Mathematical Modelling and Optimizing Design of Heliostat Field

The heliostat field is a form of solar thermal power plant, and establishing a heliostat field is of great significance for reducing carbon emissions. This paper focuses on the mathematical modelling and optimization design problem of the heliostat field. For three different design scenarios, firstly, based on the data provided by the China Undergraduate Mathematical Contest in Modeling 2023, a geometric optical efficiency model is proposed to determine the annual average optical efficiency and output thermal power under specific heliostat field design parameters. Secondly, aiming at the optimal design parameters of the solar furnace field under the condition of rated annual average output thermal power is 60


Introduction
Nowadays, constructing a novel energy infrastructure predominantly led by renewable energy sources, is a crucial step towards reducing carbon emissions and transition to renewable energy sources.This facilitates the achievement of reaching the peak carbon emission, and subsequently progressing towards the attainment of "carbon neutrality".In this context, solar energy, as the most abundant renewable clean energy source (Liu,2022), has spurred the widespread use of solar power generation technology.Tower solar thermal power generation technology is a typical form.However, the high construction costs, low equipment utilization rates, and limited power generation capacity of tower solar power generation technology have limited its development.Therefore, the urgent research challenge is how to develop tower solar thermal power generation technology with high efficiency and low cost (Liu,2022).Currently, researchers have shifted their focus to the calculation of optical efficiency and optimization layout theory of solar fields in tower solar thermal power plants, forming the basis of this study.
Heliostat fields consist of numerous heliostats, and once a power station is constructed, the positions of the heliostats cannot be changed.Therefore, the initial arrangement of heliostats requires optimal design.Researchers have proposed numerous novel methods for the optical efficiency calculation and optimization layout theory of heliostat fields.In their research, Lin Xiaolin et al. (2016) used Monte Carlo ray tracing to simulate the optical efficiency of heliostat fields, with a particular emphasis on determining the distribution of energy flux density on the heat collector surface.Lipps et al. (1978) introduced a novel way to evaluate the overall performance of the heliostat field by dividing it into multiple regions and using the average efficiency of specific regions.Utamura et al. (2007) explored the impact of the ratio between absorber tower height and the radius of the first-row heliostats on the optical efficiency of the heliostat field.However, these methods all face challenges of balancing calculation efficiency and accuracy, necessitating substantial data support.
Against this backdrop, Duan et al. (2020) attempted to solve the contradiction between calculation accuracy and efficiency using a GPU-oriented ray-tracing simulation technique, specifically QMCRT.It can avoid many unnecessary calculations through preprocessing, improve the calculation efficiency and accuracy of optical efficiency with a relatively small amount of data.This article first establishes an efficiency model and analyzes the optical efficiency and thermal power production of the heliostat field under given design parameters.Subsequently, the PSO model combined with the efficiency model is used to analyze the optical efficiency calculation and optimization layout of heliostat fields with single objective and multiple constraint conditions.Then, for the common multi-objective optimization design problem of heliostat fields, the QMCRT model and combined with the 3D-DDA method of ray tracing is utilized.Selecting the 6282-heliostat field as an example, the maximum thermal output power and the corresponding optimal layout of the heliostat field of the heliostat field are solved through simulation.Finally, sensitivity analysis is conducted to ensure its reliability.
The rest paper is organized as follows: Section 2 provides an overview to the theoretical basis and principles of efficiency model, PSO, QMCRT, and other relevant algorithms.In Section 3, a case study of the heliostat field is conducted, utilizing the efficiency model, PSO, and QMCRT to analyze results under three different scenarios.Section 4 includes a sensitivity analysis of the newly used PSO in such case study to verify its reliability.The paper concludes with a summary in section 5.

