Dung beetle optimization for probabilistic force analysis of heliostat support structures

Abstract

In tower-concentrated solar power plants, heliostats are typically anchored through freestanding columnar pylon systems across planar terrains. These mirror assemblies sustain significant wind-induced dynamic loads during operation, resulting in critical mechanical responses within their support structures. This study conducts a systematic investigation into the stochastic distribution characteristics of mechanical parameters in heliostat support systems through multi-condition numerical simulations. By developing intelligent optimization algorithm models such as Dung Beetle Optimizer (DBO), we achieve efficient computation of kurtosis and skewness coefficients for force parameters, with model performance rigorously evaluated through key metrics. The operational conditions are classified into Gaussian/non-Gaussian distribution categories based on computational results and established criteria, elucidating the fundamental mechanisms underlying Gaussian and non-Gaussian force characteristics in support structures under specific working conditions. These findings provide theoretical guidance for optimizing wind-resistant structural design frameworks.

Introduction

Conventional practice in structural wind engineering assumes fluctuating aerodynamic loads adhere to a Gaussian probability model. Yet investigations reveal pronounced non-Gaussian distribution patterns across specific zones of heliostat reflector surfaces. Employing Gaussian approximations for these regions introduces substantial inaccuracies in pressure estimations, compromising structural wind resistance integrity. Computational modeling of transient wind pressures demands resource-intensive transient analysis, presenting significant efficiency constraints. Consequently, integrating empirically derived formulations with wind tunnel validation emerges as a pragmatic methodology for characterizing pressure distribution statistics.

Force measurement studies on heliostat assemblies represent a fundamental aspect of securing the efficient and dependable performance of solar power installations. Wind tunnel experimentation is integral to accurately quantifying aerodynamic loads on heliostats, with pioneering investigations adapting methodologies previously applied to building structure force assessments. Seminal research by Peterka (1988, 1992) provided critical insights into the aggregate wind forces acting upon heliostat configurations. However, inherent constraints in the experimental capabilities of that era led Peterka’s work to rely upon idealized theoretical models. Consequently, these models disregarded the aerodynamic influence of key structural components during wind tunnel force measurement testing^1,2. Poulain P investigated the influence of gap size between heliostat panels in an isolated configuration on wind loads, finding that increasing the gap width led to an overall increase in aerodynamic force and moment coefficients³.

Beyond flow characteristics, structural configuration parameters exert a significant influence on wind-induced loads and corresponding force coefficients. Through high-frequency force balance experimentation conducted on heliostat assemblies, Emes successfully derived peak aerodynamic load coefficients via the Equivalent Static Wind Load methodology⁴. Further analysis revealed that atmospheric turbulence phenomena induce marked spatial non-uniformity in pressure distribution across the heliostat surface. This localized pressure variation substantially amplifies bending moments experienced at critical support locations, specifically the hinge and base interfaces. Supporting these findings, shifts in the aerodynamic pressure center location directly correspond with peak hinge moment magnitudes. Crucially, operational moments for specific heliostat configurations remain beneath predetermined stow thresholds at defined terrain wind speeds, thereby providing essential criteria for optimizing configuration selection and operational wind speed limits⁵. In research utilizing field measurements within the atmospheric boundary layer, Arjomandi experimentally verified that heliostat wind loads exhibit strong dependency on specific critical scale parameters and prevailing turbulence intensity levels⁶. He also compares wind tunnel and field measurements of ABL turbulence effects on heliostat aerodynamic loads. Results reveal wind tunnels overestimate drag, underestimate lift coefficients, and field data exhibit non-Gaussian skewness, highlighting turbulence anisotropy sensitivity for low-altitude (<5m) load predictions⁷. Bakhshipour S conducted wind tunnel experiments to analyze the effects of aspect ratio (AR) and ground clearance ratio (GR) of a solar mirror on the wind load coefficient. The results showed that AR increased the hinge/overturning moment by up to 55%, while GR increased the overturning/hinge moment by 40% and 30%, respectively⁸. Yu JS examines wind loads on tandem heliostats, finding 60% lower peak drag but sevenfold higher hinge moments on downstream mirrors at 90°elevation with two-chord spacing, and tripled pressure fluctuation frequencies, requiring distinct static and dynamic design considerations⁹. Sosa-Flores P uses large eddy simulation to analyze how four rear geometries of a square heliostat affect stow-position aerodynamic load coefficients, revealing peak drag differences over 128%, lift variations up to 219%, and a direct correlation between chord-to-vertical-distance ratios and load coefficient changes¹⁰.

