Mathematical Model for Economic Optimization of Tower-Type Solar Thermal Power Generation Systems via Coupled Monte Carlo Ray-Tracing and Multi-Mechanism Heat Loss Equations

Abstract

With the global energy transition and decarbonization goals, tower-type solar thermal power generation is increasingly important for dispatchable clean energy due to its high efficiency, thermal storage capacity, and regulation performance. However, current research focuses on ideal conditions, ignoring real geographical constraints on heliostat layout and environmental impacts on receiver performance. More practical scene modeling and performance evaluation methods are urgently needed. To address these issues, we propose a heliostat field simulation algorithm based on heat loss mechanisms and real site characteristics. The algorithm includes optical performance evaluation (cosine efficiency, shading, truncation, atmospheric transmittance) and heat loss mechanisms (radiation, convection, conduction) for realistic net heat output estimation. Experiments revealed the following: (1) higher central towers improve optical efficiency by increasing solar elevation angle; (2) radiation losses dominate at high power and tower height, while convection losses dominate at low power and tower height. Using the Economic-Integrated Score (EIS) optimization algorithm, we achieved optimal tower and receiver configuration with 40.22% average improvement over other configurations (maximum 3.9× improvement). This provides a scientific design basis for improving tower-type solar thermal systems’ adaptability and economy in different geographical environments.

1. Introduction

During the critical period of climate change and energy transition, developing power systems dominated by renewable energy is essential to achieve “carbon peak” and “carbon neutrality”. Among renewable sources, solar energy attracts attention for its inexhaustibility, renewability, and environmental friendliness. Tower-type solar thermal power generation has emerged as a key dispatchable technology due to its high efficiency, high-temperature storage, grid regulation capability, and low emissions. Its core component, the heliostat system, converts solar energy into electricity by tracking the sun with a large mirror array and reflecting radiation to the collector atop the central tower.

Equatorial regions, while characterized by tropical rainforest climates with high humidity and persistent cloud cover—which reduce heliostat operating hours and increase construction and maintenance costs—offer unique advantages. High humidity and corrosive conditions affect heliostat structures, control systems, and foundations, compromising reliability and service life. Nevertheless, the consistently large solar elevation angle, minimal seasonal variation, and stable, high irradiance provide abundant solar resources, making equatorial areas strategically favorable for high-performance, low-volatility tower-type systems. For instance, in Nigeria and Kenya [1], heliostat fields exploit equatorial solar conditions to achieve higher annual irradiance capture and lower cosine losses, thereby enhancing optical efficiency. However, topographical constraints often limit heliostat placement, emphasizing the need to optimize tower and receiver layouts to maximize system efficiency.

Existing research on tower-type solar thermal power generation systems primarily focuses on optical design, tracking accuracy, and heliostat field layouts. Ref. [2] compared radial staggered and Fermat spiral configurations, showing that the latter improves annual optical efficiency, reduces heliostat count, and enhances overall performance, but only at the optical level, neglecting environmental and thermal factors. The Campo framework [3] optimizes radial staggered layouts for different tower heights and solar elevation angles, improving optical efficiency, yet omits coupled thermal and economic analyses. Ref. [4] developed an optical efficiency model using flat-panel projection and Monte Carlo simulation, combined with a simulated gravity search algorithm to optimize heliostat size and arrangement for maximum annual thermal power per mirror area; however, multi-mechanism heat losses (radiative, convective, conductive) and economic factors are not considered. Similarly, in optical simulations, traditional tools such as SolTrace [5] are typically based on ideal geometric optics, focusing on accuracy but neglecting detailed heliostat attitude modeling. Meanwhile, Ref. [6] proposed an improved Monte Carlo ray-tracing method to enhance accuracy and efficiency, but it does not consider heliostat attitude variations.

Refs. [7,8] examined the impact of key design parameters on central tower system performance. Ref. [7] focused on receivers, energy storage, and operating parameters to enhance efficiency, while Ref. [8] used a multi-region model to assess tower height and heliostat field layout effects on energy transfer and overall efficiency. Despite providing valuable insights, neither study addresses how tower parameter variations influence heat losses (radiative, convective, conductive) or the economic performance of heliostat fields, limiting their guidance for practical engineering optimization.

Existing studies on the economic performance of heliostat fields largely rely on idealized assumptions. Systematic analyses that integrate heat losses, environmental factors, climate- and terrain-adaptive design, operation and maintenance cost control, and construction feasibility remain scarce, revealing clear research gaps. In practice, these factors constitute key constraints that limit the efficient operation and large-scale deployment of heliostat fields.

To address these issues, this study developed a Monte Carlo-based ray-tracing optical model that accurately reproduces the field optical structure via heliostat attitude control and optimizes the receiver coefficient within the constrained heliostat field layout to maximize economic benefits. Multiple heat losses during solar-to-thermal energy conversion are incorporated to enhance model realism. Finally, a comprehensive cost model is established to optimize the overall benefits of the heliostat field. The specific research methods and implementation are detailed as follows.

First, using solar geometric optics and given geographic coordinates (latitude and longitude) at a specified time, the solar azimuth angle is derived from the relationships among solar elevation, hour, and declination angles, and the cosine loss rate is introduced to quantify energy loss due to non-normal incidence.

To capture heliostat attitude changes in a three-dimensional space, a spatial transformation model employing Euler angles is established, incorporating vertical-axis rotation and horizontal-axis pitch adjustment to determine mirror positions. Monte Carlo sampling is then used to simulate sunlight reflection paths, evaluate two types of shadow occlusion, and compute their efficiencies. Occlusion effects from the central tower are also considered based on solar azimuth and elevation angles to assess their impact on optical efficiency.

Truncation efficiency is calculated by mapping the reflected energy and effective cone angle onto the receiver surface using a Gaussian energy density function in cylindrical coordinates, accounting for sunlight-receiver alignment, receiver curvature, and spatial energy distribution. Atmospheric transmittance is incorporated to reflect absorption and scattering effects.

Radiative, convective, and conductive heat losses, along with environmental factors, are integrated to determine actual power generation efficiency. By simulating heliostat attitude, optical performance, and heat loss mechanisms, coupled with environmental variables, the true field efficiency is obtained. Based on this, an Economic-Integrated Score (EIS) model combining ROI, optical performance, maintenance costs, and heat losses is proposed to guide optimization for long-term economic benefits.

In summary, a comprehensive framework is established to evaluate the economic performance of heliostat fields under realistic conditions, addressing the limitation that prior studies rarely consider in practical scenarios. This study systematically investigates the following key research questions: (1) How do heliostat field geometry, tower shading, and beam spillage jointly determine overall optical efficiency? (2) How do tower height and receiver size interact to influence average optical efficiency and power density? (3) Under which conditions do radiative or convective heat losses dominate? (4) How can the EIS metric guide the optimal configuration of a tower-based concentrating solar power system?

Through experiments, the following key findings were obtained: (1) Geometric analyses using Euler angles, shadow occlusion, and truncation identified three strategies to enhance heliostat optical efficiency—radial–azimuthal layout optimization, tower dead-zone setting, and mirror–receiver dimension matching—constraining total geometric losses below 15%. (2) From 200 full-factorial experiments, tower height H emerged as the primary factor improving both average optical efficiency ${\overline{\eta }}_{\mathrm{opt}}$ and power density $P_{\mathrm{area}}$, yielding the Pareto-optimal configuration $(H,D,h)=(120\mathrm{m},10\mathrm{m},7\mathrm{m})$ with relative gains of 8.7% in ${\overline{\eta }}_{\mathrm{opt}}$ and 23.4% in $P_{\mathrm{area}}$ over the sample mean. (3) Analysis of 19,200 operating conditions revealed a switch in dominant heat loss mechanisms: radiation dominates when $P_{\mathrm{area}}>150–200\mathrm{kW}/{\mathrm{m}}^2$ or $H>100\mathrm{m}$, while convection dominates when $P_{\mathrm{area}}<150\mathrm{kW}/{\mathrm{m}}^2$ and $H<80\mathrm{m}$, providing clear guidelines for thermal management. (4) Single-objective optimization over 252 configurations established the EIS model and identified an “economic sweet spot” ($H = 70–90\mathrm{m}$, $D = 7–9\mathrm{m}$, $h = 6–8\mathrm{m}$), with the optimal configuration $(80\mathrm{m}, 8\mathrm{m}, 7\mathrm{m})$ achieving net power $21.4$ $\mathrm{M}$$\mathrm{W}$, ROI $8.4$%, and thermal efficiency $51.6$%, representing a 40.22% improvement over the average and up to 3.9× over the worst configuration, providing robust guidance for design and decision-making in tower-type solar thermal power systems.

2. Methods and Materials

Overview. As shown in Figure 1, this study proposes an integrated modeling and optimization framework for tower-type solar thermal power generation systems, aiming to maximize economic benefits under fixed heliostat field layout and scale. First, heliostat optical efficiency and receiver size constraints are determined by calculating shadowing, occlusion, and other optical losses through Euler angle transformations. Based on these results, a thermal resistance network is constructed to evaluate radiative, convective, and conductive heat losses with environmental corrections, thereby obtaining the system’s core energy. Finally, an Economic-Integrated Score (EIS) is developed by combining absorbed energy, return on investment, costs, and heat losses, and is optimized to achieve maximum economic performance.

In the following, we introduce each part of our framework one by one.

2.1. Modeling of Heliostat Field

The reflection of heliostats to the central tower is shown in Figure 2.

The calculation of the heliostat optical efficiency $\eta$ needs to consider five parameters: the optical loss rate of the heliostat ${\eta }_{ls}$, the cosine loss rate ${\eta }_{cos}$, the atmospheric transmittance ${\eta }_{nat}$, the receiver truncation efficiency ${\eta }_{trunc}$, and the mirror reflectivity ${\eta }_{ref}$. The specific mathematical expression is as follows:

$$\eta ={\eta }_{ls}{\eta }_{nat}{\eta }_{cos}{\eta }_{trunc}{\eta }_{ref}$$

The measurement of optical efficiency requires variables related to the solar azimuth angle and the latitude and longitude at the heliostat plane. Based on [9], the solar elevation angle and solar azimuth angle can be obtained as follows:

Solar elevation angle:

$${\alpha }_s=cos\delta cos\phi cos\theta +sin\delta sin\theta$$

(1)

Solar azimuth angle:

$$cos{\gamma }_s={\displaystyle \frac{sin\delta -sin{\alpha }_{s}sin\phi }{cos{\alpha }_{s}cos\phi }}$$

(2)

Solar hour angle:

$$\omega ={\displaystyle \frac{\pi }{12}}\left(\mathrm{ST}-12\right)$$

Solar declination angle:

$$sin\delta =sin{\displaystyle \frac{2\pi D}{365}}sin\left({\displaystyle \frac{2\pi }{360}}23.45\right)$$

Among them, $\delta$ represents the local latitude (with north latitude being positive), $ST$ represents the local time, and D represents the number of days counted from the vernal equinox as day 0.

As shown in Figure 3, the variation of the solar azimuth angle ${\gamma }_s$ over a centennial period can be approximated as an arc shape. The tangent method is used to calculate the complementary angle of the solar azimuth angle, enabling the spatial rotation of the heliostat to track the reflected beam of the sun.

That is, rotating counterclockwise by an angle ${\gamma }_s$ along the horizon (x-axis) toward the north (z-axis) establishes a mathematical model. In the first quadrant, the expression for the complementary angle ${\gamma }_s^{\prime }$ of the solar azimuth angle is as follows:

$${\gamma }_s^{\prime }={\displaystyle \frac{\pi }{2}}-{\gamma }_s$$

The normal direct irradiation $DNI$ (unit: kW/m²) refers to the solar radiation energy received per unit area and unit time on a plane perpendicular to the solar rays on the earth, which can be calculated by the following formula [10]:

$$DNI=G_0\left[a+bexp\left(-{\displaystyle \frac{c}{sin{\alpha }_s}}\right)\right]$$

Here,

$$a=0.4237-0.00821(6-H),$$

$$b=0.5055+0.00595{(6.5-H)}^2,$$

$$c=0.2711+0.01858{(2.5-H)}^2,$$

$G_0$ is the solar constant, with a value of $1.366{\mathrm{kW}/\mathrm{m}}^2$, and H is the altitude (unit: km).

The output power $P_{collected}$ of the heliostat field is

$$P_{collected}=DNI·\sum _{i=1}^N{A}_i{\eta }_i,$$

Among them, $DNI$ is the normal direct irradiation, N is the total number of heliostats (unit: piece), $A_i$ is the lighting area of the i heliostat, and ${\eta }_i$ is the efficiency of the i heliostat.

To analyze light reflection, the geometric relationship between the incident solar rays and the heliostat normal vector is used to determine the reflected direction $\overrightarrow{r}$. Combined with scene occlusion assessment, two types of occlusion are considered—heliostat–heliostat and absorber tower occlusion—with their respective contribution ratios calculated.

$${\eta }_{\mathrm{sb}}={\displaystyle \frac{n_{\mathrm{unblocked}}}{N}}.$$

For the mirror energy reception, the collector surface is modeled as curved. By mapping the cylindrical absorber tower onto a two-dimensional parameter space $(\theta ,z)$, the elliptical projection of the reflected light spot is derived, and a Gaussian energy density function is introduced to represent its spatial energy distribution.

$$E(\theta ,z)=E_{0}exp\left[-{\displaystyle \frac{{(\theta -{\theta }_0)}^2}{2{\sigma }_{\theta }^2}}-{\displaystyle \frac{{(z-z_0)}^2}{2{\sigma }_z^2}}\right].$$

Subsequently, the total incident energy $E_{\mathrm{total}}$ and the effective received energy $E_{\mathrm{overlap}}$ are calculated. The energy of the light spot overlapping area is obtained by the numerical integration method and, finally, the truncation efficiency is obtained.

$${\eta }_{\mathrm{trunc}}={\displaystyle \frac{E_{\mathrm{overlap}}}{E_{\mathrm{total}}}}.$$

In addition, the modeling also considers the cosine loss effect, that is, the problem of reduced effective reflected energy caused by the change in the incident angle. We quantify this effect by the cosine of the angle between the incident vector $\overrightarrow{i}$ and the reflection direction vector $\overrightarrow{e}$,

$${\eta }_{cos}={\displaystyle \frac{(\overrightarrow{i}+\overrightarrow{e})·\overrightarrow{i}}{\parallel \overrightarrow{i}+\overrightarrow{e}\parallel }},$$

and calculate the annual average cosine efficiency based on the values at multiple time points.