Efficiency model
Optical efficiency  can be defined as the ratio between the energy captured by the heat collector and the solar energy striking on the heliostat field.To calculate the average optical efficiency, annual average thermal output power and the annual average thermal output power per unit mirror area of heliostat, it is important to establish the model of optical efficiency firstly.To simplify the calculation, all the "annual average" indicators in this problem are calculated at 9: 00, 10: 30, 12: 00, 13: 30,15: 00 local time on the 21 st day of each month.The general modelling method of optical efficiency can be expressed as follow:

Calculation of heliostat parameters and vertex coordinates
Suppose absorber is situated at the center of heliostat field, and regard collector as a point.In this study, the coordinate system for the heliostat field is set up with the point of intersection between the axis line of the endothermic tower and the plane of the heliostat field as the coordinate origin.That is, the geometric center of the base of the endothermic tower is taken as the origin of the coordinate system O.The direction of the east is designated as the positive semi-X-axis of the coordinate system, the direction of north is designated as the positive semi-Y-axis of the coordinate system, and the zenith direction is designated as the positive semi-Z-axis of the coordinate system.Finally, the three-dimensional space rectangular coordinate system of the heliostat field is established (Liu,2022), as shown in Fig. 1.  (Liu,2022) Then the heat collector on the endothermic tower has coordinate is  = [0,0,   ], where   represents height of the heat collector center relative to XY-plane.The mirror surface center of the heliostat has coordinated  = [  ,   ,   ]And the mirror surface center of heliostat A has coordinated   = [ , ,  , ,  , ]., where  , is the height of the mirror center of heliostat relative to the XY-plane, which can also be expressed as ℎ 0 .The sun reflected light through the helioscope points to the center of the heat collector.The unit normal vector  , expresses the direction of light from the mirror center of heliostat towards the heat collector., that is, reflected light which can be expressed in the following ways: The unit normal vector of the sun ray at the fixed time point is determined, let the unit vector of the incident light be  , ,  , = [ ,  ,  , ], then the calculation formula of  , ⃗⃗⃗⃗⃗ , is: Where   is solar elevation angle,    solar azimuth angle.They can be calculated from following two formulas: Where  is local latitude, and north latitude is positive;  is the hour angle of the sun, which can be obtained from: Where  is the local time and  is the solar declination angle, which is determined by: where  is the count of days starting from the vernal equinox (vernal equinox is day 0).
Then, the normal vector of heliostat surface is obtained from the reflection law with Incident light normal vector and reflected light normal vector: Where Based on results obtained from above formulas, the cosine efficiency and shadow occlusion efficiency can be modelled.

Calculation of Cosine efficiency
To reflect sunlight to the heat collector, there must be a solar incidence angle  between the incident light and the normal direction of the specular reflection point on the heliostat surface.The cosine efficiency of the point is defined as  .Generally, the entire heliostat surface is regarded as an ideal plane, at which time reflection points on the surface will have the same surface normal vector.Thus, Cosine efficiency   of a single heliostat can be calculated (Liu,2022): Where  is the unit vector in the opposite direction of the incident light, expressed as:  = [− cos(  ) sin(  ) , − cos(  ) cos(  ) , − sin(  )] (11)

Calculation of shadow occlusion efficiency
The computation of shadow occlusion efficiency   is difficult in the modelling process.It is primarily affected by two types of losses: shadow loss and occlusion loss.
As shown in Fig. 2, the energy loss caused by part of the heliostat A in Fig. 2-a that cannot effectively reflect sunlight due to being in the shadow of heliostat B is called shadow loss.As illustrated in Fig. 2-b, the sunlight reflected by the heliostat A, the loss due to part of the light being occluded by the heliostat B and not being able to propagate to the collector is called the occlusion loss.

Fig. 2 Shadow and occlusion schematic diagram
In this study, the shadow occlusion efficiency of the heliostat field is determined by the method of Monte Carlo ray tracing and plane projection.Serval assumptions are made:(1) the sunlight is assumed to be parallel beams; (2) assume that the mirror center coincides with the support point of the heliostat; (3) Assume that shadows and occlusions only occur in the specific area surrounding the target heliostat.Then the general calculation process can be shown as follow (Liu,2022): Choose one of the heliostats to be our target heliostat A. Then, choose A heliostat B from the remaining heliostat that may interfere with the heliostat A.
Calculating the distance between heliostat A and B on the XY-plane and determine whether B is within shadow occlusion of A. If not, the other heliostat is selected as the problem heliostat B; If so, proceed to the next step.
Determining the surface normal of heliostats, A and B at the current time point in the simulation, as well as their plane equations in the coordinate system of the heliostat field.Calculating the vertex coordinates of the heliostat A and B at the current time and divide the mirror surface of the problem heliostat B into  ×  points of equal proportion.
Calculating the coordinate data of each dot based on the sun position at the current time and the geometric parameters of the heliostat.Then, starting from the initial point, a line is drawn along the direction of the incident ray, and calculating intersection coordinates between this line and the surface of the target heliostat.Finally, On the target heliostat plane, determine if the intersection point is within the heliostat bounding rectangle area.The number of intersections in the rectangle is counted and the ratio of the number of intersections to the total number of intersections is calculated to obtain the shadow occlusion loss   .The shadow occlusion efficiency is obtained from: = 1 −   (12)

