Real-Time Optimization of Heliostat Field Aiming Strategy via an Improved Swarm Intelligence Algorithm

Abstract

Optimizing the heliostat field aiming strategy is crucial for maximizing thermal power production in solar power tower (SPT) plants while adhering to operational constraints. Although existing approaches can yield highly optimal solutions, their considerable computational cost makes them unsuitable for real-time optimization in large-scale scenes. This study introduces an efficient, intelligent, real-time optimization method based on a meta-heuristic algorithm to effectively and reliably manage SPT plant operations under varying solar conditions, such as cloud shadowing variations. To minimize redundant calculations, the real-time optimization problem is framed in a way that captures the operational continuity of the heliostat, which can be utilized to streamline the solution process. The proposed method is tested in a simulation environment that includes a heliostat field, cylindrical receiver, and cloud movement model. The results demonstrate that the algorithm presented in this paper offers higher intercept efficiency, improved robustness, and reduced optimization time in more complex scenes.

1. Introduction

Solar energy is a promising renewable energy source; it has excellent potential for [1,2,3] in various fields. Solar thermal technology captures sunlight, converts it into thermal energy, and generates electricity. Among solar thermal power systems, Solar Power Tower (SPT) systems excel in concentration ratio and operating temperature, enhancing efficiency [4,5,6]. However, SPT systems must address durability and efficiency challenges to compete effectively in energy production [7,8,9]. The heliostat aiming strategy plays a critical role in the operation of SPT [10,11]. The aiming strategy problem aims to find the optimal alignment of heliostats that generates a desirable flux distribution. The maximum solar energy interception can be obtained when heliostats point at the center of the thermal receiver; however, it may cause overheating and steep flux gradients [12,13]. Excessive temperatures can degrade absorber coatings, deteriorate heat transfer fluids (HTF), and overheat absorbers [14,15]. Suitable aiming strategies are crucial for balancing stress limits and temperature while achieving higher efficiencies [16]. Optimal aiming strategies can improve SPT plant performance, promoting operational safety and equipment reliability.

Numerous studies have explored aiming strategy optimization for SPT systems to achieve desired flux distribution. Early methods focus on a static problem where cloud shadowing was ignored [17,18], like the static aim point processing system applied at the Solar Two plant [19]; the aim points of the heliostat are spread vertically from the edge of a cylindrical receiver via the beam radii of the heliostat. The heliostats causing the peak flux at this location can be determined and removed from optimization, which is suitable for the SPT system. However, it does not optimize the receiver power. Recent studies [20] aim to develop effective and efficient aiming strategies by modifying the early optimization methods for static aiming strategy problems to improve the performance of SPT plants and achieve a more uniform flux distribution. For example, Santana et al. introduced an aiming factor k, multiplied by the heliostats beam radii, to extend the static Vant-Hull aiming [21,22]. In addition, Collado and Guallar [23] introduced an additional factor $k_3$ to divide the field into radial sectors for those located further away from the tower to smooth the flux profile. However, heliostat fields are subject to varying meteorological conditions around the world [24], especially cloud shadows. Optimization methods for static problems cannot accommodate these dynamic changes, leading to the development of dynamic aiming strategy optimization methods to optimize the aiming strategy of the heliostat fields based on meteorological conditions.

The research into dynamic aiming strategy optimization for central receiver systems has been ongoing for some time, and various approaches have been proposed. Some of the earliest work, by Dellin et al. [25,26], involves fast heuristic aiming strategies that maximize the heat flux hitting the receiver while distributing the aim points of the heliostats in specific patterns. These strategies are part of the DELSOL optics simulation software, an optimization tool for central receiver system design. Kelly et al. [27] used a heuristic approach to approximate a predefined heat flux distribution, and Astolfi et al. [28] partitioned the heliostat field into small groups to reduce peak heat flux densities. Richter et al. [29] developed an accelerated aiming strategy that can be used for dynamic scenarios, such as short-term environmental influences.

The primary challenge of solving the aiming strategy for dynamic heliostat fields is the increased computational complexity of real-time computation and heightened calculation amount due to dynamic cloud shadowing. Dividing the full heliostat field into sector fields and using machine learning algorithms that rely on pre-training and meta-heuristic optimization can provide improved real-time solutions. The field of aiming strategy optimization is advancing with the development of machine learning algorithms [30], including supervised and reinforcement learning paradigms. The former paradigm learns mapping by minimizing the difference between the model output and the optimal label, using various methods such as Hopfield networks [18], Elastic Nets [31], Pointer Nets (PN), and fully convolutional networks to solve optimization problems [32]. While these machine learning algorithms can achieve high accuracy and speed after training on massive amounts of data, they consume significant computational resources during pre-training.

On the other hand, meta-heuristic algorithms like BA [33], Whale optimization algorithm (WOA) [34], ACO algorithm [35], and Red deer algorithm (RDA) [36,37] simulate natural population behavior and are based on mathematical theories inspired by natural laws [38]. These algorithms require fewer mathematical conditions than methods like Lyapunov Vector Field (LVF) [39,40], Unscented Information Filter (UIF) [41,42], and traditional heuristics [43]. Improved meta-heuristic optimization algorithms offer fast and accurate solutions without pre-training. The TABU algorithm [7] and GA algorithm [44] have been applied to the THEMIS flat plate receiver based on the HFLCAL convolution method to flatten the flux distribution while minimizing spillage. The ant colony optimization algorithm (ACO) [45,46] has been proposed to maximize the intercepted power and comply with the limits, regardless of the thermal receiver’s shape. According to Flesh et al. [47], this method performs $2\%$ better than the single-factor aiming strategy of the cylindrical receiver. It is also coupled with the local search algorithm (TLBO) of Cruz et al. [16] to increase convergence to the global optimum.