There are also studies on force coefficients in numerical simulations. Poulain P employs the Scale-Adaptive Simulation (SAS) model to numerically simulate the aerodynamic characteristics of an isolated heliostat under adverse wind conditions, successfully predicting both mean and peak drag forces¹¹. Blume K introduces an easy-to-apply pressure measurement system for full-scale heliostats, with field tests revealing stable mean pressure coefficients aligned with wind tunnel data, and aerodynamic admittance analysis showing reduced effectiveness of small eddies in generating loads compared to large-scale vortices¹².

In recent years, intelligent optimization algorithms have been widely applied in different engineering fields. The Dung Beetle Optimizer (DBO), as a swarm intelligence optimization algorithm, demonstrates strong optimization capabilities and rapid convergence characteristics. Zhu F conducted comprehensive comparisons between the DBO and six other swarm intelligence algorithms using 37 benchmark functions and practical engineering applications. Experimental results demonstrate that the enhanced Dung Beetle Optimizer significantly improves convergence rate and optimization accuracy while maintaining strong robustness¹³. In mechanical engine performance prediction, a neural network optimized by the Dung Beetle Optimizer achieved a correlation coefficient exceeding 95%¹⁴. This demonstrates DBO’s efficacy in identifying high-correlation model configurations, even when physical parameters are highly coupled and training data is simulation-generated. In fault diagnosis, an enhanced DBO was applied to jointly optimize Variational Mode Decomposition (VMD) parameters and a Deep Belief Network-Extreme Learning Machine (DBN-ELM) network. This approach mitigated signal interference and weak fault characteristics, achieving a diagnostic accuracy of 99.87% for control valve faults under small opening conditions¹⁵. This highlights DBO’s role in optimizing pre-feature-extraction signal decomposition and fine-tuning classifier parameters. Furthermore, in resource-constrained embedded wireless sensing, a hybrid strategy using Particle Swarm Optimization (PSO) for global initialization and DBO for local refinement improved both the convergence speed and prediction accuracy of a Backpropagation (BP) network. The model yielded correlation coefficients above 0.997 and a mean relative error below 0.25% for various combustible gases¹⁶. This indicates that combining DBO with other swarm algorithms effectively balances global exploration and local exploitation, making it suitable for highly nonlinear problems with significant sensor crosstalk. The frequent reliance on modified or hybrid versions of DBO suggests that the original algorithm may have limitations in convergence speed and global exploration within high-dimensional, non-convex hyperparameter spaces. Enhancements like sampled initialization, chaotic perturbation, or heterogeneous operators are often needed to improve its diversity and ability to escape local optima^17,18.

Heliostat wind load studies reveal that non-Gaussian pressure distributions challenge traditional Gaussian models, necessitating integrated experimental and numerical approaches for improved predictions. Early studies by Peterka et al. were constrained by experimental limitations and overlooked the complexity of heliostat support structures, such as crossbeams and columns, leading to inadequate characterization of probabilistic behavior under certain operational conditions. This study addresses these gaps by employing the Dung Beetle Optimizer algorithm to enhance wind tunnel testing methods. Through efficient multi-condition numerical simulations, DBO enables precise calculation of skewness and kurtosis coefficients, establishing a data-driven Gaussian/Non-Gaussian criterion. Importantly, DBO quantifies the influence of operational parameters (elevation, azimuth, wind velocity) on vortex coherence, systematically revealing opposing distribution trends, drag force (Fx) exhibits Gaussian behavior at low elevation angles but transitions to non-Gaussian at high angles, while overturning moment (My) follows the inverse pattern. This finding rectifies key mechanisms missed by earlier studies, which ignored structural complexity. Practically, DBO’s rapid convergence and robustness allow precise identification of critical conditions inducing non-Gaussian moments, providing clear guidance for wind-resistant design by specifying when to use computationally efficient Gaussian models versus higher-order non-Gaussian models, thereby optimizing the balance between safety and economy.

Experiments and methods

This experiment’s heliostat is located in a northwestern Chinese county. The flat, open site, mainly covered in natural grassland with scattered wind-sculpted sand pits, necessitated atmospheric boundary layer simulations derived from actual northwest regional wind field measurements. Direct on-site wind assessments were performed using specialized measurement tools. The measurement location is surrounded by large-scale PV arrays on its western, eastern, and southern sides, contrasting with the desert terrain to the north. The deployed wind measurement system integrates a tower, anemometers, wind vanes, and data acquisition hardware. Figure 1 shows the installation of a ~10-meter wind measurement tower, fitted with cup anemometers and low-inertia wind vane anemometers at 1.4-meter height increments from the base to record vertical wind profiles. For wind field testing, a Chaoyu CYD9100 data acquisition unit (32Hz sampling frequency, Table 1) was employed. Boasting 16 data channels and straightforward operation, this device excels in high-speed acquisition environments.