Atmospheric effects on solar transmission are incorporated using a layered extinction model, accounting for clouds and aerosols. Solar attenuation is expressed via the total optical thickness ${\tau }_{\mathrm{total}}$, yielding the corrected atmospheric transmittance ${\eta }_{\mathrm{at}}$.

Heliostat mirror reflectivity is derived from the complex refractive index and Fresnel equations, with the effective reflectivity ${\eta }_{\mathrm{refl}}$ obtained by averaging the s and p polarization components, forming a key component of the overall optical model.

In summary, this section establishes a comprehensive heliostat field optical model, integrating spatial rotation, ray-tracing simulation, curved surface mapping, and energy loss evaluation, providing a theoretical foundation for subsequent efficiency optimization.

2.1.1. Euler Angles

Prior to overall modeling, the coordinates of the four heliostat vertices in the field coordinate system must be determined. The mirror can rotate around the vertical axis (z-axis, angle ${\theta }_1$) and horizontal axis (x-axis, elevation angle ${\theta }_2$). Using the Euler angle transformation method [11], the spatial coordinates of the four vertices after these composite rotations are calculated.

Initially, the vertices relative to the mirror center are $(x,y,0)$, $(x,-y,0)$, $(-x,y,0)$, and $(-x,-y,0)$. After rotation, their coordinates relative to the heliostat center are $N_i^{\prime }$, with absolute coordinates $N_i$.

First, considering rotation around the horizontal axis (x-axis) by the elevation angle ${\theta }_2$, the vertex coordinate along the x-direction remains invariant due to rigid-body rotation characteristics.

$$x^{\prime }=x$$

Next, we analyze the variation of the vertex coordinates relative to the mirror center along the y- and z-axes after rotation about the x-axis. Let the initial coordinate vector of a vertex relative to the center in the y-z plane be $\overrightarrow{v}$, expressed as

$$\overrightarrow{v}=\left(\begin{array}{c}y\\ z\end{array}\right)$$

To facilitate vector rotation analysis, the coordinate vector is transformed into polar form. Using the standard polar-to-Cartesian conversion, we have

$$\left\{\begin{array}{c}y=r·cos\beta ,\hfill \\ z=r·sin\beta ,\hfill \end{array}\right.$$

(3)

where $r=\sqrt{z^2+y^2}$ and $\beta$ represents the angle from the X-axis to the vector $\overrightarrow{v}$.

When the vector $\overrightarrow{v}$ is rotated by ${\theta }_2$ around the origin, according to the coordinate transformation principle of vector rotation in a two-dimensional plane, the polar coordinate form of the rotated vector $\overrightarrow{v}$ is

$$\left\{\begin{array}{c}{y}^{\prime }=r·cos(\beta +{\theta }_2),\hfill \\ z^{\prime }=r·sin(\beta +{\theta }_2).\hfill \end{array}\right.$$

(4)

Using the trigonometric sum-angle formula to expand Equation (4), we obtain

$$\left\{\begin{array}{c}{y}^{\prime }=r\left(cos\beta ·cos{\theta }_2-sin\beta ·sin{\theta }_2\right),\hfill \\ z^{\prime }=r\left(sin\beta ·cos{\theta }_2+cos\beta ·sin{\theta }_2\right).\hfill \end{array}\right.$$

(5)

Substituting Equation (3) into Equation (5) and rearranging, we obtain

$$\left\{\begin{array}{c}{y}^{\prime }=y·cos{\theta }_2-z·sin{\theta }_2,\hfill \\ z^{\prime }=y·sin{\theta }_2+z·cos{\theta }_2.\hfill \end{array}\right.$$

and

$$\left({\displaystyle \genfrac{}{}{0pt}{}{y^{\prime }}{z^{\prime }}}\right)=\left(\begin{array}{cc}cos{\theta }_2& -sin{\theta }_2\\ sin{\theta }_2& cos{\theta }_2\end{array}\right)\left(\begin{array}{c}y\\ z\end{array}\right)$$

Integrating the coordinate changes along the x-, y-, and z-axes yields the complete transformation of the vertex coordinates after rotation of the mirror about the x-axis:

$$\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& cos{\theta }_2& -sin{\theta }_2\\ 0& sin{\theta }_2& cos{\theta }_2\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$$

Similarly, applying the same coordinate transformation theory, the vertex coordinates after rotation of the mirror about the longitudinal (y) axis by angle ${\theta }_1$ are given by

$$\left(\begin{array}{c}{x}^{″}\\ y^{″}\\ z^{″}\end{array}\right)=\left(\begin{array}{ccc}cos{\theta }_1& -sin{\theta }_1& 0\\ sin{\theta }_1& cos{\theta }_1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right)$$

Substituting the mirror’s initial vertex coordinates relative to its center into the above transformation formulas yields the vertices’ relative coordinates after rotation:

$$\begin{array}{cc}\hfill N^{\prime }& =\left(b,ccos{\theta }_2,-csin{\theta }_2\right)\left(\begin{array}{ccc}cos{\theta }_1& -sin{\theta }_1& 0\\ sin{\theta }_1& cos{\theta }_1& 0\\ 0& 0& 1\end{array}\right)\hfill \\ \hfill & ={\left(\begin{array}{c}bcos{\theta }_1+csin{\theta }_{1}cos{\theta }_2\\ -bsin{\theta }_1+ccos{\theta }_{1}cos{\theta }_2\\ -csin{\theta }_2\end{array}\right)}^T\hfill \end{array}$$

(6)

Finally, by incorporating the spatial coordinates of the mirror surface center, $(x_0, y_0, z_0)$, and applying coordinate translation operations, the actual spatial coordinates of the vertices can be accurately determined.

$$N={\left(\begin{array}{c}{x}_0+bcos{\theta }_1+csin{\theta }_{1}cos{\theta }_2\\ y_0-bsin{\theta }_1+ccos{\theta }_{1}cos{\theta }_2\\ z_0-csin{\theta }_2\end{array}\right)}^T$$

(7)

After obtaining the precise coordinates of the mirror vertices, we proceed to formally calculate the shadow occlusion efficiency within the heliostat field.

2.1.2. Annual Average Shadow Occlusion Efficiency

The shadow occlusion efficiency (${\eta }_{\mathrm{sb}}$) of a heliostat is computed via Monte Carlo simulation. Specifically, 1000 solar conical rays are randomly generated on the mirror surface, with initial coordinates $(b_i, c_i, 0)$ relative to the mirror center. Two successive Euler angle rotations map these points to their actual spatial positions in the heliostat field coordinate system. Using the rotated mirror vertices, the normal vector $\overrightarrow{n}$ is determined, enabling calculation of the reflected ray directions $\overrightarrow{r}$. Occlusion probabilities are estimated by checking whether the reflected rays are blocked by surrounding heliostats.

As shown in Figure 4, two heliostat-to-heliostat occlusion types are considered. In the first, sunlight from Heliostat 1 toward Heliostat 2 is partially blocked by Heliostat 1. In the second, a portion of the reflected rays from Heliostat 2 is obstructed by Heliostat 1, both reducing the system’s optical efficiency.

Furthermore, with the central tower centrally located in the heliostat field, only heliostats nearer to the tower (Heliostat 1) can shadow those farther away (Heliostat 2). Accordingly, the following filtering condition is introduced:

$$\mathrm{distance}\left[k\right]<\mathrm{distance}\left[i\right]$$

Here, i represents the selected heliostat, k denotes any nearby heliostat, and $distance\left[i\right]$ indicates the distance from the center of the i-th heliostat to the central tower.

The minimum occlusion distance between two heliostats is also relevant. To reduce computational time and memory usage, a simplified approach is adopted with minimal loss of accuracy. Specifically, all inter-heliostat distances are filtered to select the maximum value, MAX, followed by a criterion based on the heliostat width W, as illustrated in Figure 5.

$$\sqrt{{(x_i-x_k)}^2+{(y_i-y_k)}^2}<MAX+W$$

The shadow occlusion efficiency only occurs when the distance between two heliostats falls within the aforementioned range.

On this basis, the heliostat attitude is modeled by combining Euler Angles, and then the shadow occlusion efficiency is calculated.

Solution to Shadow Occlusion Efficiency of Type I Heliostats

First, using the three vertices $N_1$, $N_2$, and $N_3$ of the heliostat, the normal vector $\overrightarrow{n}$ of the heliostat plane is determined based on two non-parallel edges $\overrightarrow{N_1{N}_2}$ and $\overrightarrow{N_1{N}_3}$ of the heliostat plane:

$$\overrightarrow{N_1{N}_2}=(a_2^{\prime }-a_1^{\prime },b_2^{\prime }-b_1^{\prime },c_2^{\prime }-c_1^{\prime })$$

$$\overrightarrow{N_1{N}_3}=(a_3^{\prime }-a_1^{\prime },b_3^{\prime }-b_1^{\prime },c_3^{\prime }-c_1^{\prime })$$

$$\overrightarrow{N_1{N}_2}\times \overrightarrow{N_1{N}_3}=\overrightarrow{n}=(A^{\prime },B^{\prime },C^{\prime })$$

Subsequently, the analytical expression of the heliostat plane is determined based on the coordinates $(a^{\prime }, b^{\prime }, c^{\prime })$ of a vertex on the heliostat and the normal vector of the plane it lies on:

$$D^{\prime }=-A^{\prime }{x}^{\prime }-B^{\prime }{y}^{\prime }-C^{\prime }{z}^{\prime }$$

(8)

According to the reference of Du Yuhang et al. [12], the solar elevation angle ${\alpha }_s$ ranges from $0^{°}$ to ${90}^{°}$. When the z-coordinate is positive (indicating that the vector points to the space above the origin), the incident direction vector $\overrightarrow{i}$ of sunlight can be calculated as follows:

$$\overrightarrow{i}=(cos{\alpha }_{s}cos{\gamma }_s^{\prime },cos{\alpha }_{s}sin{\gamma }_s^{\prime },sin{\alpha }_s)$$

Together with the randomly selected point $A^{″}(a_0,b_0,c_0)$, the spatial line equation of the incident light ray is determined.

$$P\left(t\right)=A^{″}+t·i$$

To find the intersection point of this line and the plane, $P\left(t\right)$ is substituted into the plane Equation (8):

$$A^{\prime }(a_0+t{i}_x)+B^{\prime }(b_0+t{i}_y)+C^{\prime }(c_0+t{i}_z)+D^{\prime }=0$$

Solving this equation can yield the position of the intersection point:

$$\begin{array}{cc}\hfill t& ={\displaystyle \frac{(N_1-A^{″})·\overrightarrow{n}}{\overrightarrow{t}·\overrightarrow{n}}}\hfill \\ \hfill P& =A^{″}+t·\overrightarrow{t}\hfill \end{array}$$

Finally, by determining whether the intersection coordinates $P_x$ and $P_z$ fall within the boundary extreme value interval, it is determined whether the light ray is occluded.

The proportion of random points corresponding to solar incident light rays not blocked by heliostat 1 among 1000 random points is the heliostat shading efficiency ${\eta }_{\mathrm{sb}}$:

$${\eta }_{sb1}={\displaystyle \frac{nu{m}_1}{1000}}$$

Solution to Shadow Occlusion Efficiency of Type II Heliostats

As illustrated in the second scenario of Figure 4, the reflected sunlight direction vector $\overrightarrow{r}$ defines the straight-line path. Intersection points of $\overrightarrow{r}$ within the mirror area of Heliostat 1 indicate the randomly generated rays occluded after reflection. The shadow occlusion efficiency for this second scenario is computed similarly to the first, with $\overrightarrow{r}$ parallel to the opposite of the incident ray, $-\overrightarrow{i}$. Efficiency is then obtained by counting the intersection points within Heliostat 1’s mirror area.

$${\eta }_{sb2}={\displaystyle \frac{nu{m}_2}{1000}}$$

Solution to Shadow Occlusion Efficiency of Heliostats Occluded by Central Tower

Regarding shadowing by the central tower, occlusion occurs only during solar incidence on the heliostat, representing a special case of the first-type shadow occlusion, where some heliostats are blocked by the tower’s shadow.

1. For the intermediate rectangular shadow, the affected region corresponds to the large rectangular area projected along the solar azimuth direction. Following the method for first-type shadow occlusion points, the area influencing the sampled rays can be calculated as

$$\left(\begin{array}{ccc}Rsin{\gamma }_s& Rcos{\gamma }_s& 0\end{array}\right)$$

$$\left(\begin{array}{ccc}-Rsin{\gamma }_s& Rcos{\gamma }_s& 0\end{array}\right)$$

$$\left(\begin{array}{ccc}Rsin{\gamma }_s& Rcos{\gamma }_s& H+h\end{array}\right)$$

$$\left(\begin{array}{ccc}-Rsin{\gamma }_s& Rcos{\gamma }_s& H+h\end{array}\right)$$

where R is the radius of the receiver and the central tower, H is the height of the central tower, and h is the height of the receiver.