Calculation of atmospheric transmissivity
Atmospheric transmissivity is a physical quantity that describes the transmittance of radiation energy to the heat collector.It can be calculated by (Liu,2022): Where d is the distance between the center of the heliostat surface and the center of the heat collector (distance of reflection), which can be expressed as:

Calculation of heat collector truncation efficiency
Truncation efficiency represents the proportion of the effective solar radiation energy captured by the heat collector and the total solar radiation energy that can theoretically be received under certain conditions.It can be expressed as: Where  reflected represents the energy reflected by the heliostat,  ℎ is a loss of energy due to shadow and occlusion of the light, and  received is the energy received by the heat collector.
According to this,   can be determined numerically or analytically.The numerical method mainly adopts the Monte Carlo ray tracing (MCRT) algorithm.This approach involves the generation and tracing of numerous rays to ascertain their positions upon striking the heat collector surface following reflection by the heliostat.
The computation of the truncation efficiency or the flux density on the heat collector surface necessitates the cumulative calculation.This involves computing the coordinates of the intersection points for the multitude of rays that reach the heat absorber post-reflection by the heliostat.Despite its high accuracy, this method requires substantial computational resources.
In contrast, this study utilizes an analytical approach.The widely recognized HFLCAL model is employed to assess the truncation efficiency of the heat collector for each heliostat.A circular normal distribution is incorporated within the HFLCAL model to evaluate the truncation efficiency.The corresponding formula is articulated as follow (Besarati et al., 2014): Where (  ,   ) denotes the target point of the reflected light ray generated by each heliostat on the heat collector surface.Since the heat collector has the relatively small dimensions, a simplification is applied.This simplification assumes the target point coincides with the center of the heat collector, hence (  ,   ) = (0,0) .  is the effective deviation, which can be determined by: Where   represents the error in the sun shape, which arises from the uneven distribution of the solar intensity across the sun disk,   is the beam quality error attributed to the slope error of the heliostat surface.  is the astigmatic error accounts for additional deformation of the reflected ray when incident ray is not aligned with the heliostat surface's normal.  is the tracking error (Besarati et al., 2014).Then: Where   is the slope error,   and   represent the size of the light spot located at the heat collector in the meridian direction and the direction of sagitta of arc.

Calculation of specular reflectivity
The specular reflectivity   has strong relation with the material utilized in the heliostat.The current specular reflectivity of glass mirrors is generally between 0.93 and 0.94.But the specular reflectivity will decrease due to the aging of heliostat and the environmental factors such as a large number of dust particles or impurities may adhere to the heliostat surface.
According to the study in (Shi et al., 2019), This study will take reflection efficiency of heliostat as 0.92.

Calculation of Optical efficiency and thermal output power
Based on above formulas, The optical efficiency  of the heliostat is calculated by: The Normal direct radiation irradiance (DNI) is a measure of the solar radiation energy received in unit area and per unit time. in the plane perpendicular to the sun rays on the Earth.It is calculated by: Where  0 is the solar constant with value of 1.366 kW/m 2 , H represents the altitude (unit: km).
The thermal output power of heliostat field  field is expressed as： Where DNI is the irradiance of normal direct radiation;  is the total number of heliostats;   is the daylighting area of the  th heliostat (unit:  2 );   is the optical efficiency of  th mirror.
Then the annual average thermal output power  field  ̅̅̅̅̅̅̅ can be calculated as Where  i, ̅̅̅̅̅̅̅̅̅ is average thermal output power on the 21 st day of each month."" represents the month.And "" indicates the local time on the 21st day of each month. = 1,2,3,4,5 represent 9:00, 10:30, 12:00, 13:30 and 15:00 respectively.The annual average efficiency is calculated similarly.

