Wind-Induced Stability Identification and Safety Grade Catastrophe Evaluation of a Dish Concentrating Solar Thermal Power System

Abstract

To avoid the problem of wind-induced resonance damage in a dish concentrating solar thermal power system (DCSTPS), a fluid dynamics model and a finite element analysis model of the DCSTPS were established separately. The wind load was mapped onto the surface of the concentrator of the DCSTPS using the sequential coupling method, and the static analysis and modal analysis of the DCSTPS were established based on the fluid–structure coupling (FSC) method and the validity of the established model was verified. Based on the results, it can be concluded that the upper edge of the dish solar concentrator (DSC) of the DCSTPS and the three cantilever beams near the Stirling generator are the most vulnerable to being damaged, the DCSTPS will not experience strong resonance phenomena, and effects of the FSC will decrease the natural frequencies of each order. The results of the safety grade catastrophe evaluation of the DCSTPS showed that the safety grade of the DCSTPS was 0.2586 and 0.2819 under case 1 (α = 30°, β = 90°) and case 2 (α = 60°, β = 90°), where it was found that the membership value of the moment load was low, resulting in the stress on the connection seat of the altitude angle and the steering device of the base approaching the allowable stress of the material.

1. Introduction

Compared with fossil fuel power generation, solar thermal power generation (STPG) serves as a means to reduce carbon dioxide emissions and decrease coal resource consumption [1,2,3]. Solar thermal power generation technology can be categorized into three types: dish, tower, and trough [4,5,6,7]. Among them, the DCSTPS boasts the highest efficiency of solar thermal power generation [8,9,10,11,12], accounting for approximately 29.4% of the world’s efficiency peak of solar thermal power generation, and is attracting more and more attention due to its high efficiency of solar thermal power generation, compact structure, and convenient installation [13,14,15].

In the past 20 years, there has been a trend in development to exploit the DCSTPS as a high-efficiency solar power generation technology. Broman et al. [16] proposed a simple method for manufacturing rigid and lightweight frames and supporting mirror strips, using slightly curved strips made of flat and bendable materials. According to the necessary mirror making method, two different spotlights were made, forming a parabolic dish concentrator, and the model of the mirror has been built and successfully used for cooking and baking. Johnston et al. [17] introduced a method of installing spherical reflectors with the same parabolic direction on a dish-style spatial frame structure. When traditional parabolic dishes are used for high-throughput/high-magnification solar concentrating devices, reflective elements need to be produced. Different regions of the parabolic surface of the reflective elements have different curvatures, and the manufacturing of multi-faceted concentrators is already somewhat complex. Their method can minimize its complexity. Finally, the article compares the optical performance and manufacturing feasibility of concentrators with such combined surfaces. Kalogirou [18] conducted a relevant analysis on the optics, thermodynamics, and thermodynamics of concentrators, and introduced a method for evaluating their performance. Kumar and Reddy [19] conducted a numerical investigation on the natural convective heat loss of three kinds of receivers from a fuzzy focusing solar concentrator. The results showed that the orientation and geometric dimensions of the receivers can have a significant impact on the natural convective heat loss. Chen et al. [20] used ray tracing to design a dish solar power system concentrator with two radiating mirrors, which can achieve higher concentration ratios with tracking errors. Additionally, elliptical mirrors perform better than hyperbolic mirrors.

Wu et al. [21] proposed a fluorescent solar concentrator with high geometric dimensions to reduce the energy consumption of photovoltaics. However, experiments in hybrid solar concentrators have shown that there is no self-absorption consumption, and the emitted photons increase linearly with the increase in geometric size (from approximately 50 to 200). Mlatho et al. [22] compared a laser diode measurement with a GARDON radiometer measurement using the Sun as a light source, and the results showed good agreement between the laser diode measurement and the radiometer, indicating that laser diode technology can be used to determine the spatial range of the focal point. Munir et al. [23] provided a complete description of the design principles and construction details of a Scheffler concentrator with a surface area of 8 square meters. This design process is simple, flexible, and does not require any special calculation settings, thus providing potential application prospects in China and the industrial landscape. Scheffler fixed-focus concentrators have been successfully used in low-temperature applications around the world. Velazquez et al. [24] proposed a numerical model for using a linear Fresnel reflector condenser in an advanced solar concentrating cooling system and obtained corresponding experimental verification. Zanganeh et al. [25] proposed a solar parabolic concentrator design based on an array of polyester mirror film surfaces clamped by elliptical wheel rims along the edges of the concentrator and a slight vacuum concentration applied through the underlying film, creating an ellipsoidal polyester film solar concentrator. Christo [26] numerically predicted the velocity and pressure fields around a comprehensive parabolic DSC, as well as the trajectories of dust particles in steady-state and unsteady flows. Their study also provides a preliminary evaluation of the effectiveness of various windproof devices installed on top of disk concentrators to reduce air resistance. Liu et al. [27] described the steps for designing a planar concentrator for laboratory research on intermediate temperature processes. They used Monte Carlo ray tracing analysis to obtain the optimal size and position of each face in order to achieve the desired flux characteristics on the focal plane.

Sardeshpande and Pillai [28] provided an explanatory model for two commercialized solar concentrators in India. In the model, the evaluation of the potential acceptance capacity of solar concentrators considers micro-level technical factors and macro-level market effects and trends. Xiao et al. [29] reviewed and compared three measurement methods for measuring the surface shape and optical properties of the concentrator in a disk solar power generation system—video scanning Hartmann optical test (VSHOT), photogrammetry, and deflection—and also proposed other prospects. Ruelas et al. [30] developed and applied a new mathematical model based on the geometric and optical characteristics of the Scheffler-type solar concentrator in Cartesian coordinates to evaluate its intercept factor. The research results showed that the highest concentration was obtained at an edge angle of 45 and finally indicated that the Scheffler-type solar concentrator was suitable for areas where solar height allows for reduced convective heat loss. Cheng et al. [31] introduced a new optical fluid solar concentrator based on electrowetting tracking. This method has the potential to reduce the cost of photovoltaic concentrators and improve operational efficiency by eliminating the power consumption of mechanical tracking. Compared with traditional silicon-based photovoltaic (PV) solar cells, self-tracking technology based on electrowetting will generate 70% more green energy and reduce costs by 50%.