Fig. 1

On-site measurement system.

Table 1 Specification of the actual measuring instrument.

The on-site measurement collected wind field data over a total duration of 87 hours and 10 minutes. The wind rose diagram is shown in Fig. 2. Following the time interval classification criteria specified in China’s wind load code, the measured data were segmented into sub-samples using 10-minute averaging intervals. To avoid analytical bias caused by non-stationarity in wind field time series, the run test method was employed for stationarity assessment. At a significance level of 0.05, each sub-sample underwent stationarity verification, with non-stationary wind field samples being excluded. Statistical analysis of wind field characteristic parameters was then conducted on the validated wind speed and direction samples. All stationarity-validated measurement sub-samples were filtered according to the criterion that the 10-minute average wind speed at 10m height should not be lower than 5m/s (approximately equivalent to wind force level 4 or above), resulting in 150 valid samples for subsequent statistical analysis of wind field characteristics. The maximum instantaneous wind speeds recorded at heights of 10.0m, 8.4m, 7.0m, 5.6m, 4.2m, 2.8m, and 1.4m above ground were 16.917m/s, 15.300m/s, 14.546m/s, 14.241m/s, 13.886m/s, 13.102m/s, and 11.226m/s respectively.

Fig. 2

Wind rose diagram.

Achieving fidelity in force measurement outcomes within wind tunnel studies necessitates precise simulation of the target mean wind speed profile, an aspect deemed particularly critical. Within the atmospheric boundary layer context relevant to these experiments, the characteristic mean wind speed profile exhibits a specific functional form, commonly represented by the governing exponential function:

$$\frac{{\overline{v}(z)}}{{\overline{v}_{{\text{b}}} }} = \left( {\frac{z}{{z_{{\text{b}}} }}} \right)^{\alpha }$$

(1.1)

Where Z_b is the standard reference height, $\overline{v}_{{\text{b}}}$ is the mean wind speed at the standard reference height, Z is any height above the ground, $\overline{v}(z)$ is the wind speed at any height, and α is the ground roughness exponent. Given the measurement site’s location within an arid desert environment, terrain classification adheres to provisions specified in the Load Code for Design of Building Structures (GB50009-2012). Correlating these regulatory standards with empirical field measurements confirms designation as Terrain Category A, characterized by a surface roughness coefficient α = 0.12. This classification aligns with the observed topographical and meteorological conditions documented during on-site assessments.

Based on measurements of the northwest region’s wind field, the simulations shown in Figs 3, 4 and 5 depict the wind speed profile, turbulence intensity distribution, and power spectrum attained at the central turntable position within the wind tunnel facility. The prototype heliostat, with a height of approximately 6 meters, exhibits reduced measured wind speeds in lower-altitude regions of the wind profile due to the blocking effect. Given the field measurements conducted in a desert environment, surface roughness indices derived from terrain classifications under different national standards were used to construct turbulence intensity profiles. Turbulence intensity vertical distributions undergo comprehensive validation against internationally recognized benchmarks, including Chinese regulatory standards (GB50009)¹⁹, Japanese architectural guidelines (AIJ-2004)²⁰, and British wind engineering specifications (ESDU 85020)²¹. This comparative framework further incorporates parallel assessment against experimental wind tunnel data and field-recorded atmospheric measurements to establish methodological robustness. The wind profiles calculated by various national codes exhibit similar trends, but differences arise due to variations in the defined parameters within the power law formulations. The turbulence intensity trends from wind tunnel tests and field measurements are generally consistent, with numerically close values. However, near the ground (1.5–4.5m), the turbulence intensity shows significant variability, and the field-measured turbulence intensities fall outside the error bars of the wind tunnel test results. This discrepancy is attributed to the obstruction effects caused by the heliostat arrays near the ground in the measurement site. In contrast, at higher elevations (4.5–12m), where wind speeds are greater, the field-measured turbulence intensities fully align within the error bars of the wind tunnel test data, indicating strong measurement consistency. The wind speed power spectrum derived from our analysis was benchmarked against several established theoretical frameworks, including the Von Karman, Kaimal, and Davenport models. Furthermore, it underwent comparison with empirical data sources, specifically power spectra obtained both from controlled wind tunnel experiments and from field measurements conducted on-site²².

Fig. 3

Wind profile observed in wind tunnel experiments.

Fig. 4

Turbulence profile.