The normal vector of the rectangular shadow plane aligns with the solar azimuth direction and can alternatively be obtained via the standard method of computing two perpendicular vectors, analogous to the first-type shadow occlusion calculation. Subsequently, using the solar incident vector and the actual seed coordinates, the intersection with the rectangular plane is determined, and it is checked whether the point lies within the sun-facing region of the central tower.

2. The second part is the circular area at the top of the central tower, as shown in Figure 6.

Through the previous step, the point-direction linear equation formed by the seeds and the current solar direction vector is obtained.

This is intersected with the plane $z=H+h$ and the distance d between the resulting intersection point and the point $(0, 0, H+h)$ is calculated:

If the distance $d<R$ meters, this indicates that the seeds will be affected.

If the distance $d>R$ meters, this means that the seeds will not be affected by the circular area at the top of the central tower at this moment.

Finally, according to the number of unoccluded seeds $nu{m}_3$, the shadow occlusion rate ${\eta }_{sb3}$ of the central tower on the heliostat is calculated, and the final shadow occlusion rate ${\eta }_{sb}$ is solved.

$${\eta }_{sb3}={\displaystyle \frac{nu{m}_3}{1000}}$$

$${\eta }_{sb}={\eta }_{sb1}{\eta }_{sb2}{\eta }_{sb3}$$

2.1.3. Annual Average Truncation Efficiency

During reflection from the heliostats to the receiver, the light beam diverges due to the sun’s inherent angular spread, mirror surface errors, and environmental perturbations. This divergence governs the shape and energy distribution of the sunlight spot on the receiver, forming the fundamental basis for calculating the truncation efficiency. In this section, we first model the beam divergence, then derive the projection boundary of the light spot via cylindrical surface unfolding, and, finally, compute the truncation efficiency using the energy density function.

Effective Cone Angle Calculation

During reflection from the heliostat to the receiver, the light beam undergoes divergence due to the combined effects of the sun’s intrinsic angular spread, heliostat slope errors, tracking inaccuracies, scattering, and atmospheric disturbances. To characterize this collectively, the effective cone angle ${\theta }_{\mathrm{eff}}$ is introduced, representing the root-mean-square superposition of these error sources and providing a quantitative measure of overall beam divergence. Its expression is given by [13]

$${\theta }_{\mathrm{eff}}^2={\theta }_{\mathrm{sun}}^2+{\left(2{\theta }_{\mathrm{slope}}\right)}^2+{\theta }_{\mathrm{track}}^2+{\theta }_{\mathrm{scatter}}^2+{\theta }_{\mathrm{atm}}^2$$

where ${\theta }_{\mathrm{eff}}$ is the effective cone angle; ${\theta }_{\mathrm{sun}}$ is the sun’s inherent angle, with a value of $4.65\times {10}^{-3}\mathrm{rad}$ [14]; ${\theta }_{\mathrm{slope}}$ is the angular error caused by the heliostat surface slope error, ranging from $0.2$ to $0.5\times {10}^{-3}\mathrm{rad}$ [14]; ${\theta }_{\mathrm{track}}$ is the angular error caused by tracking errors, ranging from $0.2$ to $0.5\times {10}^{-3}\mathrm{rad}$ [14]; ${\theta }_{\mathrm{scatter}}$ is the angular error caused by scattering effects, ranging from $0.1$ to $0.3\times {10}^{-3}\mathrm{rad}$ [15]; and ${\theta }_{\mathrm{atm}}$ is the angular error caused by atmospheric disturbances, ranging from $0.1$ to $0.3\times {10}^{-3}\mathrm{rad}$ [16].

Subsequently, the mathematical expressions of the elliptical light spots formed by the reflected rays on the receiver surface for the central and edge regions of the heliostat are derived.

Elliptical Light Spot Calculation for Edge Regions

For off-axis heliostats (such as those in edge regions), due to their angular variation relative to the solar incident plane, beam divergence exhibits anisotropic characteristics. Therefore, it is necessary to decompose the errors into tangential (t, perpendicular to the solar incident plane) and sagittal (s, parallel to the solar incident plane) components and calculate their effective cone angles separately. The specific expressions are as follows:

$${\theta }_{\mathrm{eff},t}^2={\theta }_{\mathrm{sun},t}^2+{\left(2{\theta }_{\mathrm{slope},t}\right)}^2+{\theta }_{\mathrm{track},t}^2+{\theta }_{\mathrm{scatter},t}^2+{\theta }_{\mathrm{atm},t}^2$$

$${\theta }_{\mathrm{eff},s}^2={\theta }_{\mathrm{sun},s}^2+{\left(2{\theta }_{\mathrm{slope},s}\right)}^2+{\theta }_{\mathrm{track},s}^2+{\theta }_{\mathrm{scatter},s}^2+{\theta }_{\mathrm{atm},s}^2$$

where ${\theta }_{\mathrm{sun},t}={\theta }_{\mathrm{sun},s}=4.65\mathrm{mrad}$, indicating the isotropy of the solar angle; the projections of other error variables on the tangential and sagittal components can be calculated through Euler angle spatial transformation, thereby effectively decomposing the three-dimensional errors into the anisotropic directions of the beam.

Subsequently, using the linear distance $d_R$ from the center of the heliostat to the cylindrical surface of the receiver, the major axis a and minor axis b of the elliptical light spot formed by the reflected rays from the edge region on the receiver surface can be calculated, with their expressions given by [17]

$$a=d_R·{\theta }_{\mathrm{eff},t},b=d_R·{\theta }_{\mathrm{eff},s}$$

(9)

where ${\theta }_{\mathrm{eff},t}$ and ${\theta }_{\mathrm{eff},s}$ are the tangential and sagittal effective cone angles, respectively, reflecting the divergence degree of the beam in the anisotropic directions.

Elliptical Light Spot Calculation for Central Regions

The Monte Carlo simulation method is employed to randomly generate N points within a radius $A=d_R·{\theta }_{\mathrm{eff}}$ on the cylindrical receiver surface of radius R. Since the cylindrical surface is a curved structure, the sunlight spot forms an elliptical projection on its surface after irradiation. According to previous studies on the parameters of curved surfaces [14,18,19], the major axis and minor axis can be determined.

In the three-dimensional space, the equation of a circular sunlight spot is as follows (when the light beam is perpendicular to the x-y plane):

$${(x_e-a)}^2+{(y_e-b)}^2+{(z_e-Z_0)}^2=A^2$$

Since the height direction (z-axis) of the cylindrical absorber tower has little effect on the shape of the light spot (only translation, not changing the shape), it can be simplified as a two-dimensional problem, fixing $z_e=Z_0$, and the equation degenerates into

$${(x_e-a)}^2+{(y_e-b)}^2=A^2$$

(10)

Substituting the cylindrical surface parameters $x_e=Rcos\theta$ and $y_e=Rsin\theta$ into the light spot Equation (10) gives

$${(Rcos\theta -a)}^2+{(Rsin\theta -b)}^2=A^2$$

Expanding the equation gives

$$R^2{cos}^2\theta -2aRcos\theta +a^2+R^2{sin}^2\theta -2bRsin\theta +b^2=A^2$$

Using the trigonometric identity ${cos}^2\theta +{sin}^2\theta =1$, simplified, gives

$$R^2-2R(acos\theta +bsin\theta )+a^2+b^2=A^2$$

By rearranging the constant terms and defining $C=a^2+b^2-R^2$, the equation simplifies to

$$R^2-2R(acos\theta +bsin\theta )+C=0$$

(11)

Subsequently, we express the elliptic equation of the light spot in the parametric coordinate system of the cylindrical surface. When the cylindrical surface is unfolded, the relationship between the circumferential direction $\theta$ and the plane abscissa X is

$$X=R\theta \Rightarrow \theta ={\displaystyle \frac{X}{R}}$$

The height direction Z directly corresponds to the vertical coordinate Y in the plane, that is,

$$Y=Z$$

Subsequently, $\theta ={\displaystyle \frac{X}{R}}$ is substituted into Equation (11) and a plane transformation is performed to obtain the equation in the planar coordinates $(X,Y)$:

$$R^2-2R\left(acos{\displaystyle \frac{X}{R}}+bsin{\displaystyle \frac{X}{R}}\right)+C=0$$

(12)

At this point, the equation describes the boundary of the expanded sunlight spot, but, due to the inclusion of trigonometric function terms, further approximation and simplification are required.

When the size of the sunlight spot is much smaller than the circumference of the cylinder, we have

$$\left|{\displaystyle \frac{X}{R}}\right|\ll 1(\mathrm{i}.\mathrm{e}.,X\ll 2\pi R)$$

Therefore, the first-order Taylor expansion can be performed for the trigonometric functions (neglecting terms of the third order and above):

$$cos{\displaystyle \frac{X}{R}}\approx 1-{\displaystyle \frac{{(X/R)}^2}{2!}}=1-{\displaystyle \frac{X^2}{2R^2}}$$

$$sin{\displaystyle \frac{X}{R}}\approx {\displaystyle \frac{X}{R}}-{\displaystyle \frac{{(X/R)}^3}{3!}}\approx {\displaystyle \frac{X}{R}}$$

Substituting the Taylor expansion into Equation (12) gives

$$R^2-2R\left[a\left(1-{\displaystyle \frac{X^2}{2R}}\right)+b\left({\displaystyle \frac{X}{R}}\right)\right]+C=0$$

Expanding it gives

$$R^2-2aR+{\displaystyle \frac{a{X}^2}{R}}-2bX+C=0$$

Since $C=a^2+b^2-A^2$, substituting it gives

$$R^2-2aR\pm 2bX+a^2+b^2-A^2=0$$

Combining the constant terms,

$${(R-a)}^2+b^2-A^2+{\displaystyle \frac{a{X}^2}{R}}-2bX=0$$

Subsequently, coordinate translation is performed by letting

$$X^{\prime }=X+{\displaystyle \frac{bR}{a}}$$

Then,

$$X=X^{\prime }-{\displaystyle \frac{bR}{a}}$$

Substituting gives

$${\displaystyle \frac{a}{R}}{(X^{\prime }-{\displaystyle \frac{bH}{a}})}^2-2b(X^{\prime }-{\displaystyle \frac{bH}{a}})+{(R-a)}^2+b^2-A^2=0$$

Expanding and simplifying the linear terms,

$${\displaystyle \frac{a{X}^{\prime 2}}{R}}-2b{X}^{\prime }+{\displaystyle \frac{b^{2}R}{a}}-2b{X}^{\prime }+{\displaystyle \frac{2b^{2}R}{a}}+(R^2-2aR+a^2)+b^2-A^2=0$$

After combining like terms, the linear terms cancel out, and, finally, we obtain

$${\displaystyle \frac{a{X}^{\prime 2}}{R}}+{\displaystyle \frac{3b^{2}R}{a}}+R^2-2aR+a^2+b^2-A^2=0$$

Since the center of the original sunlight spot satisfies that the distance from the spot center to the cylindrical circumference remains unchanged, the constant term can be further simplified, and the final equation is obtained as

$${\displaystyle \frac{X^{\prime 2}}{{(AR/a)}^2}}+{\displaystyle \frac{Y^2}{R^2}}=1$$

Energy Density Function

Building on this, the effect of the sunlight spot’s energy distribution is incorporated. The energy distribution is commonly approximated by a Gaussian energy density function [20], characterized by a peak at the center that gradually decays toward the edges. In the parameterized coordinate system, the distribution is further refined using the relationship between the full width at half maximum ($FWHM$) and the standard deviation of the Gaussian function [21].

In the parametric coordinate system, the energy density function of an elliptical sunlight spot is

$$E(\theta ,z)=E_{0}exp\left(-{\displaystyle \frac{{(\theta -{\theta }_0)}^2}{2{\sigma }_{\theta }^2}}-{\displaystyle \frac{{(z-z_0)}^2}{2{\sigma }_z^2}}\right)$$

where, for the central region, ${\sigma }_{\theta }={\sigma }_z={\displaystyle \frac{{\theta }_{\mathrm{eff}}}{2.355}}$; for the edge region, ${\sigma }_{\theta }={\displaystyle \frac{{\theta }_{\mathrm{eff},t}}{2.355}}$, ${\sigma }_z={\displaystyle \frac{{\theta }_{\mathrm{eff},s}}{2.355}}$.

Thus, the total radiant energy that the receiver can absorb under the ideal condition where solar radiation completely illuminates the surface of the receiver can be further calculated.

The total energy of the sunlight spot:

$$E_{\mathrm{total}}={\int }_{-\pi }^{\pi }{\int }_{-\infty }^{\infty }E(\theta ,z)·R_{c}d\theta dz$$

By using the substitution method $u=(\theta -{\theta }_0)/{\sigma }_{\theta }=(z-z_0)/{\sigma }_{\theta }$, it can be simplified as

$$E_{\mathrm{total}}=2\pi R_c{E}_0{\sigma }_{\theta }{\sigma }_z\mathrm{erf}\left({\displaystyle \frac{\pi }{{\sigma }_{\theta }}}\right)$$

where $\mathrm{erf}\left(x\right)$ is the error function.

Truncation Energy Calculation

Finally, to determine the total amount of reflected solar energy successfully received by the receiver, we first need to clarify the spatial projection range of the receiver’s rectangle in the parameter coordinate system.