In this paper, we propose a novel meta-heuristic method that does not require offline pre-training but is also computationally efficient without sacrificing optimality. This paper aims to reduce Particle Swarm Optimization (PSO) [48] complexity and shorten optimization time by modeling heliostats as intelligent agents (IA). PSO is an efficient metaheuristic inspired by social animal behaviors, known for quick convergence and adaptability in complex problem-solving, where particles share information and collaboratively move toward optimal solutions. Formulated in a physically meaningful way, the IA is connected to the physical heliostat, thus enabling the utilization of operational continuity to enhance the solution process. The proposed innovative mechanism incorporates the heliostat field aiming strategy of the previous period. Therefore, reinforcements are introduced into the classical PSO algorithm, resulting in the development of a new algorithm called Individual Update PSO (IUPSO).

The paper primarily focuses on creating a dynamic aiming strategy for central receiver systems that can effectively handle the real-time impact of a cloud shadow on the heliostat field.

The aim is to offer a more precise and efficient method for calculating the real-time aiming strategy of the heliostats under cloud shadow.

2. Problem Description

2.1. Aiming Strategy Problem

2.1.1. Practical Aiming Strategy Optimization

The principle of the operation of the heliostat field is shown in Figure 1. This study optimizes aiming strategies for the thermal receiver of the heliostat plant, aiming at vertical shifts of aiming points on the cylindrical receiver, shown in Figure 2. The plant is divided into 32 concentric sections, with adjustable height H for heliostats targeting their receiver panels. The receiver is composed of 32 panels, each corresponding to the sector heliostat field among the full heliostat field. The aiming point of heliostat k can be denoted by ${\mu }_k$, with n aiming points on the thermal receiver, so that ${\mu }_k$ must satisfy ${\mu }_k\in \{1,2,\dots ,n\}$. The aiming strategy of the heliostat field can be denoted as an m-dimensional vector $\mu =[{\mu }_1,{\mu }_2,\dots ,{\mu }_m]$, where m denotes the number of decision variables in real-time optimization and will be shown in the next part.

Following the aiming strategy $\mu$, the flux of the square at coordinates $(x,y)$ can be denoted by ${\Psi }_{(x,y)}\left(\mu \right)$, and corresponding flux density $f_{x,y}\left(\mu \right)$ is evaluated by Equation (1):

$$\begin{array}{c}\hfill f_{(x,y)}\left(\mu \right)\approx {\Psi }_{(x,y)}\left(\mu \right)/Q_{(x,y)},\end{array}$$

(1)

where $Q_{(x,y)}$ denotes the square area at the coordinate $(x,y)$.

The corrosion and thermal stress limitations on the central thermal receiver are transformed into a maximum allowable flux density. Allowed flux density (AFD) depends on solar thermal receiver geometry factors: number of panels $N_p$, tube diameter d, receiver diameter D, and height H. The AFD limitation at coordinates $(x,y)$ is denoted as $AF{D}_{(x,y)}$. The aiming strategy aims to maximize thermal receiver output, modeled as ${\sum }_{(x,y)}{\Psi }_{(x,y)}\left(\mu \right)$ while adhering to AFD limitations. Thus, real-time aiming strategy optimization can be formulated as an optimization problem without additional constraints.

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \begin{array}{c}F\left(M_l\right)=max\sum _{(x,y)}{\Psi }_{(x,y)}\left(\mu \right)\hfill \\ \mathrm{s}.\mathrm{t}.f_{(x,y)}\left(\mu \right)\le AF{D}_{(x,y)}\hfill \\ {\mu }_k\in \{1,2,\dots ,n\}\hfill \\ M_l\in \{1,2,\dots ,L\}\hfill \end{array}\end{array}\end{array}$$

(2)

2.1.2. Optimization Strategy

The heliostat field is partitioned into different sectors to reduce the computational load of calculations and each corresponding to a panel on the central receiver, as depicted in Figure 3. Each sector is associated with a receiver panel, forming an entire heliostat field. The full field can be seen as a collection of multiple sector heliostat fields. This paper proposes a real-time aiming strategy optimization method for heliostat fields affected by continuously dynamically changing clouds, which mainly considers the real-time aiming strategy calculation for continuously similar heliostat field scenarios. The shape of the heliostat field has little influence on real-time calculation. Therefore, the whole heliostat field is divided into several sector heliostat fields, and selecting one of them is the application scenario of the subsequent study. We assumed that these sectors were independent and would not affect each other in the calculation process. Then, the whole heliostat field real-time aiming optimization problem can be reduced to a collection of multiple sector heliostat fields real-time aiming strategy optimization problems.

Let M be the full heliostat field, and $M_l$ (for $l=1,2,\cdots ,L$) represents the sector heliostat fields; L denotes the number of the sub heliostat field. Let $F\left(M\right)$ be the aiming objective function of the full heliostat field M, which can be simplified, and $F_l\left(M_l\right)$ be the aiming objective function of each sector heliostat field $M_l$.