The fluid–structure coupling (FSC) method had been studied in aviation, transportation, shipping, and other fields, and has practical theoretical significance [32,33]. However, there are still few FSC studies on the DCSTPS in this area. The working environment of the DCSTPS is an open flow field under wind loads, and the flowing air has a significant impact on the performance and vibration characteristics of the DCSTPS [34,35,36,37]. At the same time, the disturbance of the DCSTPS will in turn affect the airfield, which is a typical FSC model.

In order to avoid wind-induced resonance damage in the DCSTPS, it is necessary to study the FSC excitation of the DCSTPS. Therefore, a fluid dynamics model and a finite element analysis model of the DCSTPS were established separately. The wind load was mapped onto the surface of the concentrator of the DCSTPS using the sequential coupling method. Moreover, the static analysis and modal analysis of the DCSTPS were established based on the FSC method, and the validity of the established model was verified. Simultaneously, a safety grade evaluation of the DCSTPS was conducted based on the catastrophe theory [38]. The investigation’s results will be helpful for the structural design of the DCSTPS.

2. Fluid–Structure Coupling Simulation Model and Verification

To ensure that the simulation data were of high accuracy, the modeling of the large-scale dish concentrating solar thermal power system (DCSTPS) utilizes 1:1 data from actual engineering equipment. In this work, the main wind-affected structures (concentrators, support trusses, support columns, etc.) were analyzed, and thus they were appropriately simplified as in Figure 1. When the altitude angle and azimuth angle are both 0°, the concentrator satisfies the parabolic equation x² + z² = 42.735. The range of y values is (4.6 × 10⁻⁴ m, 1.698 m), and the range of x and z values is (−8.52 m, 8.52 m). The length (namely the distance from the center vertex of the concentrator to the plane at the edge of the concentrator) of the concentrating parabolic surface is about 16.98 m, and the diameter of the concentrating parabolic surface is 17.04 m. Based on the size of the DCSTPS model, a three-dimensional model composed of 180 plate units was designed with a plate unit with a thickness of 0.027 m and a gap distance of 15.5 mm. The overall height of the DCSTPS model is 18.5 m, and there are 12 main trusses and 12 auxiliary trusses supporting the truss. The main and auxiliary trusses are connected by support columns, and the support trusses are connected to the support columns through the central body. The concentrator is connected to the support truss through connecting columns, and the balance box is used to balance the moment brought by the concentrator, cantilever, collector, etc.

Additionally, fiberglass with an elastic modulus of 21.0 GPa, Poisson’s ratio of 0.3, and density of 1.7 × 10³ kg/m³ is used as the mirror material of the concentrator, while Q235 carbon steel with an elastic modulus of 210 GPa, Poisson’s ratio of 0.3, and density of 7.85 × 10³ kg/m³ is used as the supporting structure material of the DCSTPS.

As for the 38 kW DCSTPS, the three-dimensional physical model and finite element model were established as shown in Figure 1, where the support truss, connecting column, cantilever beam, and balance truss are beam elements, the center body and balance box are plate elements, and the connection between the support column and the center body is a solid element.

2.1. Conservation Equations

Due to the instantaneous and pulsating wind load on the DCSTPS, dynamic analysis must be adopted to investigate the structure inertia of the DCSTPS in this work. For the convenience of analyzing problems, we use the subscript “f” to represent fluids and the subscript “s” to represent solids.

(1)

Fluid control equations

The mass conservation equation of the DCSTPS can be expressed as follows:

$${\displaystyle \frac{𝜕{\rho }_{\mathrm{f}}}{𝜕\tau }}+\nabla \cdot \left({\rho }_{\mathrm{f}}\mathit{U}\right)=0$$

(1)

The momentum conservation equation of the DCSTPS can be expressed as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{𝜕\left(\rho \mathit{u}\right)}{𝜕\tau }}+\nabla \cdot \left(\rho \mathit{u}\mathit{U}\right)=-{\displaystyle \frac{𝜕p}{𝜕x}}+{\displaystyle \frac{𝜕{\tau }_{xx}}{𝜕x}}+{\displaystyle \frac{𝜕{\tau }_{yx}}{𝜕y}}+{\displaystyle \frac{𝜕{\tau }_{zx}}{𝜕z}}+{\mathit{F}}_{\mathrm{f}x}\\ {\displaystyle \frac{𝜕\left(\rho \mathit{v}\right)}{𝜕\tau }}+\nabla \cdot \left(\rho \mathit{v}\mathit{U}\right)=-{\displaystyle \frac{𝜕p}{𝜕y}}+{\displaystyle \frac{𝜕{\tau }_{xy}}{𝜕x}}+{\displaystyle \frac{𝜕{\tau }_{yy}}{𝜕y}}+{\displaystyle \frac{𝜕{\tau }_{zy}}{𝜕z}}+{\mathit{F}}_{\mathrm{f}y}\\ {\displaystyle \frac{𝜕\left(\rho \mathit{w}\right)}{𝜕\tau }}+\nabla \cdot \left(\rho \mathit{w}\mathit{U}\right)=-{\displaystyle \frac{𝜕p}{𝜕z}}+{\displaystyle \frac{𝜕{\tau }_{xz}}{𝜕x}}+{\displaystyle \frac{𝜕{\tau }_{yz}}{𝜕y}}+{\displaystyle \frac{𝜕{\tau }_{zz}}{𝜕z}}+{\mathit{F}}_{\mathrm{f}z}\end{array}\right.$$

(2)

(2)