PSO Model
PSO is an optimization algorithm draws inspiration from swarm intelligence.PSO is a power tool can solve optimization problems effectively.It can search the given space (called search space), where a candidate solution represented by a particle in the space.Then, it updates the velocity and position of the particles by looking at the historical best locations of individuals and groups to find the optimal solution to the problem (Shami et al., 2022).PSO is simple to implement, precise, and has fewer parameters that allows faster convergence to the optimal solution, making it an attractive optimization algorithm (Shami et al., 2022).As a result, it is widely applied in many fields, including natural science, engineering, and finance.It is newly used to find the optimal solution of the heliostat field.The general modelling method of PSO can be expressed as follow: Suppose there is D-dimensional search space, and N particles forms a swarm in this search space.Each particle has a position vector   = [ 1 ,  2 , . . .,   ] and a velocity vector   = [ 1 ,  2 , . . .,   ] , where  = 1,2, . . .,  .Firstly,   and   are initialized randomly.Then, in each iteration, the optimal position of particle found is expressed as   = ⌊1, 2, . . ., ⌋, and the optimal position of whole swarm is denoted as  = [ 1 ,  2 , . ..,   ] .When finding these two optimal solutions, the particle  updates its position and velocity according to equations ( 25) and ( 26) = 1,2, . . .,   = 1,2, . . .,  Where  1 and  2 are the cognitive and social acceleration coefficients, and  1 and  2 should fall in interval [0,4] . 1 and  2 are uniform random numbers generated between 0 and (Shami et al., 2022).To ensure sufficient global optimal solution search ability during the early phase of iteration and focus on local optimal solution search ability during the later phase of iteration, linearly varying inertia weight (LVIW) is introduced (Shami et al., 2022).It can be expressed as follow: where   and   represent the initial and final values of the inertia weight.T represents the total number of iterations possible s, and t represents the number of the current iteration.And equation ( 2) is modified: In this study, PSO with LVIW is used.

QMCRT Model
The efficiency model established previously is applicable to the ideal heliostat field.
That is, Every heliostat is mounted at a uniform height and has the same size).And the heliostat is arranged regularly, making heliostat field as a perfect shape, such as circle.However, for the actual heliostat field, the installation height, and dimensions of the heliostat are often different.The arrangement of heliostat is also more complicated.Therefore, a new algorithm is employed to determine the optimal scheme for the heliostat field, significantly helps the simulation.It is QMCRT algorithm.QMCRT algorithm is firstly proposed by Duan et al. (2020) based on MCRT algorithm, aiming at improving the efficiency and accuracy of MCRT.MCRT is computational method that can simulate light propagation and reflection in the real world by randomly sampling light paths and tracing them (Jensen et al., 2003).While the QMCRT uses precomputed random variables sets, not generating all random variables (Duan et al., 2020).Flexible combinations of elements of two variables can yield results nearly the same as those of MCRT.
The modelling of the QMCRT involves two primary steps: preprocessing, ray tracing (Duan et al., 2020).The QMCRT model is represented as follow:

Preprocessing
Suppose reflected light will converges the surface of heat collector, then compute The value of  relates to flatness of the heliostat surface (Duan et al., 2020). obtained through the inverse transform sampling (ITS) method.In this study, QMCRT mainly samples  values from 0 mrad to 9.3 mrad, since sampled  values rarely larger than 4.65 mrad.When  is smaller than 4.65 mrad,  is calculated discretely, that is to form cumulative histograms and using it to approach the Cumulative Distribution Function (CDF).Then, ITS indicates that: The CDF is sampled evenly across the range from 0 to 1, with  representing the variable used for sampling (Duan et al., 2020).When  is larger than 4.65mrad, it can be calculated using formulas below: Where  is an integral constant, and <4.65 is the CDF as  is 4.65 mrad.K and  are two parameters can be calculated from: {  = 0.9 (13.5)−0.3  = 2.2 (0.52) 0.43 − 0.1 (32) X represents the Circumsolar Ratio (CSR).R specifies the Buie sun shape.Then storing them in the MHNP and SSP (Duan et al., 2020).Finally, tessellating each micro heliostats into heliostat and find random starting index array (RSIA) to aid selection of directions from the Sun Shape Pool (SSP) and micro heliostat Normal Pool MHNP (Duan et al., 2020).The SSP and MHNP are random variables sets of the identical size.SSP contains  and  and MHNP contains  1 , .