The full heliostat field real-time aiming optimization problem can be described as:

$$\begin{array}{c}\underset{M}{max}F\left(M\right)\hfill \\ s.t.M\hfill \end{array}$$

(3)

The problem of full heliostat field real-time aiming strategy optimization can be simplified as a set of L sub-problems, which is the main concern in this paper:

$$\begin{array}{c}\underset{M_l}{max}{F}_l\left(M_l\right)\hfill \\ s.t.M_l,l\in \{1,2,...,L\}\hfill \end{array}$$

(4)

After solving these sub-problems, the optimized sector heliostat fields $M_l^{opt}$ can be combined to obtain the optimized full heliostat field aiming strategy $M^{opt}$:

$$\begin{array}{c}\hfill M^{opt}=M_1^{opt}\cap M_2^{opt},\dots ,\cap M_L^{opt}.\end{array}$$

(5)

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 $P{r}_i$ represents the flux density at the ith cell, and $num$ denotes the total cell number on the receiver panel (Figure 2). Pre-calculated individual heliostat flux maps, denoted as $p{r}_i$, are used to evaluate the optimized performance of the algorithm.

$$\begin{array}{c}\hfill Pr=\left[p{r}_1,p{r}_2,\dots ,p{r}_{num}\right]\end{array}$$

(6)

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):

$$\begin{array}{c}\hfill \begin{array}{cc}\lambda =\left[g_{1,1},g_{1,2},\dots ,g_{m,n}\right],\hfill & \\ g_{j,k}=0,\dots ,7\hfill & \forall j=1..m,\forall k=1..n\hfill \\ {\displaystyle \sum _k}{h}_{j,k}=1,\hfill & \forall j=1..m,\hfill \end{array}\end{array}$$

(7)

where n is the total number of aiming points in one panel, and $h_{j,k}$ indicates whether the $j\mathrm{th}$ heliostat aims on the $k\mathrm{th}$ 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):

$$\begin{array}{c}\begin{array}{ccc}maxF\left(M_l\right)=\hfill & \Gamma \left(\frac{{\displaystyle \sum _i}P{r}_i}{P{r}_{max}}-{\gamma }_{gen}\frac{{\displaystyle \sum _i}max\left(P{r}_i-P{r}_{max},0\right)}{P{r}_{max}}\right)\hfill & \\ \mathrm{s}.\mathrm{t}.\hfill & F_{i,j,k}=Ft\left({\mathrm{cell}}_i,{\mathrm{heliostat}}_j,{\mathrm{point}}_k\right)\hfill & \forall i,j,k\hfill \\ \hfill & P{r}_i={\displaystyle \sum _j}{\displaystyle \sum _k}{F}_{i,j,k}re{f}_j{g}_{j,k},\hfill & \forall i.\hfill \\ \hfill & h_{j,k}=(0,1),\hfill & \forall j,k.\hfill \\ \hfill & {\displaystyle \sum _k}{h}_{j,k}=1,\hfill & \forall j,k.\hfill \\ \hfill & \begin{array}{c}re{f}_j=(0,1),\hfill \\ M_l=(0,32)\hfill \end{array}\hfill & \begin{array}{c}\forall j.\hfill \\ \forall l,\hfill \end{array}\hfill \end{array}\hfill \end{array}$$

(8)

where $F_{i,j,k}$ represents the flux density generated at the ith cell of the receiver when a single heliostat j aims at the kth aiming point; $Ft\left(.\right)$ denotes a pre-calculated single heliostat flux map; $re{f}_j$ is defined as a dynamic cloud circumstance; and $P{r}_i$ denotes the flux density generated by the total flux distribution at the ith section of the receiver. $P{r}_{max}$ 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 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.

2.2. Simulation Calculation

This study uses Tonatiuh, a ray-tracker tool, to pre-calculate the flux map of single heliostats [54] to reduce the computational intensity of obtaining flux maps using Monte Carlo Ray Tracing (MCRT). However, when superimposing these pre-calculated flux maps during optimization, the shadow occlusion from surrounding heliostats can reduce accuracy. Therefore, the study considers the occlusion from nearby heliostats by including them in the calculation with their aiming points set at the receiver’s center with a reflectivity of 0. The flux map of a single heliostat is pre-calculated based on the azimuth angle of [0°, 360°] and elevation angle of [0°, 90°]. In this paper, when Tonatiuh is used to pre-calculate the flux map of a single heliostat, the influence of blocking, the reciprocal shadowing of the surrounding heliostats, is considered, and the reflectance of these slave heliostats are set as 0 to prevent the influence of the surrounding heliostat on the pre-calculated flux map of a single heliostat.

3. Meta-Heuristic Algorithm for Aiming Strategy Optimization

PSO [48] is a global optimization method based on populations, which takes inspiration from the social behavior observed in the movements of flocks of birds. This behavior directs individuals toward the best solution found by each individual and the global best solution to the optimization problem. This section presents a reinforced PSO called IUPSO to achieve real-time optimization for the aiming strategy problem stated in Equation (8), and the structure of the real-time aiming strategy optimization is shown as Figure 5.

3.1. Heliostat Intelligence Agent

In order to decrease the complexity of the modified algorithm in this paper, the individual heliostat among heliostat field $g_{j,k}$ can be modeled as an intelligence agent with physical meaning, which is called physical IA in this paper. Moreover, the aiming strategy $\lambda$ is modeled as the population of Physical IA, denoted by Equation (9).

$$\begin{array}{c}\left\{\begin{array}{c}Physical:\mathrm{IA}=g_{m,n}\hfill \\ \lambda =g_{1,1}\cap g_{1,2}\cap \dots \cap g_{m,n}\hfill \end{array}\right.,\hfill \\ \begin{array}{cc}s.t.\hfill & \\ g_{j,k}=0,\dots ,7\hfill & \forall j=1..m,\forall k=1..n\hfill \\ {\displaystyle \sum _{i=1}^n}{g}_{j,k}=1,\hfill & \forall j=1..m.\hfill \end{array}\hfill \end{array}$$

(9)