Solid control equation

The solid mass conservation equation of the DCSTPS can be expressed as follows:

$${\rho }_{\mathrm{s}}{\displaystyle \frac{{\mathrm{d}}^2{\mathit{d}}_{\mathrm{s}}}{\mathrm{d}{\tau }^2}}=\nabla \cdot {\mathit{\sigma }}_{\mathrm{s}}+{\mathit{F}}_{\mathrm{s}}$$

(3)

Considering that the energy transfer of fluids and solids of the DCSTPS cannot be ignored, an energy equation should be added, and the total enthalpy h_tot in the fluid domain of the DCSTPS can be expressed as follows:

$${\displaystyle \frac{𝜕\left(\rho h_{\mathrm{tot}}\right)}{𝜕\tau }}-{\displaystyle \frac{𝜕p}{𝜕\tau }}+\nabla \cdot \left({\rho }_{\mathrm{f}}\mathit{U}{h}_{\mathrm{tot}}\right)=\nabla \cdot (\lambda \nabla T)+\nabla \cdot (\mathit{U}\cdot {\mathit{\tau }}_{\mathrm{f}})+\mathit{U}\cdot \rho {\mathit{F}}_{\mathrm{f}}+S_E$$

(4)

(3)

Fluid–structure coupling equation

In the fluid–structure coupling region of the DCSTPS, wind load can alter the vibration mode of the DCSTPS by exerting pressure on it, which in turn affects the distribution of airflow velocity and pressure upstream and downstream of the DCSTPS. Therefore, the calculated flow velocity and pressure of the DCSTPS will be loaded onto the front and rear surfaces of the concentrator of the DCSTPS, and the fluid control equation and solid control equation will be coupled and solved.

At the fluid–structure coupling interface of the DCSTPS, some variables—such as the shear force tensor τ_f of the fluid domain of the DCSTPS and the stress tensor τ_s of the solid domain of the DCSTPS, the displacement d_f in the fluid domain and the displacement d_s in the solid domain, the heat flow q_f in the fluid domain of the DCSTPS and the heat flow q_s in the solid domain of the DCSTPS, and the temperature T_f in the fluid domain of the DCSTPS and the temperature T_s in the solid domain of the DCSTPS—should be equal, that is, they should satisfy Equation (5).

$$\left\{\begin{array}{l}{\mathit{\tau }}_{\mathrm{f}}\cdot {\mathit{n}}_{\mathrm{f}}={\mathit{\tau }}_{\mathrm{s}}\cdot {\mathit{n}}_{\mathrm{s}}\\ {\mathit{d}}_{\mathrm{f}}={\mathit{d}}_{\mathrm{s}}\\ q_{\mathrm{f}}=q_{\mathrm{s}}\\ T_{\mathrm{f}}=T_{\mathrm{s}}\end{array}\right.$$

(5)

The solution steps for fluid–structure coupling are expressed as follows:

(1)

Using computational fluid dynamics (CFD) software 2025 to solve the average wind pressure p₍₁₎ corresponding to the initial shape of the DSC of the DCSTPS;

(2)

Using a static calculation program to determine the deformation x₍₁₎ of a DSC of the DCSTPS under the average wind pressure p₍₁₎, which will cause a change in the surface wind pressure coefficient of the DSC of the DCSTPS;

(3)

Based on the deformation of x₍₁₎, using CFD software again to solve the average wind pressure p₍₂₎ of the disk solar concentrator of the DCSTPS, and obtaining the deformation x₍₂₎ of the DSC of the DCSTPS under the action of p₍₂₎. If x₍₁₎ ≈ x₍₂₎, p₍₂₎ is the average wind pressure corresponding to the average deformation of the DSC of the DCSTPS. Otherwise, repeating steps (1) and (2), and continuing to solve x_(i) and its corresponding average wind pressure p_(i) until x_(i−1) ≈ x_(i), then p_(i) is the average wind pressure corresponding to the average deformation of the DSC of the DCSTPS.

2.2. Modal Analysis Equation

(1)

Free mode of the DCSTPS

Considering that the natural frequency and vibration mode of the DCSTPS are independent of external loads, the modal analysis of the DCSTPS can be used to identify the modal parameter of the DCSTPS for the vibration analysis and the vibration prediction of the DCSTPS.

The DCSTPS is a multi degree of freedom system, and the discrete dynamics equation (namely undamped free vibration equation) of the DCSTPS can be expressed as follows:

$$\left[\mathit{M}\right]\left\{\ddot{\mathit{S}}\right\}+\left[\mathit{C}\right]\left\{\dot{\mathit{S}}\right\}+\left[\mathit{K}\right]\left\{\mathit{S}\right\}=\left\{\mathit{F}\right\}$$

(6)

In order to solve the modal characteristics of the DCSTPS, the actual DCSTPS is regarded as an undamped free vibration system when the node force F of the DCSTP is equal to zero, and its control equation can be simplified as follows:

$$\left[\mathit{M}\right]\left\{\ddot{\mathit{S}}\right\}+\left[\mathit{C}\right]\left\{\dot{\mathit{S}}\right\}+\left[\mathit{K}\right]\left\{\mathit{S}\right\}=0$$

(7)

(2)

Fluid–structure coupling mode of the DCSTPS

The influence of pulsating wind action should be considered when the modal analysis of the DCSTPS is performed, and the FSC vibration equation of the DCSTPS can be expressed as follows:

$$\left[\begin{array}{cc}\mathit{M}& 0\\ {\rho }_{\mathrm{f}}\mathit{R}& {\mathit{M}}_{\mathrm{f}}\end{array}\right]\left\{\begin{array}{c}\ddot{\mathit{S}}\\ {\ddot{\mathit{p}}}_{\mathrm{f}}\end{array}\right\}+\left[\begin{array}{cc}\mathit{C}& 0\\ 0& {\mathit{C}}_{\mathrm{f}}\end{array}\right]\left\{\begin{array}{c}\dot{\mathit{S}}\\ {\dot{\mathit{p}}}_{\mathrm{f}}\end{array}\right\}+\left[\begin{array}{cc}\mathit{K}& 0\\ 0& {\mathit{{\rm K}}}_{\mathrm{f}}\end{array}\right]\left\{\begin{array}{c}\mathit{S}\\ {\mathit{p}}_{\mathrm{f}}\end{array}\right\}=\left\{\begin{array}{c}\mathit{F}\\ 0\end{array}\right\}$$