Fig. 5

Wind speed power spectrum derived from wind tunnel experiments.

The experimental campaign was performed in Hunan University’s HD-3 atmospheric boundary layer wind tunnel (Fig. 6). This recirculating facility spans 14 meters in total length and incorporates a working section measuring 3.5 m width × 3 m height. A 1.8-meter diameter turntable permits parametric adjustment of wind direction through the full 360° azimuth range, while test section velocities can be precisely controlled within the 0–25 m/s operational envelope.

Fig. 6

Wind tunnel laboratory.

Critical infrastructure includes a retractable three-axis traversing mechanism supporting aerodynamic instrumentation. Modular arrays comprising grills, vortex-generating spires, flow-conditioning baffles, and surface roughness elements collectively enable terrain replication and wind field modulation. This integrated approach facilitates simulation of diverse site classifications and their corresponding ABL characteristics as dictated by experimental protocols. In this wind tunnel force measurement test, the height of the wedge is 1800mm, with a center-to-center spacing of 760mm between the two wedges. The distance from the wedge to the velocity measurement position is 10m, and the distance from the wedge to the heliostat is 11m. Two types of roughness elements were employed: large roughness elements are cubes with side lengths of 80mm, and small roughness elements are cubes with side lengths of 50mm. The center-to-center spacing between roughness elements in both front-rear and left-right directions is 300mm (Figs 7, 8).

Fig. 7

Force balance.

Fig. 8

Heliostat force measurement experiment and model.

Force measurements utilize the "ATI DAQ F/T USB" six-axis force/torque sensor manufactured by ATI (USA) (Fig. 7). This instrumentation resolves applied loads on the heliostat into orthogonal force vectors (Fx, Fy, Fz) and corresponding torque components (Mx, My, Mz) within its Cartesian reference frame (Table 2).

Table 2 Parameters of force balance.

Employing a geometric scaling factor of 1:50, the heliostat test model replicates full-scale prototype dimensions documented in Table 3. This reflective surface comprises a 5mm-thick glass panel measuring 232mm × 200mm, structurally reinforced by a 10mm-diameter circular hollow steel tube with 2mm wall thickness. Rising vertically 112mm from its base, this tubular support connects mechanically to the rotational axis assembly (Figs. 8, 9 and 10). Experimental reference heights were established at 20cm model scale, equating to 10m in prototype dimensions per scaling conventions.

Table 3 Heliostat prototype parameters.

Fig. 9

Heliostat force and moment directions.

Fig. 10

Heliostat model.

Experimental design parameters were identified per research objectives. Analysis of 2024 meteorological station data informed the operational framework. The heliostat’s operational envelope spans 0°–90° elevation (±15° solar tracking tolerance) and 0°–180° azimuth ranges. Consequently, azimuth and elevation were designated primary variables. Parameterization employed 15° azimuth increments (13 levels) and 10° elevation increments (10 levels), establishing a 130-condition test matrix (Table 4). Aerodynamic load quantification at each configuration was achieved via six-axis force measurements.

Table 4 Test condition design.

Pursuant to the methodologies outlined in the Guide for Wind Tunnel Testing of Buildings, aerodynamic coefficients are computationally defined through the following expressions²³. The formulas for drag coefficient (C_Fx), side force coefficient (C_Fy), lift coefficient (C_Fz), side moment coefficient (C_Mx), base overturning moment coefficient (C_My), and azimuth moment coefficient (C_Mz) are as follows:

$$C_{{F_{x} }} = \frac{{F_{x} }}{{q_{H} A}},\,\;C_{{F_{y} }} = \frac{{F_{y} }}{{q_{H} A}},\,\;C_{{F_{z} }} = \frac{{F_{z} }}{{q_{H} A}}$$

(1.2)

$$C_{{M_{x} }} = \frac{{M_{x} }}{{q_{H} AH}},\;\,C_{{M_{y} }} = \frac{{M_{y} }}{{q_{H} AH}},\,\;C_{{M_{z} }} = \frac{{M_{z} }}{{q_{H} AL}}$$

(1.3)

Where F_x, F_y, and F_z represent the mean values of drag, side force, and lift along the x, y, and z axes, respectively. M_x, M_y, and M_z represent the mean values of moments around the x, y, and z axes, respectively. $q_{H}$ is the reference wind pressure, in this case. For characterizing geometric scaling relationships, the reflective surface area of the heliostat model constitutes the characteristic area A. Similarly, the longitudinal dimension of the mirror panel defines the characteristic length L. In actual tests, C_Fx, C_Fz, and C_My values are relatively large, while C_Fy, C_Mx, and C_Mz values are relatively small and close to zero, so only the effects of F_x, F_z, and M_y on the heliostat need to be considered in the design.