Ray tracing
Firstly, determine the directions of incident solar ray directions and micro heliostat surface using the SSP and MHNP.a group of incident lights along with a set of heliostat surface normal are randomly selected from SSP and MHNP.
Next, convert the sampled rays and heliostat surface normal from their local frame to the world frame.Since the heliostats and heat collector surface are defined in the world frame, while the directions included SSP and MHNP are defined in the local frame.The transformation between them is achieved through affine transformations.
Finally do simulation to test the shadow and occlusion of rays applying 3D-DDA algorithm (He et al., 2017).Based on the research (He et al., 2017) and our improvement, it involves the following steps: The first step is preparation.Initially, the algorithm generates supplementary data as the precondition.Based on this, for each heliostat, the heliostat field is segmented into uniform 3D voxels.Then, an axis-aligned cubic bounding box is established, with its center positioned at the midpoint of the heliostat.The bounding box's edge is set to match the diagonal measurement of the heliostat surface.When a voxel intersects with a cubic bounding box, it is designated to the corresponding heliostat that overlaps.The second step is testing.It involves checking if a ray intersects with a voxel adjacent to the voxel where the ray's origin resides.If this condition is met, the algorithm proceeds to test whether the ray intersects any heliostat associated with the voxel it intersected.This process is repeated iteratively until the ray moves beyond the boundaries of the 3D voxel (Duan et al., 2020).Different from regular DDA, In the ray marching of light and grid, instead of stepping a fixed value, the 3D-DDA step to the next intersection with the grid, which is conservative.For further illustration, part of a division of a heliostat field is shown in Fig. 3. Fig. 3 The 3D voxels (including heliostats) From Fig. 3, a single voxel can be associated with multiple heliostats.Similarly, a single heliostat has the potential to intersect with numerous voxels.For the further analysis, projecting the voxels into 2D plane, then the voxels becoming grids shown in Fig. 4. where     are the lengths of grid in the x and y directions.  and   represent the x and y components of unit direction vector of the ray.Every grid can be numbered with (, ), where  represents the row and n represents the column.
Then starting from ( 0, ,  0, ).It is the starting original intersection of the line and grid.
The above convention is same for   , where "sign" represents the stepping direction of the light.Then: The calculating process is the same for that of Z-axis.Each possible intersection will appear in three axis planes.Looking for the next three intersections in the three axis planes, and get the nearest one to do the step, obviously we know the orientation of the next cell according to the axis where the intersection is located, for example, if we cross the x-y plane, it must be the Z-axis step 1.After many iterations, then All voxels that intersect the light can be found.The voxel corresponds to the heliostat, so the heliostats that intersects the light are known.

Case study
In this paper, a circular heliostat field is chosen for the case study.The circular heliostat field is located at 98.5E and 39.4N, with an elevation of 3,000m and a radius of 350m.The heliostat surface side length is 2m to 8m, and the installation height b ranges from 2m to 6m.The separation between the centers of the bases of neighboring heliostats exceeds the width of the heliostat by at least 5m.Additionally, positioned at the center of the heliostat field, there is an endothermic tower.The planned height of endothermic tower, measured from the center of the heat collector, is 80m, with no heliostat installed within 100m-radius around the tower.The heat collector on the endothermic tower utilizes a cylindrical surface heat collector measuring 7m in diameter and 8m in height.
In this case study, serval scenarios were considered.The first scenario is that the heliostat size is 6 m×6 m and installed at a height of 4m and the dataset is provided to indicate the coordinates of the center of all heliostats in the heliostat field coordinate system.Subsequently, the efficiency model was employed, utilizing dataset then calculating the annual average optical efficiency, annual average thermal output power and the annual average thermal output power per unit mirror area of the heliostat.This approach allowed us to get the outcomes precisely and find related indictors incidentally to better evaluate the performance heliostat field.
Then second scenario is that the rated annual average thermal output power (hereinafter referred to as rated power) of the heliostat field is 60 MW.All heliostats have the same size and installation height.The goal is that the heliostats field can achieve the maximum annual average output thermal power per unit mirror area of heliostat under the scenario that the rated power is reached.The following design parameters of the heliostats field: coordinate of the endothermic tower, heliostats size, installation height, number of heliostats, and heliostats positions(coordinates) need to be determined to achieve the goal.PSO model is employed with efficiency model to solve this optimization problem.This approach is parallelizable and scalable, allows us to simplify the problem.It can also provide the optimal solution precisely strikes a delicate balance between local exploitation and global exploration.The third scenario is more complex.The planner wants to achieve the highest annual average thermal output power and annual average thermal output power per unit mirror area.Meanwhile, the planner wants to maximize the utilization of the funds for the construction of heliostat field.Heliostats may have the different size and height.The parameters specified in the second scenario need determining to achieve the goal.QMCRT model is employed to handle this complex problem since the it can significantly reduce the calculation amount.Also, it can be a guide for our simulation.
Ultimately, based on above process, the optimal heliostat field for scenarios 2 and 3 can be found successfully.The process diagram of the case study is illustrated in Fig. 5.