3.2. 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 ${\lambda }_{gen-1}$;

$$\begin{array}{c}\hfill {\lambda }_{initial}={\lambda }_{gen-1}\end{array}$$

(10)

$$\begin{array}{c}\hfill {\varpi }^{gen}={\varpi }_{max}-\frac{{\varpi }_{max}-{\varpi }_{min}}{ge{n}_{max}}gen,\end{array}$$

(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; $ge{n}_{max}$ 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 $\lambda =\left[g_{1,1},g_{1,2},\dots ,g_{m,n}\right]$; the velocity of the physical IA heliostat can be set as $v{p}_j^{}$. 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]:

$$\begin{array}{c}\hfill \begin{array}{c}v{p}_{j,k}^{gen+1}=\mu \left(\begin{array}{c}{\varpi }^{gen}v{p}_{j,k}^{gen}+c_1{\alpha }_1^{gen}\left(pbes{t}_{j,k}-g_{j,k}^{gen}\right)\hfill \\ +c_2{\alpha }_2^{gen}\left(pbes{t}_{j,k}-g_k^{gen}\right)\hfill \end{array}\right)\hfill \\ \left(k=1,2,\dots ,n\right)\hfill \end{array}\end{array}$$

(12)

$$\begin{array}{c}\hfill g_{j,k}^{gen+1}=g_{j,k}^{gen}+v{p}_{j,k}^{gen+1}\left(k=1,2,\dots ,n\right)\end{array}$$

(13)

$$\begin{array}{c}\hfill {\gamma }_{\lambda }=\sqrt{gen}\sum _x\sum _y\left(\theta \left(Q_{(x,y)}\left({\lambda }_{g_{j,k}}\right)\right)Q_{(x,y)}{\left({\lambda }_{g_{j,k}}\right)}^{\varphi \left(Q_{(x,y)}\left({\lambda }_{g_{j,k}}\right)\right)}\right)\end{array}$$

(14)

$$\begin{array}{c}\hfill Q_{(x,y)}\left({\lambda }_{g_{j,k}}\right)=max\left\{0,f_{(x,y)}\left({\lambda }_{g_{j,k}}\right)-AF{D}_{(x,y)}\left({\lambda }_{g_{j,k}}\right)\right\},\end{array}$$

(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]. $Q_{(x,y)}\left({\lambda }_{g_{j,k}}\right)$ refers to the relative violated function. $\varphi \left(Q_{(x,y)}\left({\lambda }_{g_{j,k}}\right)\right)$ and $\theta \left(Q_{(x,y)}\left({\lambda }_{g_{j,k}}\right)\right)$ are the power of the penalty function and multi-stage assignment function, respectively.

Compare the fitness value $Fi{t}^{gen}$ of the population at the current iteration with them at the previous iteration $Fi{t}^{gen-1}$, and update the $pbes{t}_{j,k}$ and $gbes{t}_{j,k}$ according to the calculation via Equations (16) and (17):

$$\begin{array}{c}\hfill pbes{t}_{j,k}=\left\{\begin{array}{c}{\lambda }_{gen-1}gen=1\hfill \\ Fi{t}^{gen}ifFi{t}^{gen}>Fi{t}^{gen-1}\hfill \\ {\lambda }_{gen-1}ifFi{t}^{gen}<Fi{t}^{gen-1}\hfill \end{array}\right.\end{array}$$

(16)

$$\begin{array}{c}\hfill gbes{t}_{j,k}=\left\{\begin{array}{c}{\lambda }_{gen-1}gen=1\hfill \\ pbes{t}_{j,k}ifFi{t}^{gen}>pbes{t}_{j,k}\hfill \\ {\lambda }_{gen-1}ifFi{t}^{gen}<pbes{t}_{j,k}.\hfill \end{array}\right.\end{array}$$

(17)

3.3. Individual Customized Update Mechanism

In this section, an individually customized update mechanism was proposed, creating the IUPSO algorithm based on Modified PSO. The individually customized update mechanism contains two critical parameters, which are the individual base update step of the jth heliostat and the individual custom update parameter of the jth heliostat, which is set as following Equation (18):

$$\begin{array}{c}\hfill {\xi }_j=\left\{\begin{array}{c}0.5,ifre{f}_j=1\hfill \\ 0,ifre{f}_j=0\hfill \end{array}\right.\forall j=1,2,\dots ,m,\end{array}$$

(18)

where ${\xi }_j$ refers to the base update step, which is affected by its real-time state. The base update step of an individual heliostat is determined by Equation (18).

$$\begin{array}{c}\hfill {\vartheta }_j={\zeta }_j·{\xi }_j\end{array}$$

(19)

$$\begin{array}{c}\hfill {\zeta }_j=\sum _0^{j\in \Delta }\left\{\begin{array}{c}1,ifmin{\left(x_{j\in \Delta }-x_j\right)}^2+{\left(y_{j\in \Delta }-y_j\right)}^2\in \Delta \cap re{f}_{j\in \Delta }=0\hfill \\ 0,fmin{\left(x_{j\in \Delta }-x_j\right)}^2+{\left(y_{j\in \Delta }-y_j\right)}^2\notin \Delta \cup re{f}_{j\in \Delta }=1,\hfill \end{array}\right.\end{array}$$

(20)

where ${\vartheta }_j$ refers to the individual custom update parameter of the jth heliostat and can be calculated by Equation (20); $\Delta$ refers to the set of the surrounding slave heliostat, which is defined as the six nearest heliostats in the field of heliostats to the jth heliostat.