Distribution characterization employs multi-order statistical moments as discriminators of Gaussian/non-Gaussian sample properties. For Gaussian-distributed samples, the probability density function (PDF) is defined exclusively by the first two moments. Non-Gaussian PDF require higher-order descriptors, skewness quantifies distribution asymmetry while kurtosis measures tail weight²⁴. The skewness coefficient and kurtosis coefficient are collectively referred to as the force coefficients. Their mathematical representations follow:

$$C_{sk} = \sum\limits_{i = 1}^{N} {\left[ {\left( {C_{pi} (t) - C_{pi,mean} } \right)/C_{pi,rms} } \right]^{3} } /N$$

(1.4)

$$C_{ku} = \sum\limits_{i = 1}^{N} {\left[ {\left( {C_{pi} (t) - C_{pi,mean} } \right)/C_{pi,rms} } \right]^{4} } /N$$

(1.5)

Within these statistical formulations, C_sk and C_ku respectively quantify distribution asymmetry (skewness) and tail weighting (kurtosis). The parameter C_pi,rms corresponds to temporal pressure fluctuation records, while C_pi,mean designates the arithmetic mean of sampled pressure data. Complementarily, C_pi,rms characterizes the root-mean-square magnitude of pressure oscillations about the mean value, providing a standardized measure of aerodynamic fluctuation intensity.

Analysis

Probabilistic characteristics of forces on the heliostat support structure

Establishing valid thresholds for skewness and kurtosis coefficients is prerequisite for Gaussian distribution diagnosis across structural typologies, with domain-specific criteria prescribed by Kumar for low-rise buildings (Gaussian when -0.5≤Cₛₖ≤0.5 and Cₖᵤ≤3.5)²⁵, defined by Sun for large-span roofs (Gaussian when |Cₛₖ|≤0.2 and Cₖᵤ≤3.7)²⁶, and established by Gong for heliostat surfaces (Gaussian when -0.5≤Cₛₖ≤0.5 and Cₖᵤ≤4.0)²⁷, where threshold deviations universally indicate non-Gaussian behavior per respective experimental validations.

The heliostat structure differs from conventional low-rise and large-span structures, and the force data on the heliostat support structure under wind action differs from wind pressure data. Therefore, direct application of the aforementioned criteria is not appropriate. A refined diagnostic framework must be established to evaluate Gaussian distribution compliance for wind-induced forces acting on heliostat support structures. Figure 11 delivers critical quantitative insights by plotting kurtosis coefficients (C_ku) against skewness parameters (C_sk), with the latter represented on the abscissa and former on the ordinate. This visualization establishes an intuitive discrimination criterion through spatial distribution mapping across 130 operational regimes, enabling rapid assessment of force probability characteristics.

Fig. 11

C_sk vs. C_ku relationship for forces on the heliostat support structure.

From Fig. 11, it can be seen that Gong Bo’s criteria for Gaussian distribution have the widest range for C_sk and C_ku, followed by Kumar’s, and Sun Ying’s criteria are the strictest. Under most test conditions, the C_sk and C_ku values for F_x and F_z fall within the Gaussian distribution ranges defined by all three criteria. For M_y, under most conditions, the C_sk and C_ku values fall within the Gaussian distribution ranges defined by Gong Bo and Kumar, and under about 40 conditions, they fall within Sun Ying’s criteria. However, for the probabilistic distributions of F_x, M_y, and F_z, further analysis using probability density histograms and time history curves is needed to accurately determine whether the data follows a Gaussian distribution and establish a more accurate criterion for judging whether the shear force, axial force, and bending moment data on the heliostat support structure follow a Gaussian distribution.

Criteria for Gaussian distribution of shear force, axial force, and bending moment

Criteria for Gaussian distribution of shear force

Comprehensive characterization of shear force (F_x) Gaussian distribution properties across 130 operational configurations was achieved through acquisition of time-history waveforms, probability density histograms, and skewness/kurtosis coefficients (C_sk and C_ku) for the heliostat support structure. Given manuscript length limitations, Table 5 presents a representative subset featuring comparative visualization of temporal force fluctuations, probability distributions, and corresponding moment statistics. This tabulated data synthesis employs integrated assessment of waveform morphology, histogram conformity, and moment parameter thresholds to definitively classify each distribution profile as Gaussian (G) or non-Gaussian (NG).

Table 5 C_sk and C_ku for shear force across.