Scenario 1
According to the efficiency model, the solar elevation angle, solar azimuth angle, and direct normal irradiance (DNI) are calculated.These results, along with the efficiency model, allow for the determination of the optical efficiency and the output thermal power of the heliostat field.The visualization of these parameters is performed.It clarifies the positioning relationship between the heliostats and the absorber tower.In the Fig. 8, the red star represents the endothermic tower, and the green dots represent the heliostats.

Fig. 8 Spatial distribution of heliostats
Then, the visualization of irradiance.After the radiation intensity is visualized, it is easier to observe the change of sunlight intensity with time, which can further verify the results of thermal power output obtained from the efficiency model.

Fig. 9 Visualization of irradiance
The cosine efficiency, shadow occlusion efficiency, atmospheric transmissivity, and collector intercept efficiency can be determined utilizing the efficiency model.The average efficiency and output power on the 21st of each month are shown in Table 1.Based on the provided data, it is noticeable that that the four efficiencies fluctuate slightly over the course of the year.This is followed by the average truncation efficiency and the average cosine efficiency, while the average optical efficiency is the lowest.The optical efficiencies of May, June and July are at relatively high levels and are at lower values in November, December, and January in the whole year.The thermal output power per Unit mirror area of heliostats also shows a similar pattern of highs and lows in these months.
Then, the calculation of the annual average efficiency, annual average output thermal power, and Annual Average thermal output power per unit mirror area is performed.
The annual average indicators can then be computed from these monthly average values, providing a comprehensive measure yearly average indicators of the heliostat.The annual average efficiency and output thermal power reflects the characteristics exhibited by 21 st day in each month.The result is consisting with the theoretical predictions.
Table 2 The annual average efficiency and thermal output power Annual    Based on above figures, performance evaluation of the circular heliostat field can be effectively conducted by examining its optical efficiency.On March 21st, at 10:30, the highest optical efficiency is observed in the northeast heliostats.By 13:30, this peak efficiency shifts to the northwest heliostats.This pattern is primarily due to the distribution of cosine efficiency, which exhibits the most remarkable variations among all efficiencies of the heliostat field.
This consists with theoretical model predictions.Cosine efficiency is inversely proportional to the solar incidence angle from equation 11.In the Northern Hemisphere, the sun's east-to-west trajectory across the southern sky results in northern heliostats facing the sun more directly, thereby reducing the solar incidence angle and increasing cosine efficiency.Specifically, in the morning when the sun is in the southeast, the northwest heliostats achieve higher cosine efficiency.Conversely, in the afternoon, the sun is positioned in the southwest, the northeast heliostats exhibiting higher efficiency.These dynamics underscore the importance of solar incidence angle in determining cosine efficiency.