3.4. The Steps of IUPSO

The step of IUPSO can be summarized as follows:

Step 1: Inheri the real-time aiming strategy of the previous moment via Equation (10) and acquire related parameters of the real-time heliostat scene by Equations (24) and (25);

Step 2: Initialize some parameters that are unique to the IUPSO, such as the individual custom update parameter ${\vartheta }_j$, the reflectivity list of individual heliostats among the whole heliostat plant $re{f}_j$, and the base update step ${\xi }_j$ via Equations (19) and (20); some other parameters of IUPSO are also initialized in this part;

Step 3: Calculate the AFD penalty ${\gamma }_{\lambda }$ via Equations (14) and (15);

Step 4: Update the population of IUPSO according to the customized individual update mechanism through the following Equation (21):

$$\begin{array}{c}\hfill V{p}_{j,k}^{gen+1}={\xi }_j.\left(v{p}_{j,k}^{gen+1}+{\vartheta }_j.{\xi }_j\right)\end{array}$$

(21)

$$\begin{array}{c}\hfill G_{j,k}^{gen+1}=G_{j,k}^{gen}+V{p}_{j,k}^{gen+1}(k=1,2,\dots ,n),\end{array}$$

(22)

where $V{p}_{j,k}^{gen+1}$ refers to the update velocity of heliostat individuals among the next iteration population, and $G_{j,k}^{gen+1}$ refers to the position vector of heliostat individuals among the next iteration population;

Step 5: Compare the fitness value $Fi{t}^{gen}$ and $Fi{t}^{gen-1}$ from two successive optimizations;

Step 6: Update the $\mathrm{Pbest}{}_{j,k}$ and $\mathrm{Gbest}{}_{j,k}$ via Equations (16) and (17); if $Fi{t}^{\mathrm{gen}}>Fi{t}^{\mathrm{gen}-1}$, the optimized aiming strategy can be accepted; otherwise, return to step 2;

Step 7: Repeat steps 1–6 until the terminated condition is met; otherwise, return to step 4;

Step 8: Yield the optimal real-time aiming strategy and output it; meanwhile, record it as the inherited solution for the next iteration.

The flow chart of IUPSO is shown in Figure 6.

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. Parameters of the heliostat and the receiver [56].

Parameter	Value
Direct normal irradiation	1000 W/m2
Maximun half angle	$0.00465$ rad
Northern azimuth angle	180 deg
Elevating angle	70 deg
Number of heliostats	696
Heliostat shape	$Sphericalrectangle$
Heliostat length	$11.28$ m
Heliostat width	$10.36$ m
Tower height	200 m
Receiver center height	180 m
Receiver type	$External$
Sector angle	$11.25$ deg
Sector receiver length	$3.5$ m
Receiver height	20 m
Flux cell size	$40\times 7$ (0.5 m)
Aiming point number	8

. 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 ${696}^8$.

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 $\left(a_{ini},b\right)$, and the initial position of the cloud with the mid-line path of the X-axis is presented by $\left(a,b_{ini}\right)$. The mathematical model of cloud movement is shown as Equations (23) and (24):

$$\begin{array}{c}\hfill \begin{array}{c}\left\{\begin{array}{c}{X}_{clo}=a_{ini}-v_{clo}.sa{p}_{per}\hfill \\ Y_{clo}=b\hfill \end{array}\right.\hfill \\ s.t.Clou{d}_p=True\hfill \end{array}\end{array}$$

(23)

$$\begin{array}{c}\hfill \begin{array}{c}\left\{\begin{array}{c}{X}_{clo}=a\hfill \\ Y_{clo}=b_{ini}-v_{clo}.sa{p}_{per}\hfill \end{array}\right.\hfill \\ s.t.Clou{d}_p=False,\hfill \end{array}\end{array}$$

(24)

where $\left(X_{clo},Y_{clo}\right)$ denotes the real-time position cloud, $v_{clo}$ denotes the real-time velocity of the cloud, and $sa{p}_{per}$ denotes the sampling period of the cloud. $Clou{d}_p=True$ means the clouds move along the Y-axis of the heliostat field, and the clouds move along the X-axis mid-line when the $Clou{d}_p$ 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 $re{f}_j$ and $re{f}_j\in [0,1]$. The dynamic heliostat field condition can be modeled as Equation (25):

$$\begin{array}{c}\hfill re{f}_j=\left\{\begin{array}{c}1,\mathrm{if}\sqrt{{\left(x_j-X_{clo}\right)}^2+{\left(y_j-Y_{clo}\right)}^2}\le R\hfill \\ 0,\mathrm{if}\sqrt{{\left(x_j-X_{clo}\right)}^2+{\left(y_j-Y_{clo}\right)}^2}>R\hfill \end{array},\forall j.\right.\end{array}$$

(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. Parameters of the cloud movement model.

Parameters	Value
Azimuth (/deg)	$az\in [0,30,60,\dots ,360]$
Elevation (/deg)	$ele\in [0,20,40,\dots ,90]$
Radius of cloud (/m)	$R\in [10,20,\dots ,200]$
Velocity of cloud (/m)	$v_{clo}\in [1,2,\dots ,20]$
Sampling period (/s)	$sa{p}_{per}\in [1,2,\dots ,50]$

.

5. Results and Discussion

In this section, the simulation environment is modeled, a heliostat field based on the cloud movement model is established, and we test the modified PSO’s performance under different heliostat field scenarios. Additionally, the performance of the seven optimization algorithms for solving the real-time aiming strategy of heliostat field conditions is tested in the pre-modeled simulation environment. The root mean square error (RMSE), mean square error (MSE), and mean absolute error (MAE) are chosen to evaluate the performance improvement of the algorithms. The simulation experiment is conducted in PyCharm 2020.3 software, and the specific configuration is shown as follows: CPU 3.39 GHz, RAM 28 GB, Window11.