From Table 5 and Fig. 12, it can be seen that the time history curves of shear force (F_x) on the support structure under conditions 45-0, 135-0, 180-0, and 180-30 are symmetrical, and their probability density histograms align well with the Gaussian distribution curve, indicating that the F_x data for these four conditions follows a Gaussian distribution. For the other 16 conditions, the time history curves of F_x show significant asymmetry, many large-amplitude pulse signals, or both. The probability density histograms of F_x show tailing compared to the Gaussian distribution curve, with peaks deviating to varying degrees, or they are sharper than the Gaussian distribution curve without tailing. Therefore, the F_x data for these 16 conditions follows a non-Gaussian distribution.

Fig. 12

Time history curves and probability density histograms of heliostat shear force.

Analyzing the C_sk and C_ku values under 20 operating conditions in Table 5 and Fig. 12, it is observed that in conditions judged as NG, C_sk and C_ku exhibit the following three scenarios: (1) C_sk < -0.2 or C_sk > 0.2, and C_ku ≤ 3.2; (2) C_sk = -0.2 to 0.2 and C_ku > 3.2; (3) C_sk < -0.2 or C_sk > 0.2, and C_ku > 3.2. Conversely, in conditions judged as G, both C_sk and C_ku satisfy: C_sk = -0.2 to 0.2 and C_ku ≤ 3.2. Consequently, the Gaussian distribution criterion for heliostat support structure shear forces (F_x) is empirically established: distributions satisfying |C_sk| ≤ 0.2 and C_ku ≤ 3.2 conform to Gaussian statistics, while deviations from these thresholds indicate non-Gaussian behavior. Analysis of temporal waveforms, density histograms, and moment coefficients across all 130 operational regimes revealed compliance in 126 instances (96.92% accuracy), with only four cases exceeding the proposed boundaries. This statistically validated threshold thus constitutes our final Gaussian diagnostic criterion for loading F_x.

Criterion for determining if axial force follows a gaussian distribution

To characterize Gaussian properties of lift force (F_z) distributions, comprehensive datasets were acquired across all 130 operational regimes, including time-domain profiles, probability density estimates and higher-moment coefficients (C_sk and C_ku ) for the heliostat’s support structure. Given documentation constraints, Table 6 selectively visualizes these multivariate parameters solely for representative Fz cases, providing comparative temporal sequences, histogram distributions, and corresponding moment statistics under matching configurations.

Table 6 C_sk and C_ku for axial force.

Analysis of axial force dynamics reveals distinctive non-Gaussian manifestations under specific operational regimes (Conditions 0-30, 45-30, 135-30, and 180-30), as evidenced by both temporal records and statistical profiles in Table 6 and Fig. 13. The time-history traces display pronounced asymmetry coupled with frequent transient pulse events, indicating substantial aerodynamic disturbances. Corresponding probability distributions exhibit marked positive kurtosis relative to Gaussian benchmarks, demonstrating right-skewed tailing and peak elevation discrepancies. Particularly for Conditions 0-30 and 45-30, histogram profiles manifest leptokurtic characteristics with distribution widths substantially narrower than normative Gaussian references. Therefore, the F_z data distribution under these four conditions exhibits non-Gaussian characteristics. The time-history curves of F_z under the remaining 16 conditions are symmetrically distributed, and their probability density distribution histograms align well with the Gaussian distribution curve, indicating that the F_z data distribution under these 16 conditions follows a Gaussian distribution.

Fig. 13

Time-history curves and probability density distribution histograms of heliostat axial force.

Analyzing the C_sk and C_ku values under 20 operating conditions in Table 6, it is observed that in conditions judged as NG, C_sk and C_ku exhibit the following three scenarios: (1) C_sk < -0.2 or C_sk > 0.2, and C_ku ≤ 3.2; (2) C_sk = -0.2 to 0.2 and C_ku > 3.2; (3) C_sk < -0.2 or C_sk > 0.2, and C_ku > 3.2. Conversely, in conditions judged as G, both C_sk and C_ku satisfy: C_sk = -0.2 to 0.2 and C_ku ≤ 3.2. Based on experimental validation, the proposed Gaussian-distribution criterion for axial forces (F_z) on heliostat supports requires simultaneous satisfaction of |C_sk|≤0.2 and C_ku≤3.2. Comprehensive analysis of temporal waveforms, density histograms, and moment statistics across 130 loading scenarios confirmed compliance in 126 cases (96.92% validation rate), with four deviations observed. This empirically confirmed threshold therefore establishes the definitive decision rule: F_z distributions meeting both conditions are Gaussian, otherwise non-Gaussian.