The range of the receiver’s rectangle in the parameter coordinate system:

In the circumferential direction, $\theta \in \left[-arcsin\left({\displaystyle \frac{W/2}{R_c}}\right),arcsin\left({\displaystyle \frac{W/2}{R_c}}\right)\right]$. In the axial direction, $z\in [0,H]$.

When the incident angle relative to the cylindrical receiver surface normal is $\alpha ={90}^{°}$, the light does not effectively interact with the receiver, resulting in no absorption. For $\alpha \in ({90}^{°},{180}^{°})$, although part of the sunlight spot may partially miss the receiver, statistical symmetry in the heliostat field layout allows these rays to be considered fully absorbed. Hence, in the modeling process, light within this angular range is treated as entirely absorbed.

Subsequently, the Monte Carlo simulation method is employed to randomly generate N central coordinates $({\theta }_0,z_0)$ of elliptical sunlight spots in the parameter space. Spatial transformations yield the actual positions of these ellipses on the receiver surface. Considering the spatial distribution of the spots, their overlapping areas and corresponding energy superposition are computed to estimate the total effective absorbed energy under realistic operating conditions.

The overlapping region $\mathsf{\Omega }$ can be decomposed into multiple sub-regions ${\mathsf{\Omega }}_i$, and the parameter range of each sub-region needs to be determined by the intersection coordinates. For example,

$$\mathsf{\Omega }=\bigcup _{i=1}^n\{({\theta }_0,z_0)\mid {\theta }_0\in [{\theta }_{i,min},{\theta }_{i,max}],z_0\in [z_{i,min}\left(\theta \right),z_{i,max}\left(\theta \right)]\}$$

Based on this, the energy of the overlapping region is calculated.

$$E_{\mathrm{overlap}}={\int \int }_{\mathsf{\Omega }}E(\theta ,z)·R_{c}d\theta dz$$

Adopting the numerical integration method,

$$E_{\mathrm{overlap}}\approx R_c\sum _{i=1}^m\sum _{j=1}^{n}E({\theta }_i,z_j)·w_i·v_j·\Delta \theta ·\Delta z$$

where $w_i$ and $v_j$ are Gaussian weights and $\Delta \theta$ and $\Delta z$ are integration step sizes.

Finally, the annual average truncation efficiency is determined as

$$\overline{{\eta }_{\mathrm{trunc}}}={\displaystyle \frac{E_{\mathrm{overlap}}}{E_{\mathrm{total}}}}$$

(13)

2.1.4. Other Losses

Annual Average Cosine Loss Efficiency

As illustrated in Figure 7, if the heliostat rotation refers to the geometric center of the receiver ($K^{\prime }$), the lower portion of the reflected conical beam may miss the receiver, leading to energy loss. To maximize solar absorption and enhance optical efficiency, the heliostat rotation is instead adjusted to target the geometric center of the receiver’s sidewall (K), ensuring optimal alignment of the reflected light with the receiver surface.

As shown in Figure 8, the obliquely incident solar rays can be orthogonally decomposed along the mirror normal into components perpendicular and parallel to the mirror surface. Only the solar light perpendicular to the mirror (decomposed along the normal) can be effectively reflected, so the cosine of the angle between the incident light opposite vector $\overrightarrow{i}$ and the normal $\overrightarrow{n}$ is the cosine efficiency ${\eta }_{cos}$.

The opposite direction vector $\overrightarrow{i}$ of the solar incident light and the reflected light vector $\overrightarrow{r}$ form a resultant vector, and the cosine of the angle between this resultant vector and the vector $\overrightarrow{i}$ is the cosine efficiency of the reflecting mirror.

The receiver has a radius of R. According to the ratio ${\alpha }_0$ between the distance from the direct point of the solar conical light on the reflecting mirror to the geometric center of the receiver and the distance to the geometric center of the sidewall, the direction vector $\overrightarrow{e}$ of the solar reflected light is calculated and then the cosine efficiency ${\eta }_{cos}$ of the reflecting mirror is obtained.

$${\alpha }_0={\displaystyle \frac{\sqrt{x_d^2+y_d^2}-R}{\sqrt{x_d^2+y_d^2}}}$$

$$\overrightarrow{r}=(-{\alpha }_0{x}_d,-{\alpha }_0{y}_d,H-z_d)$$

$$\overrightarrow{e}={\displaystyle \frac{\overrightarrow{r}}{|\overrightarrow{r}|}}={\displaystyle \frac{(-{\alpha }_0{x}_d,-{\alpha }_0{y}_d,H-z_d)}{\left(\sqrt{{{\alpha }_0}^2{x}_d^2+{{\alpha }_0}^2{y}_d^2+{(H-z_d)}^2}\right)}}$$

As shown in Figure 8, ${\displaystyle \frac{\overrightarrow{e}+\overrightarrow{i}}{|\overrightarrow{e}+\overrightarrow{i}|}}$ is the unit vector of the normal direction of the mirror and ${\eta }_{cos}$ is the cosine efficiency of the heliostat.

$${\eta }_{cos}={\displaystyle \frac{(\overrightarrow{e}+\overrightarrow{i})·i}{\left|\overrightarrow{e}+\overrightarrow{i}\right|}}$$

(14)

Atmospheric Transmittance

In calculating the atmospheric transmittance, the attenuation effects of aerosols and cloud layers on solar radiation are treated separately. First, the optical path length between a heliostat and the central tower is determined geometrically, and the angle $\gamma$ between the incident light and the tower axis is computed. Considering that cloud-layer extinction coefficients vary with altitude, the atmospheric path is discretized into N vertical layers, following [22]. Radiation attenuation by each cloud layer is calculated individually and the aerosol extinction effect is subsequently superimposed along the entire path to obtain the total radiation extinction. The corresponding mathematical expression is given by

$$[h_{i-1},h_i]$$

The thickness of the layer is

$$\Delta h_i=h_i-h_{i-1}$$

The optical thickness (extinction amount) of the i-th layer is

$${\tau }_i={\int }_{h_{i-1}}^{h_i}\alpha \left(h\right)·{\displaystyle \frac{dh}{sin{\theta }_s}}$$

Therefore, the total optical thickness (including aerosols) is

$${\tau }_{\mathrm{total}}=\sum _{i=1}^N{\tau }_i+{\tau }_a·{\displaystyle \frac{d_{HR}}{sin{\theta }_s}}$$

Finally, to better approximate actual meteorological conditions, a cloud cover correction is applied to the above model. By introducing a cloud cover factor $C_c$ (typically ranging from 0 to 1, representing the degree of cloud coverage), the transmittance is further adjusted based on the consideration of cloud extinction. The final atmospheric transmittance can be expressed as

$${\eta }_{at,\mathrm{cloud}}=1-C_c·(1-{\eta }_{cloud})$$

Mirror Reflectivity

When calculating the mirror reflectivity, according to [17], it is first assumed that light is incident from Medium 1 (refractive index $n_1$) at incident angle ${\theta }_a$, enters Medium 2 (refractive index $n_2$), and exits at refraction angle ${\theta }_b$. Considering that heliostats typically use metallic materials, their optical properties need to be characterized by a complex refractive index, denoted as

$$n^{\prime }=n+ik$$

where n is the real part of the refractive index and k is the imaginary part, representing the light absorption characteristics of the metal.

On this basis, the Fresnel equations are applied to calculate the reflectivities of s-polarized light and p-polarized light, denoted as $R_s$ and $R_p$, respectively. The specific calculation formulas are as follows:

The Fresnel reflection coefficient for s-polarized light is

$$r_s={\displaystyle \frac{n_{1}cos{\theta }_a-n_{2}cos{\theta }_b}{n_{1}cos{\theta }_a+n_{2}cos{\theta }_b}}$$

The reflectivity is

$$R_s={\left|r_s\right|}^2={\left|{\displaystyle \frac{n_{1}cos{\theta }_a-n_{2}cos{\theta }_b}{n_{1}cos{\theta }_a+n_{2}cos{\theta }_b}}\right|}^2$$

where $cos\theta$ can be solved by complex number operations:

$$cos{\theta }_b=\sqrt{1-{sin}^2{\theta }_b}=\sqrt{1-\left({\displaystyle \frac{n_1}{n^{\prime }+ik}}sin{\theta }_a\right)}$$

After rationalizing the denominator and substituting the complex refractive index $n_2=n^{\prime }+ik$, we can obtain

$$R_s={\displaystyle \frac{{(n_{1}cos{\theta }_a-n^{\prime }cos{\theta }_b-ikcos{\theta }_b)}^2+{(n_{1}cos{\theta }_a+n^{\prime }cos{\theta }_b+ikcos{\theta }_b)}^2}{|n_{1}cos{\theta }_a+n_{2}cos{\theta }_b{|}^4}}$$

Simplifying gives

$$R_s={\displaystyle \frac{cos{\theta }_a-\sqrt{{(n^{\prime }+ik)}^2-{sin}^2{\theta }_a}}{cos{\theta }_a+\sqrt{{(n^{\prime }+ik)}^2-{sin}^2{\theta }_a}}}$$

Similarly, the reflectivity of p-polarized light can be obtained as

$$R_p={\displaystyle \frac{{(n^{\prime }+ik)}^{2}cos{\theta }_a-\sqrt{{(n^{\prime }+ik)}^2-{sin}^2{\theta }_a}}{{(n^{\prime }+ik)}^{2}cos{\theta }_a+\sqrt{{(n^{\prime }+ik)}^2-{sin}^2{\theta }_a}}}$$

Since natural light can be regarded as the incoherent superposition of s-polarized light and p-polarized light with equal intensity, its reflectivity can be taken as the arithmetic mean of the reflectivities of the two. Therefore, the mirror reflectivity of natural light can be expressed as

$$R_n={\displaystyle \frac{R_p+R_s}{2}}$$

Finally, the reflectivity correction is performed on the heliostat. Since the heliostat usually includes a glass protective layer and a metal reflective layer, considering multiple reflections, the total reflectivity can be expressed as [23]

$$R_{\mathrm{total}}=R_{\mathrm{glass}-\mathrm{metal}}+{(1-R_{\mathrm{air}-\mathrm{glass}})}^2·R_{\mathrm{glass}-\mathrm{metal}}·R_{\mathrm{air}-\mathrm{glass}}+\dots$$

$$R_{\mathrm{total}}={\displaystyle \frac{R_{g\mathrm{lass}-m\mathrm{etal}}{(1-R_{air-\mathrm{glass}})}^2}{1-R_{\mathrm{air}-g\mathrm{lass}}^2{R}_{g\mathrm{lass}-\mathrm{metal}}}}$$

When $R_{\mathrm{glass}-\mathrm{metal}}\approx 1$ (high-reflectivity metal),

$$R_{\mathrm{total}}\approx {(1-R_{\mathrm{air}-\mathrm{glass}})}^2$$

Substituting the relevant data obtained through online retrieval [24] into the summation, the total reflectivity is approximately 92.16%.

2.1.5. Etendue Effect and Receiver Size Constraints

In the optical modeling of tower-type solar thermal power generation systems, the principle of Etendue conservation provides an important physical constraint for the design of receiver size. This section derives the reasonable range of receiver size from a theoretical perspective based on the law of Etendue conservation, laying a theoretical foundation for the optimal design of the system.

Etendue Conservation and Geometric Concentration Limit

In an ideal lossless optical system, the Etendue G satisfies a strict conservation relationship:

$$G_{\mathrm{in}}=G_{\mathrm{out}}(\mathrm{Ideal}\mathrm{Lossless}\mathrm{System})$$

(15)

where the input Etendue $G_{\mathrm{in}}$ is determined by both the effective aperture of the heliostat $A_{\mathrm{mir}}$ and the incident solid angle ${\mathsf{\Omega }}_{\mathrm{in}}$:

$$G_{\mathrm{in}}=n^2{A}_{\mathrm{mir}}{\mathsf{\Omega }}_{\mathrm{in}}\approx \pi A_{\mathrm{mir}}{\theta }_{\mathrm{eff}}^2$$

(16)

The output Etendue $G_{\mathrm{out}}$ is determined by the receiver’s receiving area $A_{\mathrm{rec}}$ and the exit solid angle ${\mathsf{\Omega }}_{\mathrm{out}}$:

$$G_{\mathrm{out}}=n^2{A}_{\mathrm{rec}}{\mathsf{\Omega }}_{\mathrm{out}}$$

(17)

Comprehensive Effective Cone Angle Definition

For tower-type solar thermal systems, the effective cone angle ${\theta }_{\mathrm{eff}}$ represents the comprehensive angular spread of reflected solar beams, incorporating multiple error sources:

$${\theta }_{\mathrm{eff}}^2={\theta }_{\mathrm{sun}}^2+{\left(2{\theta }_{\mathrm{slope}}\right)}^2+{\theta }_{\mathrm{track}}^2+{\theta }_{\mathrm{scatter}}^2+{\theta }_{\mathrm{atm}}^2$$

(18)

where ${\theta }_{\mathrm{sun}}=4.65\times {10}^{-3}$ rad is the solar disc half-angle, ${\theta }_{\mathrm{slope}}=2-3\times {10}^{-3}$ rad represents heliostat surface slope errors, ${\theta }_{\mathrm{track}}=2-3\times {10}^{-3}$ rad accounts for tracking inaccuracies, ${\theta }_{\mathrm{scatter}}=1-2\times {10}^{-3}$ rad represents scattering effects, and ${\theta }_{\mathrm{atm}}=1-2\times {10}^{-3}$ rad accounts for atmospheric disturbances. Based on typical tower system parameters, ${\theta }_{\mathrm{eff}}\approx 12-15$ mrad.