(8)

Free vibration of the DCSTPS is generally in harmonic form, and then

$$\mathit{S}={\mathit{\Phi }}_x\cos {\omega }_{x}t$$

(9)

Substituting Equation (9) into Equation (8), and then

$$(\mathit{K}-{\omega }_x^2\mathit{M}){\mathit{\Phi }}_x=0$$

(10)

The condition for Equation (10) to have non-zero solutions (where Φ_x cannot all be zero) is that the determinant (K − Φ_x²M) must be zero, and the characteristic equation can be expressed as follows:

$$\mathit{K}-{\omega }_x^2\mathit{M}=0$$

(11)

The eigenvalue ${\omega }_x^2$ can be solved by Equation (11), and by substituting ${\omega }_x^2$ into Equation (9), the corresponding eigenvector Φ_x can be obtained.

2.3. Simulation Conditions

The setting of the computational domain (CD) model for the fluid domain needs to be performed before setting the boundary conditions of the FSC field. The selection of the physical model is for static steady-state analysis. The selected fluid in the fluid domain of the DCSTPS is air with a constant density of 1.18415 kg/m³. The turbulence model of the DCSTPS is the standard K-Epsilon model. Since the fluid flows at low speed in this fluid domain, a separation solver was used to set simulation conditions in the CD of the DCSTPS.

(1)

Inlet of the DCSTPS: We consider that the fluid flow in this area of the DCSTPS is incompressible, and the initial wind speed is divided into constant wind speeds of 20 m/s, 22 m/s, and 25 m/s.

(2)

Boundary condition at the outlet of the DCSTPS: The boundary condition at the outlet is the pressure outlet, and the pressure is set to one standard atmospheric pressure, and the ambient temperature and ambient pressure are 298 K and 101.325 kPa, respectively.

(3)

Wall conditions in the CD of the DCSTPS: The radius of the DSC is 8.42 m and the maximum orthographic projection area is 222.5102 m². The boundary conditions of the DCSTPS between the bottom of the CD and the surface of the DSC of the DCSTPS are set to smooth wall, and there is no velocity slip between the viscous fluids and smooth wall. The surface and the ground of the DSC are fixed and do not move, so a non-slip wall condition is set, and slip boundary conditions of the DCSTPS are used at the top, the front, and the back of the CD of the DCSTPS.

2.4. Grid Division and Grid Independence Verification

The DCSTPS is located in the atmospheric boundary layer. When wind flows around the DCSTPS, it is equivalent to the DCSTPS being in a completely open flow field. The wind loads effects on the DCSTPS have a definite range in the DCSTPS, so a definite three-dimensional calculation area in the DCSTPS and corresponding boundary conditions in the numerical simulation of the DCSTPS need to be given. An excessively large fluid region means an excessive number of grids, resulting in a larger computational load and longer computation time. On the other hand, too small a fluid region leads to insufficient flow development and can easily cause distortion of the calculation results. So, it is important to designate a suitable fluid calculation area to reduce the calculation time and ensure calculation accuracy.

The diameter of the concentrator in the DCSTPS is about 17 m and its thickness is 0.27 m. To ensure that the fluid flow in the CD of the DCSTPS reaches a fully developed state, the length of the CD of the DCSTPS is about 10 times the size of the concentrator of the DCSTPS, and the height and the width of the CD of the DCSTPS are about 5 times the size of the concentrator of the DCSTPS. After multiple modeling simulations, 170 × 80 × 80 m is set to be the size of the CD of the DCSTPS, with the center of the DCSTPS at a height of 10 m from the ground of the DCSTPS and the inlet of the DCSTPS at a distance of 55 m from the center of the DCSTPS. The finite element model of the CD for the altitude angle α = 0° and azimuth angle β = 0° of the DSC in the DCSTPS is shown in Figure 1.

The calculation area is divided into three encryption zones using a step-by-step encryption method. As shown in Figure 2, the zones were gradually encrypted from far to near, maintaining a certain degree of continuity. The DCSTPS was individually encrypted to ensure the quality of the CD’s mesh. In order to enhance the accuracy of the calculation results, the polyhedral grids were adopted in the CD of the DCSTPS, which will make it easy to perform grid adaptation in the CD of the DCSTPS.

For balancing and meeting the accuracy requirement and computational complexity of CD of the DCSTPS, the grid independence verification must be performed to determine the most suitable grid number for simulation calculations. Figure 3 reveals the change trend of the drag coefficient of the DCSTPS at a wind speed of 20 m/s under different grid numbers. At 1,500,000 grids, the change rate of the drag coefficient is less than 1.0%, and the variation tends to be stable. Therefore, it can be considered that the simulation calculation can converge under this grid number condition. Therefore, under the premise of ensuring computational accuracy, in this work, as shown in Figure 2c, the total number of grids is 1,538,715 under this condition.

2.5. Model Validity Verification

To ensure the accuracy of the simulation research, it is necessary to verify the validity of the established model. Selecting experimental data from Uzair et al. [39], the drag coefficient and lift coefficient of the DCSTPS were compared at an azimuth angle of 0° with previous research data, and we observed whether the changes in the two coefficients with altitude angle were consistent. Figure 4a,b show a comparison of the results of the drag coefficient and lift force coefficient in this work with previous studies [39], and the results show that the behavior of the two coefficients with respect to altitude angle is consistent with previous research results from Uzair et al. [39]. The drag coefficient decreases uniformly and reaches its maximum at an altitude angle of 0°, while the lift force coefficient reaches its trough value at an altitude angle of 60°. Also, deviations between the simulation results and Uzair’s experimental data were 4.91–6.69% for drag coefficient and 3.92–7.28% for lift force coefficient.