5.1. Simulation Scenarios

Various parameters of the cloud movement model affect the dynamic heliostat field scene, including shape, size, speed, and sampling period. However, the number of heliostats shadowed by clouds can describe their impact. Similar cloud shadowing effects can result from different parameter combinations. In order to avoid duplicate calculations and better illustrate the impact of the cloud movement model on the heliostat field scene, an equivalent velocities parameter has been proposed. This parameter equates the effects of various cloud parameters and provides an intuitive description of different dynamic heliostat field scenes.

To illustrate the dynamic cloud cover changes in the heliostat field, we sampled the cloud pathways above it along the mid-line of the Y-axis based on the cloud movement model. Due to varying parameter combinations, the number of shadowed heliostats differs. Thus, we used different cloud equivalent velocities to depict these changes, as shown in Figure 8, where black circles indicate the heliostat field and red circles represent shadowed heliostats. To enhance the accuracy in depicting heliostat field scenarios under various cloud equivalent velocities, we calculated and charted the number of heliostats covered by the moving cloud per unit sampling period for each of the cloud equivalent velocities, which is shown in Figure 9.

The performance of Modified PSO in various heliostat field scenarios is depicted in Figure 10. Our tests across different cloud equivalent velocity heliostat fields reveal that the performance of a heliostat for aiming strategy real-time optimization deteriorates when the cloud equivalent velocity exceeds 4.0. This decline is primarily due to the classical PSO algorithm’s population mechanism, which, as the number of covered heliostats per unit sampling period increases, leads to higher computation demands. The heliostat struggles to complete optimization within the iterative constraint period, indicating significant potential for its further optimization.

5.2. Simulation Results between Different Algorithms

This section tests the performance of the IUPSO algorithm in different heliostat field scenarios. Three scenarios with different cloud equivalent velocities (5.0, 6.0, and 10.0) from the former section are selected, and seven optimization algorithms are compared: IUPSO, Modified PSO, PSO, BA, GA, WOA, and RDA. The parameters of these parameters are set in

Table 3. Some parameters are set in seven algorithms.

Method	Parameter
GA	$sizepop=1000$, $maxgen=200$, the mutation factor is $F=0.5$, the crossover factor is $CR=0.7$ [11].
BA	$sizepop=1000$, $maxgen=200$, $V=[0,5]$, $f=[0,100]$, $f=[0,100]$, $A=[0,2]$ [57].
WOA	$sizepop=1000$, $maxgen=200$, $\overrightarrow{a}$ is decreased from 2 to 0, $\overrightarrow{A}\in \left[-1,1\right]$, $p\in \left[0,1\right]$ and is set as $50\%$ [34].
RDA	$sizepop=1000$, $maxgen=200$, the Red deer is set as $\left[X_1,X_2,...,X_{N\mathrm{var}}\right]$, $a_1,a_2,a_3\in [0,1]$,$b_1,b_2\in [0,1]$, the initial parameter $\alpha ,\beta \in \left[0,1\right]$ [36].
PSO	$sizepop=1000$, $maxgen=200$, $c_1=c_2=[0,0.7]$ [58].
heliostat	$sizepop=1000$, $maxgen=200$, the inertia constant is $\varpi \in \left[0.3,1.2\right]$, $c_1,c_2\in [0,1]$.
IUPSO	$sizepop=1000$, $maxgen=200$, base update step is ${\xi }_j\in \left[0,0.5\right]$, surrounding slave individual is ${\zeta }_j\in \left[0,6\right]$, the innovation parameter is $a=[0,1]$, $c_1,c_2\in [0,1]$.

. The simulation results are shown in Figure 10, Figure 11, Figure 12 and Figure 13 and

Table 4. Performance indexes [33].

Metric	Equation
MAE	$MAE=\frac{1}{N}{\displaystyle \sum _{i=1}^N}\left|Y_i-{\widehat{Y}}_i\right|$
MSE	$MSE=\frac{1}{N}{\displaystyle \sum _{i=1}^N}{\left(Y_i-{\widehat{Y}}_i\right)}^2$
RMSE	$RMSE=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N}{\left(Y_i-{\widehat{Y}}_i\right)}^2}$

.

In Table 4, $Y_i$ and ${\widehat{Y}}_i$ denote the optimization results of different algorithms. From Figure 11, Figure 12 and Figure 13, it is evident that with the population inheritance mechanism, IUPSO and MPSO outperform GA, BA, WOA, and RDA by providing a better initial solution for optimizing the aiming strategy. Specifically, IUPSO, through its custom update mechanism for heliostat individuals, surpasses Modified PSO in solution efficiency, reducing over 100 iterations on average in tests across three heliostat scenarios. While WOA achieves convergence in all scenarios due to its spiral update mechanism, its performance degrades as heliostat complexity increases. In contrast, IUPSO demonstrates consistent performance and robustness, proving its superior real-time aiming strategy for heliostats. Additionally, from Figure 13 and

Table 5. Data comparison between seven algorithms.