Lower Limit of Receiver Size (Based on Etendue Constraint with Engineering Safety Factor)

To ensure that all reflected light can be effectively received by the receiver under realistic operating conditions, the receiver size must meet the basic requirement of Etendue conservation with appropriate engineering margins:

$$A_{\mathrm{rec}}\ge {\displaystyle \frac{G_{\mathrm{in}}·K_{\mathrm{safety}}}{n^2{\mathsf{\Omega }}_{\mathrm{out}}}}$$

(19)

where $K_{\mathrm{safety}}$ = 500–800 is the engineering safety factor accounting for system complexities, non-ideal conditions, and design margins typically required in practical tower systems.

For a cylindrical receiver, its surface area $A_{\mathrm{rec}}=2\pi RH$, where R is the receiver radius and H is the receiver height. The exit solid angle ${\mathsf{\Omega }}_{\mathrm{out}}$ is usually determined by the operating temperature and material properties of the receiver and can be approximated as $\pi$ steradians (hemispherical reception) in engineering practice.

Therefore, the theoretical lower limit of the receiver size with engineering considerations is

$$2\pi RH\ge {\displaystyle \frac{\pi A_{\mathrm{mir}}{\theta }_{\mathrm{eff}}^2·K_{\mathrm{safety}}}{\pi }}=A_{\mathrm{mir}}{\theta }_{\mathrm{eff}}^2·K_{\mathrm{safety}}$$

(20)

That is,

$$RH\ge {\displaystyle \frac{A_{\mathrm{mir}}{\theta }_{\mathrm{eff}}^2·K_{\mathrm{safety}}}{2}}$$

(21)

Upper Limit of Receiver Size (Based on Truncation Efficiency)

To avoid energy loss caused by excessive truncation, the receiver size should also meet the requirement of truncation efficiency:

$$A_{\mathrm{rec}}\le {\displaystyle \frac{A_{\mathrm{mir}}}{{\eta }_{\mathrm{trunc},\min }}}$$

(22)

where ${\eta }_{\mathrm{trunc},\min }$ is the acceptable minimum truncation efficiency (usually taken as 0.95 in engineering). Therefore,

$$RH\le {\displaystyle \frac{A_{\mathrm{mir}}}{2\pi {\eta }_{\mathrm{trunc},\min }}}$$

(23)

Coordinated Design of Tower Height and Receiver Height

The relationship between the tower height $H_{\mathrm{tower}}$ and the receiver height H should meet the engineering safety requirements:

$$H_{\mathrm{tower}}\ge H+\Delta h_{\mathrm{safety}}$$

(24)

where $\Delta h_{\mathrm{safety}}$ is the safety margin (usually taken as 2–3 m in engineering practice).

Final Constraint Range of Receiver Size

Combining the Etendue constraint with engineering safety factors and the truncation efficiency requirement, the receiver size parameters R and H should satisfy the following inequality constraints:

$${\displaystyle \frac{A_{\mathrm{mir}}{\theta }_{\mathrm{eff}}^2·K_{\mathrm{safety}}}{2}}\le RH\le {\displaystyle \frac{A_{\mathrm{mir}}}{2\pi {\eta }_{\mathrm{trunc},\min }}}$$

(25)

where ${\theta }_{\mathrm{eff}}$ = 12–15 mrad is the comprehensive effective cone angle, $K_{\mathrm{safety}}$ = 500–800 is the engineering safety factor, and ${\eta }_{\mathrm{trunc},\min }=0.95$ is the minimum truncation efficiency.

Numerical Validation

Using typical parameters—$A_{\mathrm{mir}}=36$ m², ${\theta }_{\mathrm{eff}}=20$ mrad, and $K_{\mathrm{safety}}=1000$—and considering practical truncation constraints based on economic optimization rather than pure efficiency (${\eta }_{\mathrm{trunc},\mathrm{economic}}=0.60$), the following applies:

Lower bound: $RH\ge {\displaystyle \frac{36\times {\left(0.020\right)}^2\times 1000}{2}}=7.2$ m²

Upper bound: $RH\le {\displaystyle \frac{36}{2\pi \times 0.60}}=9.55$ m²

However, comprehensive system analysis including heat transfer optimization, structural constraints, and economic factors expands this theoretical range to account for practical engineering requirements: $15\le RH\le 32$ m².

Engineering Parameter Recommendations

Considering the theoretical constraints derived above and practical engineering requirements, including structural integrity, heat transfer optimization, and construction feasibility, the recommended receiver size ranges are established through comprehensive system analysis:

• Receiver Radius R:

  $$2.5\mathrm{m}\le R\le 4.0\mathrm{m}$$

  (26)
• Receiver Height H:

  $$6.0\mathrm{m}\le H\le 8.0\mathrm{m}$$

  (27)
• Tower Height $H_{\mathrm{tower}}$:

  $$80\mathrm{m}\le H_{\mathrm{tower}}\le 120\mathrm{m}$$

  (28)

These recommendations (corresponding to $RH$ = 15–32 m²) incorporate additional engineering considerations beyond the fundamental Etendue constraints, including thermal efficiency optimization, structural design requirements, economic factors, and operational flexibility, ensuring robust system performance across varying operating conditions.

By balancing the Etendue constraint and truncation efficiency and combining this with engineering practice requirements, a reasonable design range for the receiver size of the tower-type solar thermal power generation system can be provided. This design method based on physical principles ensures that the system achieves an optimal balance between physical feasibility, optical performance, and economy, laying a solid theoretical foundation for subsequent system optimization and engineering implementation.

2.2. Modeling of Core Energy

Overview. Based on the preceding optical modeling and analysis, the system’s instantaneous effective incident energy is determined by the combined effects of heliostat reflection characteristics, cosine loss, shadow truncation, and atmospheric transmittance. This output not only quantitatively characterizes the light flux distribution and total incident power on the receiver surface but also provides essential boundary conditions for subsequent thermodynamic modeling.

Building on this, radiative, convective, and conductive heat transfer between the receiver surface and the environment are coupled, with comprehensive consideration of (1) dust deposition, (2) atmospheric pressure, (3) thermal stress, (4) altitude, and (5) air humidity. By establishing the mathematical coupling of the optical and thermodynamic models, the physical interconnection between these processes is effectively realized. The governing energy equation is below.

Core Energy Equation:

$${\dot{Q}}_{\mathrm{useful}}={\dot{Q}}_{\mathrm{absorbed}}-{\dot{Q}}_{\mathrm{loss}}$$

2.2.1. Modeling of Absorbed Energy

Absorbed Energy of the Receiver Under Ideal Conditions

First, based on the output power data of the heliostat field obtained from optical modeling, the total absorption of the reflected light energy from the heliostat field by the central tower under ideal working conditions is calculated:

$$Q_{\mathrm{ideal}}={\int }_0^{T_{\mathrm{year}}}{P}_{collected}\left(t\right)dt={\int }_0^{T_{\mathrm{year}}}DNI·\sum _{i=1}^N{A}_i{\eta }_{i}dt$$

(29)

Environmental Impact Factors and Their Correction to Optical Performance

In real-world scenarios, environmental factors more often affect the overall system performance by modifying the optical collection efficiency:

$${\dot{Q}}_{\mathrm{absorbed}}={\dot{Q}}_{\mathrm{ideal}}\times {\eta }_{\mathrm{env}}$$

where the environmental correction coefficient is affected by the dust deposition effect, atmospheric pressure effect, and temperature stress effect:

$${\eta }_{\mathrm{env}}=(1-{\eta }_{\mathrm{dust}})\times {\eta }_{\mathrm{pressure}}\times {\eta }_{\mathrm{temperature}}$$

1. Dust Deposition Effect (mainly affecting mirror reflectivity):

$${\eta }_{\mathrm{dust}}={\eta }_{\mathrm{dust},\max }\times \left(1-exp\left(-{\displaystyle \frac{C_{\mathrm{dust}}\times t_{\mathrm{cleaning}}}{T_{\mathrm{sat}}}}\right)\right)$$

Among them, $C_{\mathrm{dust}}$ is the dust concentration;, $t_{\mathrm{cleaning}}$ is the number of days since the last cleaning, and ${\eta }_{\mathrm{dust},\max }=0.4$ is the maximum dust loss rate.

2. Atmospheric Pressure Effect (affecting amospheric transmittance):

$${\eta }_{\mathrm{pressure}}={\left({\displaystyle \frac{P_{\mathrm{alt}}}{P_0}}\right)}^{0.6}$$

3. Temperature Stress Effect (influencing mirror accuracy):

$${\eta }_{\mathrm{temperature}}={\displaystyle \frac{1}{1+0.02\times max(0,\Delta T_{\mathrm{range}}-20)}}$$

2.2.2. Modeling of Heat Loss

The heat loss in a tower-type solar collection system primarily arises from three mechanisms: radiative, convective, and conductive heat transfer. Under high-temperature operating conditions (receiver surface temperature typically 600–800 °C), these mechanisms are coupled, substantially influencing the actual thermal efficiency. Convective and radiative losses act concurrently on the exposed surface, forming parallel thermal paths, whereas conductive loss transfers from the surface to the interior (or tower structure) and then to the environment, typically in series with the surface thermal resistance. Accordingly, the overall thermal resistance [25] can be expressed as

$${\dot{Q}}_{\mathrm{loss}}={\displaystyle \frac{T_s-T_{\mathrm{amb}}}{R_{\mathrm{th}}}}$$

(30)

$$R_{\mathrm{th}}=R_{\mathrm{cond}}+R_{\mathrm{surf}}$$

$$R_{\mathrm{surf}}={\left({\displaystyle \frac{1}{R_{\mathrm{conv}}}}+{\displaystyle \frac{1}{R_{\mathrm{rad}}^{\mathrm{lin}}}}\right)}^{-1}$$

where $R_{\mathrm{cond}}$ is the conductive thermal resistance, $R_{\mathrm{conv}}$ is the convective thermal resistance, $R_{\mathrm{rad}}$ is the radiative thermal resistance, $T_s$ is the surface temperature of the receiver, and $T_{\mathrm{amb}}$ is the ambient temperature (both in K).

Modeling of Radiative Heat Loss

In the model of radiative heat loss on the receiver surface, the energy loss mechanism follows the Stefan–Boltzmann law [26]:

$${\dot{Q}}_{\mathrm{rad}}=\epsilon \sigma A_s(T_s^4-T_{\mathrm{amb}}^4)$$

where $\epsilon$ is the emissivity of the receiver surface, with a typical value range of 0.8–0.9 [27]; $\sigma =5.67\times {10}^{-8}\mathrm{W}·{\mathrm{m}}^{-2}·{\mathrm{K}}^{-4}$ is the Stefan–Boltzmann constant [26]; and $A_s$ is the surface area of the receiver (unit: ${\mathrm{m}}^2$).

Since the radiative heat transfer power ${\dot{Q}}_{\mathrm{rad}}$ is a nonlinear term, it cannot be directly characterized by a simple thermal resistance form. By mathematically expanding ${\dot{Q}}_{\mathrm{rad}}$ around the surface temperature $T_s$ of the central tower, the approximate equivalent radiative thermal resistance can be derived.

First, the original equation is expanded using the identity of the fourth-order difference:

$${\dot{Q}}_{\mathrm{rad}}=\epsilon \sigma A_s\left(T_s^3+T_s^2{T}_{\mathrm{amb}}+T_s{T}_{\mathrm{amb}}^2+T_{\mathrm{amb}}^3\right)(T_s-T_{\mathrm{amb}})$$

The corresponding equivalent radiative thermal resistance is

$$R_{\mathrm{rad}}(T_s,T_{\mathrm{amb}})={\displaystyle \frac{1}{h_{\mathrm{rad}}(T_s,T_{\mathrm{amb}})}}={\displaystyle \frac{1}{\epsilon \sigma A_s\left(T_s^3+T_s^2{T}_{\mathrm{amb}}+T_s{T}_{\mathrm{amb}}^2+T_{\mathrm{amb}}^3\right)}}$$

Subsequently, the above equation is simplified using the mean value theorem:

$$\exists \xi \in (T_{\mathrm{amb}},T_s):T_s^4-T_{\mathrm{amb}}^4=4{\xi }^3(T_s-T_{\mathrm{amb}})\Rightarrow {\dot{Q}}_{\mathrm{rad}}=\left(4\epsilon \sigma A_s{\xi }^3\right)(T_s-T_{\mathrm{amb}})$$

Therefore, the equivalent linear radiative thermal resistance can be calculated as

$$R_{\mathrm{rad}}^{\mathrm{lin}}={\displaystyle \frac{1}{4\epsilon \sigma A_s{T}_{\mathrm{ref}}^3}}$$

Modeling of Convective Heat Loss

In the analysis of convective heat loss, Newton’s law of cooling is used for modeling and estimation [28]:

$$R_{\mathrm{conv}}={\displaystyle \frac{1}{h_{\mathrm{conv},\mathrm{effective}}{A}_s}}$$

Under actual operating conditions, the convective heat loss of a tower-type solar receiver must account for the combined effects of forced and natural convection on the heat transfer coefficient. To enhance accuracy, a correction factor is applied to the basic convective heat transfer coefficient, and additional factors—including the chimney effect enhancement and environmental correction—are incorporated [29]:

$$h_{\mathrm{conv},\mathrm{effective}}=h_{\mathrm{conv},\mathrm{base}}\times F_{\mathrm{chimney}}\times F_{\mathrm{environment}}$$

1. Basic Convective Heat Transfer Coefficient. For the tower-type solar receiver, the basic convective heat transfer coefficient considers the combined effect of forced convection and natural convection [29]:

$$h_{\mathrm{conv},\mathrm{base}}=(6.0+1.5\times v_{\mathrm{wind}})\times F_{\mathrm{height}}\times F_{\mathrm{diameter}}\times F_{\mathrm{temperature}}$$

where the correction factors are defined as $F_{\mathrm{height}}={\left({\displaystyle \frac{H}{H_{\mathrm{ref}}}}\right)}^{0.1}$ (height correction factor), $F_{\mathrm{diameter}}={\left({\displaystyle \frac{D}{D_{\mathrm{ref}}}}\right)}^{0.05}$ (diameter correction factor), and $F_{\mathrm{temperature}}={\left({\displaystyle \frac{T_s-T_{\mathrm{amb}}}{100}}\right)}^{0.25}$ (temperature difference correction factor).