Potential causes include the following: Due to the differences in the size and shape of the research object compared with previous studies from Uzair et al. [35], and the simplification of the model during modeling; there are no components such as the base, collector, or other supporting trusses. The simplified model base and other structures are the result of streamlining, which can reduce the wind load coefficient of the entire system to a certain extent. Therefore, the wind load coefficient in this paper is greater in absolute value than in previous studies. Therefore, it can be considered that the established computational model is valid.

3. Results and Discussion

The wind load was mapped onto the surface of the concentrator of the DCSTPS using the sequential coupling method. The specific operation process was expressed as follows: (1) extracting the boundary of the concentrator that needs to be coupled in the finite element analysis model; (2) importing it into the computational fluid dynamics model and making the fluid boundary of the concentrator basically coincide with the structural boundary; (3) coupling the pressure data obtained from the above calculation to each node element on the surface of the concentrator; (4) exporting the INP file back to the finite element analysis model for strength calculation.

3.1. Static Analysis of the Dish Concentrating Solar Thermal Power System

Conducting static analysis on the DCSTPS can help evaluate its structural performance, stability, and safety, and ensure that the DCSTPS can meet design and lifespan requirements under actual working conditions. By analyzing the load situation, reasonable structural optimization can be carried out to evenly distribute the stress and deformation of different components of the DCSTPS, thereby improving the overall performance and lifespan of the DCSTPS.

To obtain the stress concentration and location in the DCSTPS model, without considering the influence of wind load on the model, the main loads on the model are the gravity of the DCSTPS and the pressure of the Stirling generator on the cantilever structure of the DCSTPS.

The gravity effect of the DCSTPS can be obtained by defining the material properties and adding gravitational acceleration. The Stirling generator weighs about 1500 kg, so a downward pressure of 15.0 kN is applied at the connection between the cantilever structure and the Stirling generator to simulate the influence of the gravity of the Stirling generator on the entire model.

In order to balance the gravity effect on both sides of the supporting components of the DCSTPS, the structure of the balance box was redesigned. Unlike the original structure, which balances the torque produced by the Stirling generator, concentrator, and supporting structure through counterweights, the balance box can flexibly and effectively balance the torque under various working conditions by injecting different medium masses, which can effectively reduce the vibration of the overall structure under wind load and provide new ideas for subsequent anti-vibration research in this project.

The DCSTPS has many daily operating conditions, but under the maximum wind load, the DCSTPS is generally at an altitude angle (vertical rotation angle) of 0° and an azimuth angle (horizontal rotation angle) of 0°. Therefore, only the DCSTPS with an altitude angle of 0° and an azimuth angle of 0° is subjected to static analysis.

According to the specific working conditions, for the structure of the balance box, the maximum torque that needs to be balanced is about 4 × 10⁶ N·m, so the force that needs to be balanced is about 300 kN. Therefore, at intervals of 50 kN, the balance forces of 50 kN, 100 kN, 150 kN, 200 kN, 250 kN, and 300 kN were applied, totaling six working conditions, to verify the balance effect of the structure design of the balance box. Figure 5 shows the displacement cloud map of the DCSTPS under various operating conditions, and Figure 6 shows the maximum displacement of the DCSTPS under various operating conditions.

Under the condition of an altitude angle of 0 ° and an azimuth angle of 0°, based on Figure 5 and Figure 6, the following analysis and explanation can be performed:

(1) The maximum displacement of the DCSTPS is located at the connection between the cantilever and the Stirling generator, while the maximum deformation of the support structure is mainly distributed at the edges of the upper and lower parts of the concentrator. (2) Applying different balance forces to the balance box has a significant impact on the overall deformation of the DCSTPS model, and the maximum deformation of the DCSTPS is 0.75847 mm, 1.25763 mm, 1.88587 mm, 2.93343 mm, 4.51384 mm, and 5.23154 mm. This proves that in practical working conditions, this design can flexibly and effectively adjust the DCSTPS’s stress, which is beneficial for balancing the DCSTPS’s stress under different working conditions and increasing the safety and stability of the DCSTPS structure.

3.2. Modal Analysis

A dual-axis tracking system is adopted in the dish solar thermal power generation system under the daily operating conditions, and generally the maximum wind load altitude angle α = 0° and the azimuth angle β = 0°. Therefore, due to the fact that the DCSTPS is subjected to the maximum wind load, a modal analysis of the DCSTPS at an altitude angle α = 0° and an azimuth angle β = 0° was calculated in this work.

When calculating the natural frequency and completing the corresponding modal analysis, considering that the high-order modes of the DCSTPS will not have a significant impact on the DCSTPS, only the first six modes of the DCSTPS were calculated.

The constraints to the bottom of the support column of the DCSTPS model were applied, and FSC was used to investigate the natural frequencies and maximum deformation corresponding to the first six modes of the DCSTPS. The comparisons of the modal diagrams, maximum deformations, and natural frequency errors of the DCSTPS with the first- to sixth-order modal shape under α = 0° and β = 0° are shown in Figure 7 and Figure 8, respectively.

Based on Figure 7, the following analysis and explanation can be performed: (1) the first modal shape of the DCSTPS is the rotation of the DCSTPS downwards around the X-axis; (2) the second modal shape is the rotation of the DCSTPS upwards around the X-axis; (3) the third modal shape is the rotation of the DCSTPS upwards around the X-axis; (4) the fourth modal shape is the rotation of the DCSTPS downwards around the X-axis; (5) the fifth modal shape is the rotation of the DCSTPS downwards around the X-axis; (6) the sixth modal shape is the rotation of the DCSTPS left and right around the Y-axis.

In addition, as shown in Figure 7c, the deformation of the entire DCSTPS is relatively large during the third-order modal vibration process, especially at the upper edge of the concentrator, where the deformation reaches 3.2076 mm, which makes it prone to damage; therefore, the stiffness of the DCSTPS needs to be improved. From Figure 7d, it can be seen that during the fourth mode vibration process, the three cantilever beams supporting the Stirling generator undergo significant deformation, and the part near the Stirling generator reached 5.4318 mm, which makes it prone to damage; therefore, the stiffness of the parts of the three cantilever beams near the Stirling generator needs to be increased.