Equivalent Velocities 5.0
Algorithm	MSE	RMSE	MAE	Difference of Mean Iteration Periods
IUPSO/Modified PSO	1923.9562	43.8629	14.5977	110
IUPSO/GA	8953.7463	94.6242	75.3990	110
IUPSO/BA	32,429.9280	180.0831	185.5373	>130
IUPSO/WOA	4191.5491	64.7422	219.0136	10
IUPSO/RDA	8390.6510	91.6005	92.0474	——
Equivalent Velocities 6.0
IUPSO/Modified PSO	1351.7197	36.7657	31.8737	80
IUPSO/GA	7517.8218	86.7054	78.4167	70
IUPSO/BA	28,827.2781	169.7860	168.2572	>130
IUPSO/WOA	3993.9796	63.1979	39.4935	60
IUPSO/RDA	6635.3528	81.4577	69.1545	——
Equivalent Velocities 10.0
IUPSO/Modified PSO	9724.3286	98.6120	93.7950	>130
IUPSO/GA	22,535.1572	150.1171	145.2421	>130
IUPSO/BA	37,189.3496	192.8454	190.0358	>130
IUPSO/WOA	5710.2145	75.5660	60.3263	70
IUPSO/RDA	7077.4914	84.1278	76.5827	——

, Modified PSO, GA, and BA fail to converge within the iteration constraint period. Although RDA achieves convergence quickly, it falls into a local optimum with a significantly lower output of the optimal solution than WOA and IUPSO. WOA requires more than 70 iterations to complete the optimization compared to IUPSO, highlighting IUPSO’s ability to converge quickly. It also suggests that GA and BA have weaker problem-solving skills, while Modified PSO and RDA may become trapped in a local optimum. IUPSO, Modified PSO, and WOA outperformed classical algorithms such as BA, GA, and RDA, regarding MSE, RMSE, and MAE under the same iteration constraint in three test scenes. IUPSO and WOA demonstrated higher global optimization capabilities by completing optimization before the iteration constraints. RDA showed good local optimization capabilities, but its global optimization ability was weaker than the other algorithms, resulting in inferior optimality. Modified PSO had similar local optimization capabilities as IUPSO but was more robust, taking longer to optimize in more complex scenes and surpassing the iteration constraints.

5.3. Large-Scale Simulation Validation Experiments

In order to test the performance of the Modified PSO and IUPSO algorithms in more complex scenarios, 400 scenarios were generated using equal probability random sampling based on the parameters of the cloud movement model. The simulation results are shown in Figure 14, and the difference in simulation data between the IUPSO and Modified PSO algorithms is illustrated in Figure 15.

The purpose of the real-time scheduling optimization of the heliostat aiming strategy is to maximize the output power of the heliostat field under the premise of satisfying the heliostat field scheduling constraints, and thus, the fitness value function is proposed in Section 2.1.2. From Figure 14 and Figure 15, for the heliostat, IUPSO achieve the real-time scheduling of the heliostat aiming strategy under dynamic cloud cover, satisfying the scheduling constraints of the heliostat aiming strategy. Compared with Modified PSO, the IUPSO adaptation value function is improved by 19%, and the real-time mirror field output power is improved by 16%. Modified PSO exhibits two performance degradations after 40–60 obscured heliostats compared to more than 90, indicating its limitations.

On the other hand, IUPSO demonstrates better overall simulation robustness. However, specific dispersion points are observed after obscured heliostats exceed 160, highlighting its limitations under some real-time dynamic cloud shadow scenarios. Furthermore, dispersion points are along other axes, particularly noticeable in Figure 14a. This dispersion is attributed to calculation errors arising from varying direct irradiation intensities on the heliostats positioned across different locations within the sector region. As a cloud moves closer to the sector region’s center, the direct sunlight angle increases, resulting in a steeper heliostat solar irradiance gradient. When clouds traverse the central axis of the X-axis, passing through the sector’s center, the neighboring heliostats experience significant variations in irradiation intensity. This can lead some heliostats to converge into local optima, resulting in the presence of dispersion points.

In Figure 14b, IUPSO lacks dispersion points, primarily due to minimal variation in the solar irradiance gradient at the Y-axis mid-line of the sector heliostat field. It efficiently avoids dispersion by employing an individualized update mechanism tailored to heliostat modeling, enhancing aiming strategy global optimization efficiency. However, as indicated in Figure 14a, IUPSO still encounters limitations in certain specific heliostat field scenarios.

Table 6. Data analysis and comparison based on the mid-line cloud path of the Y-axis and X-axis.

Parameters	Mid-Line Cloud Path of Y-Axis
	Modified PSO	IUPSO
Mean Intercept ratio score	0.7525	0.9131
Mean Fitness Value	335.0197	394.9755
Optimization time per-period (/s)	0.52	0.15
Mean Optimization periods	——	87
Mean Optimization time (/s)	——	12.89
Parameters	Mid-Line Cloud Path of X-Axis
Mean Intercept ratio score	0.8028	0.8310
Mean Fitness Value	297.6034	387.6110
Optimization time per-period (/s)	0.66	0.31
Mean Optimization periods	——	82
Mean Optimization time (/s)	——	25.96

compares simulation data of IUPSO and Modified PSO algorithms for 400 sets of simulated heliostat fields. The average energy intercept efficiency obtained by the IUPSO algorithm in real-time is $91.3\%$, which is a $16\%$ improvement over Modified PSO when the clouds move along the Y-axis mid-line. The average optimization time for a single optimization of IUPSO and Modified PSO is $0.15$ s and $0.52$ s, respectively. The average time taken to complete the optimization of the aiming strategy is $12.89$ s, which is much smaller than the average sampling time of $69.4$ s for the cloud. However, when the clouds move along the X-axis mid-axis, the average energy intercept efficiency of IUPSO drops to $83.1\%$, which is a $3\%$ improvement compared to the Modified PSO. The average optimization time to achieve convergence is $25.96$ s, much lower than the cloud’s average sampling time of $86.08$ s.