Scenario 2
Selecting appropriate values for the parameters in the PSO model is of great importance.As the value of  increases, the global optimization capability strengthens, albeit at the expense of local optimization capability.Thus      are chosen based on previous research of linearly varying inertia weight (LVIW), which are 0.9 and 0.4.Some typical values of cognitive and social acceleration coefficients  1 and  2 are tried, which are  1 =  2 = 2,  1 = 1.6,  2 = 1.8,  1 = 1.6,  2 = 2 to see which may produce best results.The maximum iteration limit is established at 60.The typical iteration process of the PSO is illustrated in the Fig. 20.Then the 3 different results of optimal design parameters for the heliostat field are derived.They are presented in Table 3.
Table 3 The optimal design parameters under different  1 and  2 .
No From Table 3, irrespective of the variations in the parameters  1 and  2 ., the endothermic tower consistently maintains its central position within the heliostat field.This observation aligns with real-world scenarios, where the majority of global heliostat fields are designed with the endothermic tower at their core.Furthermore, the installation heights across all results exhibit a remarkable uniformity.To maximize thermal output power per unit mirror area of the heliostat, smaller total area is preferred.This can reflect in derivation process.For further clarity, the detailed derivation process corresponding to the result with the smallest total area, specifically Result No. 1, is shown in Fig. 20.The "Function value" denotes the local optimal thermal output power per unit mirror area of the heliostat, while the 'Best Function value' indicates the global optimal.Then, the annual average thermal output power per unit mirror area of heliostat can be determined from the PSO model.They are (0.6413,0.6410,0.6030)/ 2 .Thus, to achieve t annual average thermal output power per unit mirror area of heliostat, the NO. 1 result, where  1 =  2 = 2 , is the optimal scheme of the heliostat field.when  1 =  2 = 2 the average efficiency and output power on the 21st of each month are presented in Table 4.  4, the annual averge efficiencies and output power are calculated and shown in the Table 5.
Table 5 The annual average efficiency and output thermal power when  1 =  2 = 2 Annual The layouts of the optimal heliostat field are presented in the Fig. 21.In Fig. 21, the heliostats located in the southern region generally exhibit superior optical efficiency compared to their counterparts in other directions.This observation implies a significant disparity in the optical efficiency across different sections of the heliostat field, thereby suggesting that a circular layout may not suffice to maximize the thermal output power.Thus, layouts that more commonly seen in the real world is further explored in the scenario 3.

Fig. 21
Optimal heliostat layout and optical efficiency distribution

Scenario 3
The analysis of scenario 3 is based on a kind of circular heliostat field in the reality called 6282-heliostat field.It contains 6282 heliostats, and there is passage in the south of endothermic tower.The distribution and partition of the heliostats is of this heliostat field is presented in the Fig. 22.

Fig. 22
The distribution and partition of the heliostats The partition of heliostat field depends on the distribution density of heliostats.From Fig. 22, there are seven ring-shape subregions of the heliostat field.And they are numbered 1-7 from inside to outside.Assuming the subregions are independent and has no effect on each other.The dimensions of the heliostats in the 6282-heliostat field are 4m long and 3.2m wide or 6m long and 4.8m wide.The parameters of each subregion can be calculated by considering the heliostat as a sphere with a diagonal diameter.For instance, the result of the heliostat with the dimension of 4 × 3.2 (from inside to outside) is shown in Table 6.Given that the radius of the heliostat field earmarked for optimization measures 350m, , the F and G regions are not considered.
Then layouts of heliostat field with different heliostat size can be generated based on above process.There are 2 kinds of heliostat field meets the requirements of scenario 3 are shown in the Figures below.The Fig. 25 indicates that the annual average output thermal power in each subregion increases with the increase of heliostat size, but suddenly decreases at some points.This is caused by the change of the subregion with the increasing heliostat size.At these points, the energy increased by the heliostat size is less than the energy lost by reducing a subregion of heliostat field.Additionally, except for the seventh region, the annual average output thermal power in other subregions increases with the increase of subregion number when the heliostat size is the same.This shows that in this certain heliosphere field, the heliosphere in the outer subregion can generate more thermal power.The decrease of annual average output thermal power in seventh subregion is mainly because the heliostat is too far away from the heat collector.Thus, to generate more thermal output power, a larger size heliostat should be selected, and arranging in the outer subregion (but not outmost) of the heliostat field.
The annual average output thermal power per unit mirror area in each subregion, this is related to the total area of the heliostats in the subregion and the annual average output thermal power.The inner subregion has smaller total area, while the outer subregion has larger total area.Then, do the simulation by changing the parameters of the QMCRT model.It is concluded that the inner subregions have higher annual average output thermal power per unit mirror area.This shows that for the heliostat closer to the heat collector, there is a decrease in energy loss and an enhancement in efficiency.Then the cost is further considered.In the simulation, there are two categories of expenses: the cost of construction and maintenance.Cost ratio is defined as proportion of the cost of subregion relative to the overall cost of the heliostat field.From the simulation, in each subregion, the total energy of mirror field per unit cost increases first and then decreases.Cost ratio is the Combing cost ratio of each subregion, the optimal size (Length*width in m) and installation height of heliostat field are shown in the Table 7.The variation trend of annual average output thermal power of each subregion is similar to result obtained in Fig. 25.This indicates that simulation method of QMCRT model is often able to provide highly accurate simulation data, reflecting the light in the heliostat field more realistically.And normal disturbances can be roughly ignored under most situations since the deviations between results in Table 8 and 26,27 Bar plots of the sensitivity for PSO and QMCRT model Thus, the annual average thermal output power is generally insensitive to changes in key design parameters.Therefore, the PSO and QMCRT model adopted in this paper has strong stability.However, both two models exhibit the relative high sensitivity to the width mirror width.This needs further improvement.