Based on the natural frequencies and maximum deformation of the first- to sixth-order modal shapes shown in Figure 8, the following analysis and explanation can be performed:

(1)

Compared with previous experimental values [40], the relative error is above 2.514–7.167%, indicating that the design of the balance box will reduce the natural frequency of the DCSTPS to a certain extent. At the same time, there are small differences in the material, structure, and other parameters between the established DCSTPS model and the experimental model [26], resulting in a small relative error, but the trend fluctuation of this relative error is not significant, which also proves the validity of the established DCSTPS model.

(2)

The fundamental natural frequency of the DCSTPS structure is 0.67910 Hz, and the natural frequency is much higher than the dominant frequency of the pulsating wind load (0.001~0.01 Hz), so the DCSTPS structure will not experience resonance phenomenon.

(3)

The most strongly vibrating parts of the DCSTPS structure are the edge area of the concentrator grid and the front and rear ends of the truss. The individual positions with protruding deformation are caused by an uneven distribution of model vibration, which belongs to the design problem of the DCSTPS model structure and has no impact on the overall mode of the DCSTPS in practice.

(4)

According to the modal analysis, the subsequent safety evaluation of the wind-induced vibration characteristic parameters of the DCSTPS model should focus on analyzing the edge position of the concentrator and the position with the strongest natural vibration, which will provide a basis for node selection.

3.3. Effects of Fluid–Structure Coupling on Natural Frequency and Maximum Deformation

When considering the effect of FSC and not considering the effect of FSC, the 1st–6th-order modal shapes of the DCSTPS are basically the same, but there are significant differences in the 1st–6th-order natural frequencies and their maximum deformations. Moreover, there is little effect on the modal vibration of the DCSTPS from the constant wind speed.

As for ith modal shape, the natural frequencies and maximum deformations of the DCSTPS while considering the effect of the FSC are f_si and δ_si, respectively, and the natural frequencies and maximum deformations of the DCSTPS without considering the effect of FSC are f_ci and δ_ci. The relative errors of the natural frequencies and maximum deformations of the DCSTPS are η_f = (f_si − f_ci) × 100%/f_ci and η_δ = (δ_si − δ_ci) × 100%/δ_ci, respectively.

A comparison of the natural frequencies and maximum deformations at the 1–6-order modal under α = 0° and β = 0° are shown in Figure 9.

According to Figure 9a, it can be seen that the 1st–6th-order natural frequencies of the DCSTPS considering the effect of FSC are all lower than those obtained without considering the effect of FSC, and the relative errors are −9.89%, −4.39%, −2.69%, −1.72%, −2.86%, and −2.26%, respectively. Additionally, considering the influence of FSC, the natural frequencies of all orders have decreased, with the first-order natural frequency showing the most significant decrease. For the above results, the main reason is that the fluid has a greater effect on the lower order natural frequency of the DCSTPS, while having a smaller effect on the higher order natural frequency of the DCSTPS; therefore, FSC has the effect of suppressing the natural mode coefficient of the DCSTPS.

In summary, to avoid the premature failure of the DCSTPS due to its resonance from external excitation, the influence of FSC on the change in natural frequencies should not be ignored.

According to Figure 9b, for the 1st to 6th-order modal, compared with the maximum deformation of the DCSTPS obtained without considering the effects of the FSC, the relative errors of the maximum deformation of the DCSTPS considering the effects of the FSC are 9.27%, 6.41%, 3.63%, 4.53%, 3.50, and 3.59%, respectively. It is evident that the relative error of the maximum deformation of the DCSTPS decreases overall with the increase in the order.

This is mainly because fluid impact has the effect of suppressing the higher order modal deformation of the DCSTPS. The fluid has a greater impact on the maximum deformation of the DCSTPS at a lower order modal, while having a smaller impact on the maximum deformation of the DCSTPS at a higher order modal.

3.4. Safety Grade Evaluation of the Dish Concentrating Solar Thermal Power System Based on Catastrophe Modeling

In this work, the catastrophe modeling (for example, cusp point type and dovetail type) [38,41], as shown in

Table 1. Cusp point-type model and dovetail-type model.

Type of Catastrophe	Control Variable	State Variable	Primary Potential Functions	Divergence Equation
Cusp point-type model	2	1	V(x) = x4 + Ux2 + Vx	U = 6x2, V = 8x3
Dovetail-type model	3	1	V(x) = x5 + Ux3 + Vx2 + Wx	U = 6x2, V = 8x3, W = 3x4

, will be used to evaluate the safety grade of the DCSTPS.

(1)

Evaluation standards for safety grade of the DCSTPS

The safety grade of the DCSTPS was divided into four categories, low safety grade of the DCSTPS, average safety grade of the DCSTPS, high safety grade of the DCSTPS, and very safety high level of the DCSTPS, and the evaluation criteria of the safety grade were [0.00, 0.35], [0.35, 0.60], [0.60, 0.85], and [0.85, 1.00].

(2)

Safety grades of the DCSTPS normalized by Catastrophe models

Considering that the raw data of the different operating condition parameters in the DCSTPS is of generally different significance, it is meaningless to compare the raw data of the different operating condition parameters directly. Therefore, after the raw data of the different operating condition parameters in the DCSTPS is normalized in the interval [0, 1] by catastrophe models, the safety grade of the DCSTPS can be evaluated based on the catastrophe models and the evaluation criteria of the safety grade.

