  
\[\lambda_{initial}=\lambda_{gen-1} \tag{10}\]  
\[\varpi^{gen}=\varpi_{\max}-\frac{\varpi_{\max}-\varpi_{\min}}{gen_{\max}}gen, \tag{11}\]  
where \(\varpi_{\max}\) refers to the maximum inertia constant and is set as 1.2, while \(\varpi_{\min}\) is the minimum and set as 0.3; \(gen_{\max}\) refers to the max iteration limitation number.  
The physical IA heliostat \(j\) has the position vector \(g_{j,k}\), and the whole aiming strategy can be presented by \(\lambda=[g_{1,1},g_{1,2},\ldots,g_{m,n}]\); the velocity of the physical IA heliostat can be set as \(vp_{j}\). The position vector \(g_{j,k}\) is updated according to Equations (12) and (13), and the penalty \(\gamma_{\lambda}\) is modeled in Equations (14) and (15) [55]:  
\[\begin{split} vp_{j,k}^{gen+1}=\mu\begin{pmatrix}\varpi^{gen} vp_{j,k}^{gen}+c_{1}\alpha_{1}^{gen}\left(pbest_{j,k}-g_{j,k}^{gen} \right)\\ +c_{2}\alpha_{2}^{gen}\left(pbest_{j,k}-g_{k}^{gen}\right)\end{pmatrix} \\ (k=1,2,\ldots,n)\\ \qquad g_{j,k}^{gen+1}=g_{j,k}^{gen}+vp_{j,k}^{gen+1}\quad(k=1,2, \ldots,n)\end{split} \tag{12}\]  
\[\gamma_{\lambda}=\sqrt{gen}\sum_{x}\sum_{y}\Bigg{(}\theta\Big{(}Q_{(x,y)} \Big{(}\lambda_{g_{j,k}}\Big{)}\Big{)}Q_{(x,y)}\Big{(}\lambda_{g_{j,k}} \Big{)}^{\phi\big{(}Q_{(x,y)}\big{(}\lambda_{g_{j,k}}\big{)}\big{)}}\Bigg{)} \tag{14}\]  
\[Q_{(x,y)}\Big{(}\lambda_{g_{j,k}}\Big{)}=\max\Big{\{}0,f_{(x,y)}\Big{(} \lambda_{g_{j,k}}\Big{)}-AFD_{(x,y)}\Big{(}\lambda_{g_{j,k}}\Big{)}\Big{\}}, \tag{15}\]  
where \(\alpha_{1}^{gen}\) and \(\alpha_{2}^{gen}\) are randomly generated from the range of \([0,1]\), and \(\mu\) represents the constriction factor [55]. \(Q_{(x,y)}\Big{(}\lambda_{g_{j,k}}\Big{)}\) refers to the relative violated function. \(\phi\Big{(}Q_{(x,y)}\Big{(}\lambda_{g_{j,k}}\Big{)}\Big{)}\) and \(\theta\Big{(}Q_{(x,y)}\Big{(}\lambda_{g_{j,k}}\Big{)}\Big{)}\) are the power of the penalty function and multi-stage assignment function, respectively.  
Compare the fitness value \(Fit^{gen}\) of the population at the current iteration with them at the previous iteration \(Fit^{gen-1}\), and update the \(pbest_{j,k}\) and \(gbest_{j,k}\) according to the calculation via Equations (16) and (17):  
\[\begin{split} pbest_{j,k}=\begin{cases}\lambda_{gen-1}&gen=1\\ Fit^{gen}&if\quad Fit^{gen}>Fit^{gen-1}\\ \lambda_{gen-1}&if\quad Fit^{gen}<Fit^{gen-1}\end{cases}\\ \qquad gbest_{j,k}=\begin{cases}\lambda_{gen-1}&gen=1\\ pbest_{j,k}&if\quad Fit^{gen}>pbest_{j,k}\\ \lambda_{gen-1}&if\quad Fit^{gen}<pbest_{j,k}.\end{cases}\end{split} \tag{16}\]