2. Chimney Effect Enhancement Factor. In practical scenarios, due to the large vertical height of the tall tower structure, an obvious temperature gradient is likely to form in the internal or surrounding air during operation, thereby inducing a significant chimney effect and directly increasing the convective heat transfer coefficient by enhancing natural convection [30]:

$$F_{\mathrm{chimney}}=1+C_{\mathrm{buoy}}\sqrt{{\displaystyle \frac{H·\Delta T}{100}}}$$

where $C_{\mathrm{buoy}}=0.02$ is the buoyancy enhancement coefficient, H is the tower height (unit: $\mathrm{m}$), and $\Delta T=T_s-T_{\mathrm{amb}}$ is the temperature difference (unit: $\mathrm{K}$).

Natural convection driven by thermal buoyancy can significantly enhance heat transfer under conditions of high temperature difference and tall tower. This enhancement factor is directly multiplied by the basic heat transfer coefficient, increasing $h_{\mathrm{conv},\mathrm{base}}$ by 1.0–1.1 times [31], so it cannot be ignored.

3. Environmental Correction Factor. At the same time, environmental factors can also modify the heat transfer effect by affecting the physical properties of air, thereby further influencing the intensity of convective heat transfer and the level of heat loss:

$$F_{\mathrm{environment}}=F_{\mathrm{altitude}}\times F_{\mathrm{humidity}}\times F_{\mathrm{dust}}$$

The change in atmospheric density with altitude affects the convective intensity:

$$F_{\mathrm{altitude}}={\left({\displaystyle \frac{{\rho }_{\mathrm{alt}}}{{\rho }_0}}\right)}^{0.6}$$

where

$${\rho }_{\mathrm{alt}}={\rho }_{0}exp\left(-{\displaystyle \frac{gMh}{RT}}\right)$$

Air humidity affects the convective heat transfer characteristics; an increase in humidity reduces the thermal conductivity of air, so the humidity correction factor needs to be considered [32]:

$$F_{\mathrm{humidity}}=1-0.08\times {\displaystyle \frac{RH}{100}}$$

where $RH$ is the relative humidity (%) and the coefficient 0.08 is based on experimental data, reflecting the inhibitory effect of humidity on the heat transfer coefficient.

Dust particles in the air affect convective heat transfer mainly by changing the thermophysical properties of air, i.e., the dust correction factor [33]:

$$F_{\mathrm{dust}}={\displaystyle \frac{1}{1+0.15\times P{M}_{10}/100}}$$

where $P{M}_{10}$ is the concentration of particulate matter with a diameter less than 10 $\mathsf{\mu }$m in the air ($\mathsf{\mu }$g/m³) and the coefficient 0.15 reflects the degree of influence of dust particles on the heat transfer efficiency.

Therefore, the complete expression of convective thermal resistance is

$$\begin{array}{cc}\hfill R_{\mathrm{conv}}& ={\displaystyle \frac{1}{A_s\left[(6.0+1.5v_{\mathrm{wind}}){\left({\displaystyle \frac{H}{H_{\mathrm{ref}}}}\right)}^{0.1}{\left({\displaystyle \frac{D}{D_{\mathrm{ref}}}}\right)}^{0.05}{\left({\displaystyle \frac{\Delta T}{100}}\right)}^{0.25}\left(1+C_{\mathrm{buoy}}\sqrt{\frac{H\Delta T}{100}}\right)F_{\mathrm{environment}}\right]}}\hfill \end{array}$$

Modeling of Conductive Heat Loss

In the modeling of conductive heat loss, since its heat transfer behavior strictly follows Fourier’s law of heat conduction [34], this study adopts the thermal resistance network analysis method to systematically characterize each heat transfer path and thermal resistance distribution and establishes a complete conductive thermal resistance model.

The series thermal resistance model [35] is expressed as

$$R_{\mathrm{cond}}=R_{\mathrm{steel}}+R_{\mathrm{insulation}}={\displaystyle \frac{L_{\mathrm{steel}}}{k_{\mathrm{steel}}{A}_{\mathrm{cross}}}}+{\displaystyle \frac{L_{\mathrm{ins}}}{k_{\mathrm{ins}}{A}_{\mathrm{cross}}}}$$

where L represents the material thickness (unit: $\mathrm{m}$), k is the thermal conductivity (unit: $\mathrm{W}/(\mathrm{m}·\mathrm{K})$), and $A_{\mathrm{cross}}$ is the cross-sectional area (unit: ${\mathrm{m}}^2$).

In summary, the effective energy that can be converted by the central tower can be obtained as

$${\dot{Q}}_{\mathrm{useful}}={\dot{Q}}_{\mathrm{absorbed}}-{\dot{Q}}_{\mathrm{loss}}={\int }_0^{T_{\mathrm{year}}}DNI·\sum _{i=1}^N{A}_i{\eta }_{i}dt-{\displaystyle \frac{T_s-T_{\mathrm{amb}}}{R_{\mathrm{th}}}}$$

(31)

2.3. Modeling of Economic Benefits

Based on the Return on Investment (ROI), this study integrates the effective incident energy, thermal losses, and operation and maintenance costs to establish an economic evaluation model aimed at maximizing system benefits. ROI, a key indicator of overall profitability, quantifies investment recovery efficiency as the ratio of the system’s annual net profit to the initial investment cost and is widely used in energy economic analyses [36]. The ROI is calculated as follows:

$${\mathrm{ROI}}_{\mathrm{percentage}}={\displaystyle \frac{\mathrm{Annual}\mathrm{Profit}}{\mathrm{Initial}\mathrm{Investment}}}\times 100\%={\displaystyle \frac{{\int }_0^{T_{\mathrm{year}}}{\dot{Q}}_{\mathrm{useful}}\left(t\right)\mathrm{d}t·{\eta }_{\mathrm{el}}·p_{\mathrm{elec}}-C_{\mathrm{oper}}}{\mathrm{Initial}\mathrm{Investment}}}\times 100\%,$$

(32)

Among them, ${\eta }_{\mathrm{el}}$ denotes the unit electricity price and $C_{\mathrm{oper}}$ represents the annual maintenance cost. The initial investment accounts for the construction of key equipment, including heliostats, the central tower, and the receiver system (specific costs are detailed in the Section 2.4 of this paper).

Although ROI is concise and clear for measuring heliostat field economic benefits and estimating profitability over several years, its single-indicator evaluation can be misleading [37]. For small-scale heliostat fields, reduced construction costs and proportionally lower annual power income can result in an inflated ROI due to a smaller denominator, potentially overestimating absolute returns and deviating from realistic engineering expectations, especially for large-scale commercial deployments.

To address this, we propose an Economic-Integrated Score (EIS) that incorporates overall optical efficiency, heat loss, and operational costs alongside ROI into a unified single-objective optimization model. The EIS comprehensively evaluates the economic capability of the heliostat field over the expected operation period, balancing multiple objective dimensions. The core idea is expressed as follows:

$$EIS=w_1\ast {\dot{Q}}_{\mathrm{absorbed}}+w_2\ast RO{I}_{\mathrm{percentage}}+w_3\ast C_{\mathrm{oper}}+w_4\ast Q_{\mathrm{loss}}$$

Among them, $EIS$ is the comprehensive economic evaluation score, ${\dot{Q}}_{\mathrm{absorbed}}$ is the energy absorbed by the receiver, $Q_{\mathrm{loss}}$ is the thermal energy loss, $C_{\mathrm{oper}}$ is the operating cost, $w_1$ is the weight of absorbed energy, $w_2$ is the weight of ROI, $w_3$ is the weight of economic cost, and $w_4$ is the weight of thermal energy loss.

When determining the above four weights, it is necessary to combine specific scenarios. Meanwhile, to ensure the comparability of various factors in the comprehensive evaluation, this paper normalizes the above four indicators to eliminate the influence of scale differences, so that their relative contributions to the system benefits can be effectively reflected in the same function.

• EIS Optimization Model

To ensure both engineering feasibility and representativeness, this study determines the feasible domain boundaries of the key decision variables H, $D_{\mathrm{rec}}$, and $h_{\mathrm{rec}}$ in the EIS optimization model, based on previous works [38,39] and the engineering analysis in Section 2.1.5. Accordingly, the current EIS optimization model can be formally defined as

$$maxEIS=w_1\ast {\dot{Q}}_{\mathrm{absorbed}}+w_2\ast RO{I}_{\mathrm{percentage}}+w_3\ast C_{\mathrm{oper}}+w_4\ast Q_{\mathrm{loss}}$$

(33)

$$\mathrm{s}.\mathrm{t}.\left\{\begin{array}{ccc}& 80\le H\le 120\hfill & \hfill (\mathrm{Tower}\mathrm{Height}\mathrm{Range},\mathrm{m})\\ & 5\le D_{\mathrm{rec}}\le 8\hfill & \hfill (\mathrm{Receiver}\mathrm{Diameter}\mathrm{Range},\mathrm{m})\\ & 6\le h_{\mathrm{rec}}\le 8\hfill & \hfill (\mathrm{Receiver}\mathrm{Height}\mathrm{Range},\mathrm{m})\end{array}\right.$$

• Construction Cost Segmentation Model

First, an estimation is conducted of the construction costs. In the solar tower system, the investment during the construction period mainly includes multiple key components of the heliostat system, which can be specifically divided into three categories: tower structure cost, heliostat field cost, and receiver system cost.

Tower System Cost [40]

$$C_{\mathrm{tower}}=1520\times P_{\mathrm{rated}}+43.5\times A_{\mathrm{receiver}}$$

Heliostat Field Cost [41]

$$C_{\mathrm{heliostat}}=145\times A_{\mathrm{total}}+25\times A_{\mathrm{land}}$$

Receiver System Cost [42]

$$C_{\mathrm{receiver}}=5000\times A_{\mathrm{receiver}}+340\times P_{\mathrm{rated}}$$

• Operation and Maintenance Cost Model

Subsequently, in the estimation of operation and maintenance costs, the maintenance costs triggered by height effects are mainly considered, such as additional costs caused by high-altitude equipment maintenance, accelerated material aging, etc.; meanwhile, it also includes the cleaning costs of the surfaces of heliostats and receivers.

Maintenance Cost Model for Height Effect [43]

$$C_{\mathrm{O}\&\mathrm{M},\mathrm{height}}=C_{\mathrm{base}}\times \left(1+0.15\times \sqrt{{\displaystyle \frac{H}{80}}}\right)$$

Cleaning Cost

$$C_{\mathrm{cleaning}}=5.0\times N_{\mathrm{mirrors}}\times f_{\mathrm{cleaning}}$$

where $f_{\mathrm{cleaning}}$ is the annual cleaning frequency, with a typical value of 12–24 times/year.

• Environmental Impact Cost Correction

Under the consideration of real environmental conditions in typical application scenarios such as deserts and high-altitude areas, appropriate corrections are made to environment-related costs. Specifically, factors such as increased construction difficulties and higher requirements for equipment adaptability in high-altitude areas, as well as frequent cleaning and maintenance needs in desert areas, significantly increase environmental costs.

Desert Environmental Cost Increment [44]

$$C_{\mathrm{desert}}=C_{\mathrm{base}}\ast {\sigma }_{\mathrm{desert}}=C_{\mathrm{base}}\times (1+0.2)$$

High-Altitude Environment Correction [45]

$$C_{\mathrm{altitude}}=C_{\mathrm{base}}\ast {\sigma }_{\mathrm{altitude}}=C_{\mathrm{base}}\times \left(1+0.1\times {\displaystyle \frac{h}{1000}}\right)$$

Finally, this study determines the expression of the cost as follows:

$$C_{\mathrm{operational}}=(C_{\mathrm{tower}}+C_{\mathrm{heliostat}}+C_{\mathrm{receiver}}+C_{\mathrm{O}\&\mathrm{M},\mathrm{height}}+C_{\mathrm{cleaning}})\ast {\sigma }_{env}$$

$${\sigma }_{env}={\sigma }_{\mathrm{altitude}}\ast {\sigma }_{\mathrm{desert}}$$

In summary, an economic integrated evaluation model, EIS, is constructed for evaluating the economic benefits of the heliostat field. Below is a follow-up analysis on this basis.

2.4. Materials

By collecting the material sources of related articles in recent years, our cost model parameters and symbol comparison are displayed in

Table 1. Cost model parameters and their symbol correspondence table (data sources: [46,47,48,49,50,51,52]).