Criterion for determining if bending moment follows a Gaussian distribution

Table 7 delineates skewness (C_sk) and kurtosis (C_ku) coefficients for bending moments (M_y) under identical operational parameters to those documented in Table 5 and Table 6. This tabular presentation is complemented by parametric analysis of M_y distribution characteristics through temporal waveform examination and probability density histogram assessment.

Table 7 C_sk and C_ku for overturning moment.

From Table 7 and Fig. 14, it can be seen that the time-history curves of the M_y acting on the heliostat support structure under conditions 45-0, 135-0, 180-0, and 45-60 are symmetrically distributed, and their probability density distribution histograms align well with the Gaussian distribution curve, indicating that the M_y data under these four conditions follow a Gaussian distribution. The time-history curves of M_y under the remaining 16 conditions exhibit significant asymmetry or have many large-amplitude pulse signals, or both; the probability density distribution histograms of M_y show tailing compared to the Gaussian distribution curve, with varying degrees of deviation in the peaks, or they may not show tailing but are sharper compared to the Gaussian distribution curve. Therefore, the M_y data distribution under the remaining 16 conditions exhibits non-Gaussian characteristics.

Fig. 14

Time-history curves and probability density distribution histograms of heliostat bending moment.

Analyzing the C_sk and C_ku values under 20 operating conditions in Table 7, it is observed that in conditions judged as NG, C_sk and C_ku exhibit the three scenarios mentioned earlier. In conditions judged as G, C_sk and C_ku generally satisfy: C_sk = -0.2 to 0.2 and C_ku ≤ 3.2. Although the C_sk and C_ku values for the M_y under conditions 135-0 and 45-90 do not meet the aforementioned criterion, 16 out of the 20 conditions still adhere to it, with an accuracy rate of 96.92%. Consequently, analysis confirms consistent Gaussian-distribution criteria across all structural force components, Lift force (F_z), Base overturning moment force(M_y), and Drag force(F_x) acting on heliostat supports exhibit Gaussian characteristics when satisfying dual conditions, skewness within |C_sk|≤0.2 and kurtosis C_ku≤3.2. This unified diagnostic framework, validated through comprehensive analysis of 130 operational regimes per force type, any structural force dataset conforming to these threshold parameters follows Gaussian statistics, all others exhibit non-Gaussian behavior.

The proposed Gaussian-distribution thresholds (|C_sk|≤0.2, C_ku≤3.2) are empirically grounded through statistical rigor, domain-specific consensus, and physical justification. Statistically, these thresholds represent a practical tolerance that accommodates natural variability and measurement uncertainties in real-world engineering data, while maintaining discriminative power against the ideal Gaussian parameters (C_sk=0, C_ku=3). Through comparative analysis with established wind engineering criteria, the proposed thresholds are stricter than those of Kumar and Gong, yet align closely with the stringent standard proposed by Sun for large-span roofs. This demonstrates that the thresholds are not arbitrary but are refined to suit the specific wind-structure interaction characteristics of heliostat support systems. Most significantly, the thresholds are physically justified by the vortex superposition model, values within the thresholds indicate wind loads resulting from statistically independent vortex actions, whereas exceedances signal the dominance of spatially correlated vortex structures leading to non-Gaussian behavior. This mechanistic alignment corroborates the validity of the thresholds beyond purely statistical considerations.

Calculation of force and moment coefficients for the heliostat support structure based on the dung beetle optimization algorithm

Principle of the dung beetle optimization algorithm

Introduced by Xue et al. in 2022²⁸, the Dung Beetle Optimization (DBO) metaheuristic algorithm emulates multifaceted dung beetle ethology, including ball-rolling navigation, celestial orientation dances, foraging strategies, brood-ball theft, and reproductive behaviors, to achieve balanced global exploration and local exploitation capabilities. This biologically inspired optimizer has demonstrated competitive performance across benchmark mathematical functions, exhibiting rapid convergence characteristics. The algorithmic workflow comprises the following phases:

(1) Algorithm initialization involves configuring key parameters: maximum iterations as $T$ and population size as $N$. The initial population is randomly generated with concurrent computation of individual fitness metrics.