The safety evaluation index system of the DCSTPS is divided into wind load, force load, and moment load, which are defined by u, v, and w, respectively. The safety evaluation index system of the DCSTPS is a first-level indicator, namely, the target layer. Wind load (u), force load (v), and moment load (w) are the secondary indicator layer. Wind load (u) includes two third indicator layers such as the maximum surface wind pressure (u₁) and the air velocity (u₂); force load (v) includes three third indicator layers such as the lateral force (v₁), the drag (v₂), and the lift force (v₃); and moment load (w) includes three third indicator layers such as the pitch moment (w₁), rolling moment (w₂), and azimuth moment (w₃). Among them, the parameters of all indicators in the same layer were sorted from top to bottom based on their importance as shown in

Table 2. Safety evaluation index system of the DCSTPS.

Target Layer	Secondary Indicator Layer	Third Indicator Layer
Dish concentrating solar thermal power system safety evaluation index system	Wind load, u	Maximum surface wind pressure p (Pa), u1
		Air velocity (m/s), u2
	Force load, v	Lateral force (kN), v1
		Drag (kN), v2
		Lift force (kN), v3
	Moment load, w	Pitch moment (N·m), w1
		Rolling moment (N·m), w2
		Azimuth moment (N·m), w3

.

Considering that Figure 1c provides these angles and their relative positional relationship (altitude angle α and azimuth angle β), simulation cases for the safety performance of the DCSTPS are shown in

Table 3. Simulation cases for the safety performance of the DCSTPS.

Case	Altitude Angle α/°	Azimuth Angle β/°
1	30	90
2	60	90
3	90	135
4	30	180
5	60	180
6	0	180
7	0	0

. Based on the simulation results, the safety performance of the DCSTPS will replace the evaluation scores as in

Table 4. Simulation results for the safety performance of the DCSTPS.

Case	u1/Pa	u2/m·s−1	v1/N	v2/N	v3/N	w1/N·m	w2/N·m	w3/N·m
1	3.38 × 102	20.0	1.46 × 104	4.60 × 103	−8.46 × 103	4.50 × 104	−8.24 × 102	7.80 × 104
2	3.30 × 102	20.0	8.47 × 103	4.65 × 103	−1.47 × 104	7.83 × 104	−45.8	4.52 × 104
3	3.84 × 102	22.0	−6.46	5.80 × 103	−1.82 × 105	1.13 × 105	−1.45 × 102	11.7
4	3.91 × 102	22.0	4.06	8.20 × 104	4.09 × 104	1.07 × 105	50.7	68. 9
5	4.73 × 102	25.0	−22.4	4.24 × 104	5.50 × 104	1.76 × 105	−4.07 × 102	−2.61 × 102
6	4.96 × 102	25.0	−84.0	1.28 × 105	2.80 × 103	−5.60 × 104	−16.3	−1.80 × 103
7	4.97 × 102	25.0	21.9	1.55 × 105	−1.83 × 103	−4.16 × 104	−29.4	−4.03 × 102

.

Due to the negative correlation between safety performance and the indicators u₁, u₂, v₁, v₂, v₃, w₁, w₂, and w₃ of the DCSTP, the indicators u₁, u₂ in the third indicator layers were normalized using Equation (11), and the indicators v₁, v₂, v₃, w₁, w₂, and w₃ in the third indicator layers were normalized using Equation (12).

$${\displaystyle \frac{\min \left|X_i\right|}{\left|X_i\right|}}$$

(12)

The cusp point-type model for the normalized formula is

$$\left\{\begin{array}{l}{x}_u=\sqrt{u}\\ x_v=\sqrt[3]{v}\end{array}\right.$$

(13)

The dovetail-type model for the normalized formula is

$$\left\{\begin{array}{l}{x}_u=\sqrt{u}\\ x_v=\sqrt[3]{v}\\ x_w=\sqrt[4]{w}\end{array}\right.$$

(14)

According to the safety evaluation index system of the DCSTPS established in Table 2, the normalized results on safety evaluation index system of the DCSTPS are given in

Table 5. The normalized results on safety evaluation index system of the DCSTPS.

Case	u1 (−)	u2 (−)	v1 (−)	v2 (−)	v3 (−)	w1 (−)	w2 (−)	w3 (−)
1	0.9755	1.0000	0.0003	1.0000	0.2160	0.9242	0.0198	0.0002
2	1.0000	1.0000	0.0005	0.9894	0.1246	0.5314	0.3557	0.0003
3	0.8579	0.9091	0.6285	0.7935	0.1003	0.3674	0.1119	1.0000
4	0.8424	0.9091	1.0000	0.0561	0.0446	0.3883	0.3210	0.1702
5	0.6976	0.8000	0.1812	0.1085	0.0332	0.2363	0.0400	0.0450
6	0.6643	0.8000	0.0483	0.0360	0.6537	0.7436	1.0000	0.0065
7	0.6640	0.8000	0.1851	0.0297	1.0000	1.0000	0.5530	0.0291

.

As for the third indicator layer in case 1, the indicators u₁, u₂, v₁, v₂, v₃, w₁, w₂, and w₃ of the DCSTP were normalized using the cusp point-type model and dovetail-type model.

$$\left\{\begin{array}{l}{x}_{u1}=\sqrt{u_1}=\sqrt{0.9755}=0.9877\\ x_{u2}=\sqrt[3]{u_2}=\sqrt[3]{1.0000}=1.0000\end{array}\right.$$

$$\left\{\begin{array}{l}{x}_{v1}=\sqrt{v_1}=\sqrt{0.0003}=0.0173\\ x_{v2}=\sqrt[3]{v_2}=\sqrt[3]{1.0000}=1.0000\\ x_{v3}=\sqrt[4]{v_3}=\sqrt[4]{0.2160}=0.6817\end{array}\right.$$

$$\left\{\begin{array}{l}{x}_{w1}=\sqrt{w_1}=\sqrt{0.9242}=0.9614\\ x_{w2}=\sqrt[3]{w_2}=\sqrt[3]{0.0198}=0.2705\\ x_{w3}=\sqrt[4]{w_3}=\sqrt[4]{0.0002}=0.1189\end{array}\right.$$

Considering that there is no interconnection between the u₁, u₂, v₁, v₂, v₃, w₁, w₂, and w₃ of the DCSTP, the minimum normalized value was selected as the value of the secondary indicator layer in the DCSTP.