Conclusion
This paper proposes an efficiency model grounded in geometrical optics.Itis used to calculate the annual average optical efficiency and thermal output power under specific design parameters of a heliostat field.By combining the efficiency model with a PSO model, we successfully determine the optimal design parameters that maximize the annual average output thermal power per unit mirror area of the heliostat field while ensuring the rated power, effectively improving the efficiency and performance of the heliostat field.Furthermore, simulations are conducted using the QMCRT model to explore methods for obtaining the optimal design parameters of the heliostat field that maximize the annual average output thermal power and ensure high efficiency, and the annual average output thermal power under different normal disturbances is obtained.In this study, due to space constraints, we focus solely on scenarios featuring the most typical arrangement of the heliostat field (6282-heliostats) for the case study,.Future work will encompass an exploration of additional heliostat field layouts.Additionally, while some idealizations in this paper reduce computational complexity and make the results representative, they also introduce some computational errors.Therefore, researchers should consider the correlation between these idealizations and the accuracy of simulation future work.Another direction for future work is trying use different optimization algorithms, such as genetic algorithms and simulated annealing algorithm, and compare them with PSO to determine the better choice.This will make the results more convincing and reliable.Finally, both PSO and QMCRT model can be further improved to reduce their sensitivity to the key design parameters, increasing their stability.

FirstlyFig. 6
Fig. 6 Visualization of solar azimuth angle utilizing the model, the efficiency visualization of the heliostat can be achieved.The optical efficiency and its components are shown in the figures below.

Fig. 20
Fig. 20 Example iteration process of PSO (where  1 =  2 = 2)The "Function value" denotes the local optimal thermal output power per unit mirror area of the heliostat, while the 'Best Function value' indicates the global optimal.Then, the annual average thermal output power per unit mirror area of heliostat can be determined from the PSO model.They are (0.6413,0.6410,0.6030)/ 2 .Thus,

Fig. 25
Fig. 25 Annual average thermal output power (W) of each subregion theoretical results in Fig. 25 are small.4Testing of the modelThe Sensitivity analysis is performed to evaluate the effects of three key design parameters on the annual average thermal output power obtained in the PSO and QMCRT model.Three important parameters are installation height of the heliostat, mirror width and mirror length of the heliostat.The sensitivity is characterized as al., 2021).Where  represents output of the model, parameter  represents the input of the model.Here,  is annual average thermal output power  represents installation height, mirror width and mirror length, that is 3 key design parameters.For the PSO and QMCRT model, the results are shown in the tables and bar plots below.
Then, elevation Angle   and azimuth Angle   of the heliostat can be obtained from:  2 ⋅  2 (  ) − 2 (  ) ⋅  ⋅ ( , ⋅ (  ) −  , ⋅ (  )) Then reaching the ( , ,  , ).It is next original intersection point of the line and another grid.And we should reach one that is smaller in  , and  , .

Table 1
The average efficiency and output power on the 21 st day in each month

Table 4
The average efficiency and output power on the 21 st day in each month when  1 =  2 = 2

Table 7
Optimal heliostat size (m × m) at different cost ratios annual average output thermal power for each subregion is presented in the Table8.

Table 8
Annual average output power (W) in each subregion at different normal

Table 9
Sensitivity of key design parameters of the PSO model