5.4. Full-Heliostat Field Simulation Experiment

The 3-dimensional display of the heliostat field receiver unfolded diagram is shown in Figure 16. A full-heliostat field simulation with a cloud movement path is presented in this section, and the clouds move according to the paths in Figure 17. The cloud positions at three different periods are selected chronologically to demonstrate the performance of the proposed method, where the aiming strategy of each heliostat field sector is obtained independently by IUPSO.

The velocity of the cloud in the case of full heliostat field experiments is 5 m/s, the radius of the cloud is 200 m, the azimuth angle is 180°, and the elevation angle is 70°. The flux map of the cylindrical receiver is unfolded to a flat surface according to Figure 17. The flux map of the three locations in the full heliostat optimized by IUPSO proposed in this paper is shown in Figure 17 and Figure 18. All experiments were performed ten times; the data are reported in figures and tables as the mean values.

From Figure 18a–c, the flux gap indicates the influence on the flux map caused by the obscuration of the heliostat field caused by clouds at different locations. The mean optimization time in location A is $15.36$ s, in location B is $23.23$ s, and location C is $18.53$ s. The IUPSO effectively addresses the real-time aiming strategy for the full heliostat field. The results show it is equally effective for the sector heliostat aiming strategy optimization algorithm. Additionally, the IUPSO’s average optimization time in the three scenes is shorter than the heliostat field sampling period, indicating its capability for real-time aiming strategy optimization.

5.5. Atmospheric Attenuation Inaccuracy Estimation

In the initial study, we ignored the effect of atmospheric attenuation on solar irradiance, mainly for the sake of simplifying the model and reducing computational complexity. In order to more accurately assess the effect of atmospheric attenuation on the real-time scheduling uncertainty of the heliostat field aiming strategy, we propose a simplified atmospheric attenuation model by considering the effects of atmospheric scattering and absorption as Equations (26)–(28), shown below:

$$DN{I}_{\mathrm{scatter}}=DN{I}_0·e^{-k_{scatter}·m\left(az\right)}$$

(26)

$$DN{I}_{\mathrm{absorb}}=DN{I}_{\mathrm{scatter}}·e^{-k_{absorb}·m\left(az\right)}$$

(27)

$$DN{I}_{\mathrm{total}}=DN{I}_0·e^{-\left(k_{scatter}+k_{absorb}\right)·m\left(az\right)},$$

(28)

where $DN{I}_0$ denotes the initial irradiance, $k_{scatter}=1\times {10}^{-5}$ and $k_{absorb}=1\times {10}^{-5}$ [59] denote the scattering coefficient and absorption coefficient, which are related to the number and type of particles in the atmosphere, respectively. $m\left(az\right)=1/sin\left(az\right)$ denotes the air quality factor. The impact of the inaccuracy caused by omitting atmospheric attenuation is estimated based on the experiments in Section 5.3, and the results are shown as Figure 19.

In Figure 19a, the impact of neglecting atmospheric attenuation on the intercepted power of the heliostat field, optimized by IUPSO, is analyzed. The figure demonstrates that in the scenario of an X-axis cloud trajectory across a multi-heliostat field, the gradient of decreasing irradiation intensity grows near the center of the sector heliostat field, influenced by the incidence angle. This results in an average reduction of 4.8% in the heliostat field’s energy after introducing the atmospheric attenuation model. Conversely, Figure 19b examines the Y-axis cloud trajectory scenario. Here, the decrease in irradiation intensity is more uniform, leading to a smaller average energy reduction of 2.7% in the heliostat field. Therefore, integrating a simple atmospheric attenuation model slightly affects the optimization of real-time aiming strategies in heliostat fields.

6. Conclusions

This paper proposed a method for solving the real-time aiming strategy in SPT plants, utilizing an improved meta-heuristic optimization algorithm. Two new mechanisms were introduced into the classical PSO based on the mathematical model of a heliostat’s physical IA. These mechanisms allowed the intelligence agent, directly associated with the physical heliostat, to receive status information from others in the region, adaptively update its state, and leverage the physical nature of heliostat operational continuity to facilitate the solution process. A simulation environment of a heliostat field is established as a case test to verify the performance of the algorithm proposed in this paper by comparing the optimization performance of six swarm intelligence optimization algorithms’ dynamic mirror field real-time aiming strategy. The results demonstrated that our method achieved real-time optimization in heliostat field scenes with continuously changing cloud shadows.

The population inheritance mechanism obtains 10 times the initial solution for Modified PSO and IUPSO, which greatly improves the solution efficiency of the algorithms, but the heliostat field scene becomes more complex, and the performance of the Modified PSO appears to be degraded. IUPSO physically models the heliostat individuals, as well as proposes a customized updating mechanism of the heliostat individuals to solve the problem and successfully realizes the complex. It successfully realizes the real-time optimization of the aiming strategy in complex heliostat field scenarios and reduces the computational cost. Tests on 400 real heliostat field scenarios show that IUPSO outperforms the improved PSO in all aspects and ensures robust real-time optimization even under complex conditions. A model for atmospheric attenuation was developed and implemented to assess its impact on the real-time scheduling optimization of heliostat field aiming strategies. Testing on a full-heliostat field simulation environment with a cloud movement path and heliostat scenes based on three different cloud locations confirmed the effectiveness of the IUPSO real-time aiming strategy. Future work should consider more realistic and complex cloud conditions. Combining machine learning with swarm intelligence optimization algorithms may significantly mitigate real-time scheduling errors in heliostat fields caused by cloud prediction inaccuracies. Additionally, the current atmospheric attenuation model only accounts for scattering and absorption. A more sophisticated model should be developed to enhance the realism and reliability of future studies.