(1)

For case 1: x_u = 0.9877; x_v = 0.0173; x_w = 0.1189.

(2)

Similarly, for case 2: x_u = 1.0000, x_v = 0.0224, x_w = 0.1316; case 3: x_u = 0.9262, x_v = 0.5628, x_w = 0.4819; case 4: x_u = 0.9178, x_v = 0.3828, x_w = 0.6231; case 5: x_u = 0.8352, x_v = 0.4257, x_w = 0.3420; case 6: x_u = 0.8150, x_v = 0.2198, x_w = 0.2839; and case 7: x_u = 0.8149, x_v = 0.3097, x_w = 0.4130.

The safety evaluation values of the indicator grade of the DCSTPS are expressed in Figure 10.

Wind load, force load, are moment load are normalized according to Equation (14). Since wind load, force load, and moment load cannot be substitutable with each other, taking case 1 as an example, wind load (x_u = 0.9877), force load (x_v = 0.0173), and moment load (x_w = 0.1189) are normalized using a dovetail-type model to obtain

$$\left\{\begin{array}{l}{x}_1=\sqrt{x_u}=\sqrt{0.9877}=0.9938\\ x_2=\sqrt[3]{x_v}=\sqrt[3]{0.0173}=0.2586\\ x_3=\sqrt[4]{x_w}=\sqrt[4]{0.1189}=0.5872\end{array}\right.$$

Therefore, the minimum value for the {0.9938, 0.2586, 0.5872} was chosen as the safety evaluation value of the target grade of the DCSTPS as follows:

x = min{0.9938 0.2586 0.5872} = 0.2586

Similarly, based on the above calculation process, the other safety evaluation values of the target grade of the DCSTPS can also be obtained as case 2: x = 0.2819; case 3: x = 0.8256; case 4: x = 0.7261; case 5: x = 0.7523; case 6: x = 0.6035; and case 7: x = 0.6766. Thus, the safety evaluation values of the target level of the DCSTPS can be expressed as in Figure 11.

From the above, it can be seen that the safety evaluation values of the target grade are 0.2586, 0.2819, 0.8256, 0.7261, 0.7523, 0.6035, and 0.6766.

Due to the established evaluation criteria of the safety grade, the results indicate that the safety evaluation values of the DCSTPS were 0.2586 and 0.2819 under case 1 and case 2, which were in the low safety grade, and the safety evaluation values of the DCSTPS under case 3, case 4, case 5, case 6, and case 7 are in the high safety grade. Among them, the highest safety evaluation value of the DCSTPS is found in case 3 in relation to the other cases.

Figure 12 shows the calculation results for the stress on the connection seat of the altitude angle in the DCSTPS and the steering device of the base in the DCSTPS under the seven operating conditions. It can be seen that the maximum stress on the connection seat of the altitude angle under the seven operating conditions is 246.81 MPa, and the maximum stress on the steering device of the base is 169.5 MPa.

The material used in the connection seat of the altitude angle in the DCSTPS is Q345 and the material used in the steering device of the base in the DCSTPS is Q235. The yield limit of Q345 is σ_s1 = 345 MPa, the yield limit of Q235 is σ_s2 = 235 MPa, and the safety factor of the material is γ = 1.375. Therefore, the allowable stresses of Q345 and Q235 are [σ₁] = σ_s1/γ = 250.9 MPa and [σ₂] = σ_s2/γ = 170.9 MPa, respectively.

According to the maximum stress values of σ_max1 = 246.81 MPa < [σ₁] = 250.9 MPa and σ_max2 = 169.5 MPa < [σ₂] = 170.9 MPa, the maximum stress value on the connection seat of the altitude angle and the steering device of the base under working condition 1 is close to the allowable stress of Q345 and Q235, and there is a high possibility of the plastic deformation or the failure during operation. In future research, design optimization will be carried out to minimize the stress concentration near the connection seat of the altitude angle and the steering device of the base.

4. Conclusions

(1)

According to the static analysis results, applying different forces to the balance box has a significant impact on the overall deformation of the DCSTPS model. Under α = 0° and β = 0°, the maximum displacement of the DCSTPS is basically at the connection between the cantilever and the Stirling generator, while the deformation of the support structure is mainly at the edges of the upper and lower parts of the concentrator.

(2)

According to the modal analysis results, the most strongly vibrating parts of the DCSTPS are the edge area of the concentrator grid and the front and rear ends of the truss grid. The fundamental natural frequency of the DCSTPS is 0.67910 Hz, and the natural frequency of the DCSTPS is much higher than the dominant frequency of the pulsating wind load (0.001~0.01 Hz), so the DCSTPS will not experience strong resonance phenomena.

(3)

The absolute values of the relative errors of the natural frequency of the DCSTPS and the maximum deformation of the DCSTPS decrease with the increase in the modal order considering the effect of FSC. This is mainly because the FSC has the effect of suppressing the deformation of higher order modes of the DCSTPS. Therefore, the fluid has a greater effect on the maximum deformation of a lower order DCSTPS, while having a smaller effect on the maximum deformation of a higher order DCSTPS. Additionally, considering the effect of FSC, the natural frequencies of all orders decrease, with the first-order natural frequency showing the most significant decrease.

(4)

Taking the safety grade evaluation index of the DCSTPS as the target layer, wind load, force load, and moment load as the criterion layer, and establishing relevant index layers below the criterion layer, a safety grade evaluation index system for the DCSTPS was ultimately established. On this basis, a safety grade evaluation of the DCSTPS was conducted based on the catastrophe theory. The results showed that the safety grade of the DCSTPS was 0.2586 and 0.2819 under case 1 and case 2. Through analysis and calculation, it was found that the membership value of the moment load was low, resulting in the stress on the connection seat of the altitude angle and the steering device of the base approaching the allowable stress of the material. This indicates that there are weak links in the system’s structural design, and such safety issues should be avoided in future design optimization work.