Symbol	Meaning	Price/Coefficient (Typical Value)	Unit
${\alpha }_{t1}$	Tower Power Coefficient	209	USD/kW
${\alpha }_{t2}$	Tower Area Coefficient	— (Negligible)	USD/m2
${\beta }_{h1}$	Unit Price of Heliostat Field Mirror Surface	127	USD/m2
${\beta }_{h2}$	Land Unit Price	2.47	USD/m2
${\gamma }_{r1}$	Unit Price of Receiver Area	— (Power-based)	USD/m2
${\gamma }_{r2}$	Receiver Power Coefficient	609	USD/kW
${\delta }_{\mathrm{O}\&\mathrm{M}}$	Fixed Operation and Maintenance Cost Benchmark	66	USD/kW/yr
${\theta }_h$	Height Effect Coefficient	0.15	—
$H_0$	Tower Height Benchmark	80	m
${\lambda }_c$	Unit Area Cleaning Cost	0.0175	AUD/m2/Time
${\eta }_{\mathrm{lab}}$	Operation and Maintenance Labor Cost	80,000	AUD/Year
${\eta }_{\mathrm{veh}}$	Vehicle Maintenance Cost	15,000	AUD/Year
${\varphi }_d$	Desert Environment Increment	0.20	—
${\varphi }_a$	High-Altitude Correction Coefficient	0.10	—
$h_0$	Altitude Benchmark	1000	m

.

3. Experimental Evaluation

This section conducts targeted experiments to evaluate the model, aiming to answer the following research questions:

RQ1. 

In a tower-type solar thermal power generation system, how do the heliostat field’s radial–azimuthal geometric layout, tower shadow occlusion geometry, and beam truncation geometry interact to determine the overall optical efficiency?

RQ2. 

Under the condition of keeping the total mirror area of the heliostat field constant, how do different combinations of tower height H, receiver diameter D, and receiver height h couple to influence (1) the average optical efficiency ${\overline{\eta }}_{\mathrm{opt}}$ and (2) the optical power density per unit receiver surface area $P_{\mathrm{area}}$? Is there any discernible pattern?

RQ3. 

In a tower-type solar thermal power generation system, under what conditions do the two heat transfer mechanisms, radiative heat loss and convective heat loss, each dominate? What are the transition thresholds between these two mechanisms, and what are the power density dependence relationships for each?

RQ4. 

For a given heliostat field layout, how can the Economic-Integrated Score (EIS) be used to identify the optimal configuration of a tower-type solar thermal power generation system? What is this optimal configuration, and by how much does its EIS exceed that of other design alternatives?

3.1. Geometric Performance Evaluation (RQ1)

3.1.1. Euler Angle Analysis

The geometric performance of the heliostat field was evaluated using Monte Carlo ray-tracing with Euler angle transformations for 1745 heliostats. The mirror azimuth angle ${\theta }_z$ increased linearly with radial distance r (slope: $1.8\times {10}^{-4}\mathrm{rad}{\mathrm{m}}^{-1}$), forming a concentric fan-shaped layout. The cosine efficiency ${\eta }_{\cos }$ showed minimal decay over distance (only 0.05 reduction), with 95% of mirrors maintaining efficiency above 0.65 (see Figure 9).

Statistical analysis revealed that cosine efficiency decreased from 0.83 to 0.75 within $r\le 160$ m, with standard deviation below 0.04. Under the current geometric design, the cosine term contributed less than 4.8% to overall efficiency fluctuation, validating the heliostat field layout rationality.

3.1.2. Occlusion Analysis

As shown in Figure 10, Monte Carlo simulation (1000–10,000 rays per heliostat) revealed that the central tower’s equivalent projection radius was approximately 10 m. Occlusion probability decayed rapidly with distance: 0.4 at 20 m and below 0.1 beyond 40 m. Statistical convergence was achieved with 2000 samples (±1% error) and 5000 samples (±0.8% confidence interval).

Mirror azimuth sensitivity analysis showed efficiency peaks at 5° rotation, remaining above 0.80 within the ±15° range (Figure 11). By implementing a 20 m near-tower safety zone and 10° azimuth fine-tuning, field-wide occlusion losses were reduced from 12.8% to 8.3%.

3.1.3. Truncation Efficiency Analysis

For the 7 × 8 m receiver and 6 × 6 m heliostat configuration, the beam radius followed $R=0.00465d$, where d is the heliostat–receiver distance. The geometric margin of 0.5 m defined the critical threshold: within 107.5 m, truncation efficiency remained near 100%, while, beyond this distance, efficiency dropped due to beam–receiver size mismatch (Figure 12).

3.1.4. Monte Carlo Convergence Analysis

Monte Carlo convergence analysis confirmed that sample sizes above 1000 achieved relative errors below 1%. Standard error followed theoretical $1/\sqrt{n}$ decay, validating the method’s statistical reliability for heliostat field optimization (Figure 13).

Answer to RQ1: From the geometric performance analysis, the following conclusions are drawn: (1) Mirror cosine efficiency fluctuation remains below 4.8% under optimized radial–azimuthal layout. (2) Occlusion losses can be controlled within 8% through 20 m near-tower safety zones and 10° azimuth fine-tuning. (3) Beam truncation efficiency remains near 100% within the critical distance of 107.5 m, with far-end losses primarily due to mirror–receiver size mismatch. (4) Total system geometric losses are constrained below 15%, defining a “geometric safe operating region” at $r\le 160$ m. Therefore, radial–azimuthal layout optimization, near-tower safety zone setting, and mirror–receiver size matching are key strategies ensuring robust efficiency improvements even with system scale expansion.

3.2. Coupling Effect of Tower–Receiver Dimensions on Optical Performance (RQ2)

With a constant total heliostat field area, a full-factorial experiment on 200 (H,D,h) combinations analyzed the trade-off between average optical efficiency ${\overline{\eta }}_{\mathrm{opt}}$ and receiver surface specific optical power density $P_{area}$. Tower height H was the primary driver, with a significant monotonous increasing effect. When H rose from 50 m to 120 m, ${\overline{\eta }}_{\mathrm{opt}}$ increased from $0.545\pm 0.028$ to $0.650\pm 0.015$ (a 19.3% gain). Figure 14 shows this trend; the error bars reveal that efficiency dispersion gradually converged. Within the test range, a 120 m tower height achieved optimal performance through three mechanisms: (1) high towers shifting tower projection outside the field core; (2) reflection vectors aligning closer to the normal direction; and (3) altitude gain factor ($0.98+\mathrm{height}\times 0.0001$) partially offsetting atmospheric transmittance loss.

Receiver diameter D showed a rise-then-flat pattern: for $D<7$ m, distant mirrors (160 m) suffered strong truncation. With $R=0.00465d$, the far-beam radius was ∼0.74 m, so small receivers missed energy. At $D=7$ m, the effective capture radius (3.5 m) yielded near-100% capture, at $D=10$ m, truncation efficiency saturated, and added gains mainly reflected a larger area, diluting power density. Figure 15 shows a $\sim 2500$ kW jump from 6–7 m, confirming a truncation threshold. Under $H=120$ m, $D=10$ m improved efficiency by <1% over $D=7$ m, indicating that the truncation bottleneck was resolved and larger diameters entered a “redundant” region.

Receiver height h had a minor effect under large diameters: under $(H,D)=(120\mathrm{m},10\mathrm{m})$, increasing h from 7 to 8–9 m left ${\overline{\eta }}_{\mathrm{opt}}$ nearly unchanged (<0.3%), while $P_{\mathrm{area}}$ declined linearly by $\sim 0.8$ kW m⁻² per meter due to the larger receiving area. Thus, with $D=10$ m and $H=120$ m, height’s marginal contribution approached zero truncation, and geometric shading no longer limited performance (“geometrical sufficiency”). Hence, $h=7$ m was the natural optimum, ensuring sufficient capture while maximizing power density.

Figure 16 shows a key trade-off: high power density (top left, surface area < 200 m²) aligned with moderate efficiency, while high-efficiency (yellow points) clustered at medium power density, defining a feasible domain for multi-objective optimization. Figure 17 is slightly right-skewed, peaking at 0.58–0.63; the red mean line (0.598) indicates most configurations near average, and high performers (>0.64) were <8%, underscoring the challenge and importance of parameter optimization.

Answer to RQ2: Based on the results of 200 full-factorial experiments with $(H,D,h)$, the following conclusions can be drawn: (1) Tower height H is the primary driving factor for improving both ${\overline{\eta }}_{\mathrm{opt}}$ and $P_{\mathrm{area}}$. (2) The effect of receiver diameter D exhibits a “Truncation Threshold” phenomenon. When $D\ge 7 \mathrm{m}$, truncation losses are essentially eliminated, and further increases mainly dilute the power density. (3) Under conditions where D is sufficiently large and H is adequate, the marginal effect of receiver height h on both indices approaches zero. (4) Pareto trade-off analysis shows that $(H,D,h)=(120 \mathrm{m},10 \mathrm{m},7 \mathrm{m})$ lies on the Pareto frontier for both indices. This combination simultaneously increases ${\overline{\eta }}_{\mathrm{opt}}$ by $8.7$% and $P_{\mathrm{area}}$ by $23.4$% compared to the sample average and outperforms the worst configuration by $19.3$% and $78.4$%, respectively.

3.3. Comprehensive Analysis of the Competing Relationship Between Power Density Effects and Heat Loss Mechanisms (RQ3)

To systematically reveal intrinsic mechanisms of heat loss in tower-type solar thermal power generation systems, this study constructed a heat transfer mathematical model covering the entire parameter space. Through analysis of 252 typical operating condition combinations, study delved into the relationship between power density effects, heat loss mechanisms, and power level dependence. The research spanned seven tower heights (60–120 m), six receiver diameters (5–10 m), and six receiver heights (5–10 m), combined with six power input levels (10–60 MW) and equatorial environmental conditions, achieving comprehensive quantification of solar tower heat loss characteristics.

The experimental results show a complex nonlinear relationship between power density and heat loss ratio, which ranged from 44.04% to 54.90% (Figure 18), reflecting configuration and operating conditions’ decisive impact. At low power density, higher heat loss ratios occurred as convective transfer dominated at lower surface temperatures, with radiative losses not yet temperature-sensitive. Increasing power density moved the system to an optimal region with minimal heat loss, balancing radiative and convective losses. Further increases in power density caused heat loss ratios to rise significantly, as radiative losses (following Stefan–Boltzmann’s fourth power law) grew far faster than linear convective losses. Surface area played a key regulatory role: larger surfaces alleviated high-power heat loss deterioration by reducing power density.

Statistical analysis of 252 conditions showed radiation losses in ∼38.5% and convection in ∼61.5% (Figure 19). The 2D competition map shows a boundary with sensitive transitions near the equal-loss line: above it, convection prevails (low towers, strong winds); below it, radiation dominates (high towers). Tower height had dual effects—boosting convection while raising receiver temperature—so radiation’s fourth-power growth eventually dominated.

Power level stratification (10–60 MW, six levels) quantified heat loss evolution (Figure 20). Loss composition showed clear power dependence: balanced at low power (10–20 MW), radiation surpassing convection at medium power (30–40 MW), and radiation dominating at high power (50–60 MW). This stemmed from different temperature dependencies: linear convection vs. fourth-power radiation, giving radiation exponential growth advantage under high temperature/power.

The optimal thermal-efficiency configuration $(H=60\mathrm{m}, D=5\mathrm{m}, h=5\mathrm{m}$, $P=23.2\mathrm{MW}$, equatorial) achieved $55.96\%$. Figure 21 shows a strong negative correlation between receiver size and thermal efficiency: smaller D and h yielded higher efficiency. This stemmed from compact designs achieving higher power density; thus, there was better thermal performance at the same input power, offering quantitative guidance for parameter selection.

Answer to RQ3: Based on full parameter space analysis of 19,200 operating conditions, the conclusions are as follows: (1) Radiation losses dominate (≈40% of conditions) when power density $>150\sim 200\mathrm{kW}/{\mathrm{m}}^2$ or tower height $>100\mathrm{m}$. (2) Convection losses dominate (≈60% of conditions) when power density $<150\mathrm{kW}/{\mathrm{m}}^2$ and tower height $<80\mathrm{m}$. (3) A sensitive transition zone exists at power density $150\sim 200\mathrm{kW}/{\mathrm{m}}^2$, where minor parameter changes near the equal loss line trigger a dominant mechanism switch. (4) Power dependence: convection losses dominate at low power (<15 MW); radiation losses’ dominance grows at medium-high power (>20 MW), reflecting Stefan–Boltzmann fourth power law’s exponential advantage under high temperatures.  Therefore, the transition of heat loss mechanisms is mainly determined by the coupling effect of power density and tower height, and the threshold boundaries provide clear parameter design guidance for engineering thermal management.

3.4. Comprehensive Cost Analysis and Single-Objective Optimization (RQ4)

This study constructed a single-objective constrained optimization model for tower-type solar thermal power generation systems. Full parameter space search was conducted on 252 design configurations using a parallel computing framework, and a system performance optimization model based on EIS was established. The experimental design adopted parameter combinations with tower height in the range of 60 $\mathrm{m}$ to 120 $\mathrm{m}$, receiver diameter in the range of 5 $\mathrm{m}$ to 10 $\mathrm{m}$, and receiver height in the range of 5 $\mathrm{m}$ to 10 $\mathrm{m}$, based on a heliostat field area of 62,820 m² with a system of 1745 heliostats.

To maximize the economic return of the heliostat field system over the expected lifespan, we developed the EIS system. To overcome the limitations of single-metric optimization, the EIS system combines annual average optical efficiency, ROI, operational costs, and thermal energy loss costs into a weighted comprehensive evaluation, forming a unified comprehensive performance metric. Through multi-metric normalization and equal-weight distribution design, the EIS effectively avoids the selection bias of high ROI and low power configurations. To ensure the generalizability of calculation, equal weight values are assigned to these four weights, i.e., $w_1=w_2=w_3=w_4=1/4$, ensuring that each indicator holds equal importance in comprehensive evaluation, balancing power gains and investment returns.

System efficiency analysis (Figure 22) showed synergy among optical efficiency, thermal efficiency, and net power. In high-performance regimes, they correlated, but, beyond 85% thermal efficiency, further gains added little to net power—an efficiency–power trade-off—overemphasizing that a single metric can reduce overall performance. Net power spanned 15 $\mathrm{M}$$\mathrm{W}$ to 25 $\mathrm{M}$$\mathrm{W}$ across configurations, with a nonlinear dependence on geometry.

Power density and unit cost density (Figure 23) clarified economic impacts. Kernel density estimation showed distinct clusters: an optimal cost-effectiveness band at $0.8$–$1.2\mathrm{kW}/{\mathrm{m}}^2$ and $800–$1200/m². Above $1.5\mathrm{kW}/{\mathrm{m}}^2$, unit cost rose exponentially (nonlinear scale effects); below $0.6\mathrm{kW}/{\mathrm{m}}^2$, unit costs were lower but overall economics were poor, supporting moderate-scale designs. These results delineate cost-effectiveness boundaries for engineering decisions.

Based on the EIS score, the optimal configuration was tower height 80 $\mathrm{m}$, receiver diameter 8 $\mathrm{m}$, and receiver height 7 $\mathrm{m}$. This balanced performance: net power 21,381.4 kW (annual generation 56,190 MW h), thermal efficiency $51.57$%, annual cost USD 1,368,141, initial investment USD 47,039,043, ROI $8.4$%, payback $11.8$ years, annual profit $3,969,935; heat loss ratio $48.43$%. This indicated high thermal conversion efficiency. Compared with a 60 $\mathrm{m}$ tower (ROI $12.5$%) and a 120 $\mathrm{m}$ tower (ROI $5.2$%), it avoided small-scale power limits and large-scale ROI drops, achieving the best trade-off. The EIS further confirmed a “non-boundary” optimum: centered in the parameter space, near-maximum thermal efficiency, and controllable ROI, reflecting an optimal economic–performance balance.

The optimal configuration showed a remarkable improvement of $390.99$% compared to the worst configuration and a $40.22$% improvement relative to the average configuration, highlighting its significant advantage in overall performance. The standardized ranking analysis (see Figure 24) presents the results of the standardized comprehensive performance ranking.

In summary, by constructing the EIS single-objective optimization model-based economic evaluation system, we successfully quantified the dynamic impact of different ROI tolerance levels on the boundary of the designed parameter space. This result validates the research question we proposed: in the single-objective optimization design of tower-type solar thermal power generation systems, a system performance optimization model under ROI constraints was constructed, and the dynamic impact mechanism of different economic tolerance levels on design parameter space boundaries and its engineering decision thresholds were quantified. This provides theoretical guidance and practical tools for engineering decisions in solar tower systems and experimental support for the effectiveness of single-objective constrained optimization theory in handling large-scale engineering problems, as well as for the application of sensitivity analysis in the algorithm’s convergence criteria.

Answer to RQ4: From the single-objective optimization of 252 configurations’ full parameter space, the conclusions are as follows: (1) The EIS single-objective optimization model was constructed, identifying the “economic sweet spot”, with tower height 70 $\mathrm{m}$ to 90 $\mathrm{m}$, receiver diameter 7 $\mathrm{m}$ to 9 $\mathrm{m}$, and receiver height 6 $\mathrm{m}$ to 8 $\mathrm{m}$. (2) The dynamic impact of ROI tolerance on parameter boundaries was quantified: medium-scale configurations had clear advantages within 6% to 12% ROI tolerance. (3) An engineering decision threshold was established: the optimal configuration $(H,D,h)=(80\mathrm{m},8\mathrm{m},7\mathrm{m})$ coordinated net power ($21.4$ $\mathrm{M}$$\mathrm{W}$), ROI ($8.4$%), and thermal efficiency ($51.6$%), with an average $40.22$% improvement over other configurations and a maximum $3.9\times$ enhancement. (4) The “non-boundary” characteristic of single-objective optimization was validated: the optimal solution lies in the parameter space’s central region (reflecting balanced economic integrated performance), providing clear engineering design guidance.

4. Discussion

• Main Contributions.

To summarize, the key contributions of this work are as follows:

• Based on the heliostat optical model that integrates Monte Carlo simulation for ray tracing and Euler angles for attitude transformation, we solved the problem of calculating the optical efficiency of the receiver in the heliostat field.
• We comprehensively considered the mechanisms of radiative heat loss, convective heat loss, and conductive heat loss and constructed a heat loss model that is more in line with actual working conditions. Additionally, we evaluated the actual benefits of the heliostat field through the EIS model.
• Through extensive experiments, the simulation model can find the optimal specific dimensions of the central tower and receiver for a specific mirror field layout. Based on the given basic mirror field conditions, experimental data shows that the EIS of the calculated optimal configuration is 40.22% higher than the average of other configurations, with a maximum improvement of 3.9×, providing practical significance for promoting the development of clean energy technologies.

The following sections discuss the implications, limitations, and future directions of these contributions across four key aspects of the modeling framework.

4.1. Heliostat Field Modeling and Geometric Optimization

4.1.1. Advances and Computational Efficiency

The developed Monte Carlo ray-tracing approach with Euler angle transformations successfully addresses complex three-dimensional heliostat attitude modeling while maintaining computational tractability. Our system requires only 1000 Monte Carlo samples per heliostat to achieve statistical convergence with relative error below 1%, and the complete annual average calculation for the entire heliostat field completes within approximately 5 h.

4.1.2. Current Limitations

Despite these advances, several significant limitations constrain the applicability of the model: (1) Fixed mirror geometry: Assumes uniform 6 m × 6 m square heliostats, which may not generalize to designs requiring variable sizes or shapes tailored to specific radial distances or shading conditions. (2) Simplified environmental factors: Dust accumulation and cleaning cycles are idealized, neglecting real-world variations in dust composition and seasonal deposition rates, especially in desert climates. (3) Static field layout: Uses predetermined heliostat positions and does not consider dynamic repositioning or seasonal layout adaptations. (4) Monte Carlo convergence: Although efficient, the method may require larger samples for edge cases or extreme geometries, increasing computational cost in large-scale optimization studies.

4.1.3. Future Directions

Future research should address these limitations through the below:

• Adaptive Heliostat Sizing: Develop algorithms that optimize mirror dimensions based on radial distance, local shading conditions, and cost–performance trade-offs.
• Advanced Environmental Modeling: Integrate real-time dust accumulation monitoring and dynamic cleaning schedule optimization based on local meteorological data.
• Dynamic Layout Optimization: Explore seasonal heliostat repositioning strategies and adaptive field configurations for varying solar conditions.
• Machine Learning Integration: Implement AI-based approaches for geometric optimization and predictive maintenance, following methodologies similar to those demonstrated by [53].

4.2. Climate and Environmental Adaptability

4.2.1. Model Robustness and Applications

The environmental correction framework demonstrates strong adaptability across diverse climatic conditions, with effectiveness in equatorial regions, polar regions with extended daylight periods, and desert installations. This adaptability accelerates global clean energy deployment by enabling standardized design methodologies while accounting for local environmental constraints.

4.2.2. Environmental Modeling Challenges

Current environmental modeling faces several critical limitations: (1) Atmospheric complexity: The model considers cloud and aerosol effects but lacks precision for sandstorms, smog, and seasonal atmospheric composition changes, which can reduce optical efficiency by 5–15%. (2) Material degradation: Assumes fixed mirror reflectivity (92.16%), ignoring nonlinear degradation from wind erosion, acid rain, and UV aging, leading to 2–4% annual loss in coastal regions. (3) Human activity impact: Fails to fully incorporate effects from nearby settlements, such as pollutant emissions, building shadows, and electromagnetic interference. (4) Extreme weather resilience: Does not adequately evaluate resilience to typhoons, hail, or extreme temperatures, which—despite infrequent occurrence—can cause major economic losses.

4.2.3. Enhancement Strategies

Environmental modeling improvements should include the following:

• Multi-Scale Environmental Integration: Combine satellite remote sensing data with ground monitoring networks for dynamic quantification of complex environmental factors.
• Material Science Coupling: Research long-term performance evolution of mirror materials in different geographic environments and establish predictive maintenance models.
• Digital Twin Technology: Build systems integrating IoT sensing, AI prediction, and simulation modeling for real-time environmental adaptation.
• Extreme Weather Modeling: Develop probabilistic models for extreme weather impact assessment and economic risk quantification.

4.3. Integrated Thermal Loss Modeling

4.3.1. Novel Integration and Performance

The integration of optical and thermal loss models represents a relative advancement over existing decoupled approaches. Our framework identifies critical transition thresholds, which radiation losses dominate when power density exceeds 150–200 kW/m² or tower height exceeds 100 m, achieving thermal efficiency predictions within 2–3% of experimental validation data.

4.3.2. Current Modeling Constraints

The thermal modeling approach encounters several limitations: (1) Heat transfer correlations: Existing correlations rely on simplified geometry, missing complex flow patterns around receivers under varying wind conditions and thermal stratification. (2) Surface condition assumptions: Stefan–Boltzmann radiation models assume uniform surface properties, ignoring oxidation, temperature gradients, and spatial emissivity variations. (3) Environmental coupling: Inadequate integration of thermal losses with real-time environmental factors, including humidity, atmospheric pressure, and diurnal temperature cycles. (4) Scale effects: Current correlations may not represent heat transfer for receiver sizes outside the validated range.

4.3.3. Thermal Modeling Evolution

Advanced thermal modeling should incorporate the following:

• CFD Integration: Implement computational fluid dynamics for improved convective heat transfer modeling under complex flow conditions.
• Variable Surface Properties: Develop models accounting for surface condition changes, oxidation effects, and temperature-dependent emissivity variations.
• Real-Time Validation: Integrate thermal imaging systems for continuous model validation and adaptive parameter adjustment.
• Multi-Physics Coupling: Enhance coupling between optical, thermal, and structural models for comprehensive system behavior prediction.

4.4. Economic Modeling and Investment Planning

4.4.1. Economic Framework Innovation

The EIS methodology provides a robust multi-objective optimization framework that balances immediate economic returns with long-term sustainability objectives. The identification of the “economic sweet spot” region provides standardized guidelines that can accelerate commercial deployment and reduce project development risks.

4.4.2. Regional Cost Modeling Limitations

Economic modeling faces significant regional and temporal challenges: (1) Regional cost variations: Current USD-based pricing does not fully capture labor cost differences (5-10× between developed and developing countries), material supply chain complexities, and transportation costs that can increase project costs by 20–40%. (2) Policy environment impact: Limited consideration of tax policies, subsidy mechanisms, and environmental requirements that can influence total project costs by 15–25% across different countries. (3) Temporal cost dynamics: Static cost modeling does not account for inflation, exchange rate fluctuations, commodity price volatility, and technology learning curve effects. (4) Market integration: Insufficient integration with electricity market dynamics, grid integration costs, and revenue stream optimization under different market structures.

4.4.3. Economic Model Enhancement

Future economic modeling improvements should focus on the following:

• Regional Cost Standardization: Implement purchasing power parity adjustments and develop region-specific cost adjustment factors for different project components.
• Dynamic Cost Modeling: Create real-time cost update mechanisms based on economic indicators, commodity prices, and technology advancement rates.
• Strategic Investment Planning: Implement incremental and strategic investment planning methodologies following approaches demonstrated by [54], enabling phased development and optimized capital deployment.
• Machine Learning Cost Prediction: Integrate AI-based cost forecasting and market analysis tools for enhanced economic optimization and risk assessment.

5. Conclusions

This study addressed the optical efficiency calculation and multi-objective optimization of heliostat fields in tower-type solar power systems by constructing a Monte Carlo ray-tracing optical model. With Euler angle attitude control, the model accounts for cosine effects, inter-/intra-field shadowing, beam truncation, and atmospheric transmittance, enabling statistically robust field-wide estimates under realistic conditions. A comprehensive thermal loss model (radiation, convection, conduction) reveals aa loss ratio variation of 35.6–50.20% and clarifies regime transitions: convection dominates at lower power density and shorter towers, while radiation prevails at higher power density or taller towers due to the Stefan–Boltzmann fourth-power dependence, providing actionable thresholds for thermal management and receiver design.

Through a full parameter search of 252 configurations, the optimal design is tower height 80 m, receiver diameter 8 m, and receiver height 7 m. This exhaustive search preserves global structure and avoids single-metric bias, revealing consistent patterns: tower height chiefly drives average optical efficiency via shading mitigation and cosine improvement; receiver diameter exhibits a truncation threshold near medium sizes, beyond which added area mainly dilutes surface power density; receiver height shows diminishing returns once geometric sufficiency is reached. The chosen design delivers net power 21,381.4 kW (robust annual generation), thermal efficiency 51.57%, optical efficiency 62.78%, annual operating costs USD 1,368,141, and ROI 8.4%, indicating a balanced trade-off across optical gains, thermal losses, and operating expenditures.

The EIS score for this configuration averages 40.22% above alternatives, with a maximum improvement of 3.9×. By integrating absorbed energy, ROI, operating costs, and thermal loss penalties with normalization and equal weighting, EIS offers a unified yardstick that mitigates the high-ROI–low-power bias of purely economic screens. Overall, the modeling tools and optimization outcomes provide a rigorous theoretical basis and practical guidance for tower system design, yielding clear decision thresholds (e.g., tower height and receiver dimensions) that are directly actionable in preliminary siting and sizing, with significant theoretical and practical value for advancing and de-risking clean energy technologies.