(2) Update the positions of ball-rolling dung beetles. If $\lambda < \gamma$ indicates an obstacle-free state, update the position using Equation (1.6); otherwise, update it according to Equation (1.7) in an obstacle state, where $\lambda$ is a random number and $\lambda \in [0,1][0,1]$, $\gamma = 0.9$.

$$R_{{{\text{new }},e}}^{t + 1} = R_{e}^{t} + \alpha \times k \times R_{e}^{t - 1} + u \times \Delta x$$

(1.6)

$$\Delta x = \left| {R_{e}^{t} - X^{w} } \right|$$

(1.7)

Where $t$ represents the current iteration number, $R_{e}^{t}$ represents the position of the $e$ th ball-rolling dung beetle after the $t$ th iteration. $k$ is a constant and $k \in [0,0.2]$, representing the position deflection coefficient. $a$ is a natural coefficient used to simulate the influence of some natural factors on the direction of movement. $X^{w}$ represents the globally worst position, and $\Delta x$ is used to simulate changes in light intensity, with higher $\Delta x$ values indicating weaker light sources.

(3) Update the position of the brooding ball using Equation (1.8) and apply the upper and lower bounds in Equation (1.9) to constrain the new position.

$$B_{{{\text{new }},m}}^{t + 1} = X^{b*} + a_{1} \times \left( {B_{m}^{t} - Lb^{*} } \right) + a_{2} \times \left( {B_{m}^{t} - Ub^{*} } \right)$$

(1.8)

$$\begin{array}{*{20}l} {Lb^{*} = \max \left( {X^{b*} \times (1 - Q),Lb} \right)} \hfill \\ {Ub^{*} = \min \left( {X^{b*} \times (1 + Q),Ub} \right)} \hfill \\ \end{array}$$

(1.9)

Where $B_{m}^{t}$ represents the position of the $m$ th brooding ball after the $t$ th iteration, $a_{1}$ and $a_{2}$ are two independent random vectors of size $1 \times D$, and $D$ is the dimension of the optimization problem’s solution. $Ub^{*}$ and $Lb^{*}$ represent the upper and lower bounds of the spawning area, respectively. $X^{b*}$ represents the current best position of all dung beetles in the population, and $Q = 1 - t/T$, $T$ represent the maximum number of iterations. $Ub$ and $Lb$ represent the upper and lower bounds of the optimization problem.

(4) Update the positions of small dung beetles and thief dung beetles using Equations (1.10) and (1.11).

$$L_{{{\text{new }},h}}^{t + 1} = L_{h}^{t} + C_{1} \times \left( {L_{h}^{t} - Lb^{\prime } } \right) + C_{2} \times \left( {L_{h}^{t} - Ub^{\prime } } \right)$$

(1.10)

$$T_{{{\text{new }},z}}^{t + 1} = X^{b} + S \times g \times \left( {\left| {T_{z}^{t} - X^{b*} } \right| + \left| {T_{z}^{t} - X^{b} } \right|} \right)$$

(1.11)

Among them, $C_{1}$ represents the position $Lb^{*}$ of the $h$ th dung beetle after the $t$ th iteration, $C_{1}$ is a randomly generated number following a normal distribution and $C_{1} \in [0,1]$, and $C_{2}$ is a random vector with components ranging between 0 and 1.

(5) The optimal position is $X^{b}$ and the worst position is $X^{w}$.

(6) The algorithm’s iteration cycle terminates upon reaching the prescribed maximum iteration count, at which point execution ceases and the current optimal position is output as the solution. Should the iteration threshold remain unmet, computational processing reverts to Step 2. Fig. 15 schematically illustrates the complete DBO workflow, with its operational parameters exhaustively detailed in Table 8.

Fig. 15

The flowchart of DBO.

Table 8 Parameter of DBO.

The parameters for the DBO listed in Table 8 were determined through a systematic methodology to ensure algorithmic performance and convergence. The key parameters such as the learning rate (0.01) and population size (60) were initially informed by common defaults in metaheuristic algorithms. The number of hidden layer nodes (5) was determined through a trial-and-error approach to prevent overfitting, which aligns with the relatively small input dimension (4 nodes). All algorithms were applied to identical datasets and preprocessing pipelines, including a 90% training set ratio and 1000 training iterations . Parameters for the competing algorithms (PSO, GWO, BP) were similarly configured based on recommendations from their respective seminal literature or optimized through the same pre-experimental procedure, ensuring all methods operated under their optimal settings. Monitoring of the training loss curves confirmed that the selected parameters enabled stable convergence before reaching the maximum iteration count, with no significant increase in validation loss, as corroborated by the high R² values (>0.8) and low errors reported in Table 9.

Table 9 Errors of training and testing sets for each coefficient.

Comparison between other intelligent algorithms and dung beetle optimization algorithm

This study employs four computational algorithms, namely Grey Wolf Optimizer (GWO), Particle Swarm Optimization (PSO), Back Propagation (BP) neural network, and (Dung Beetle Optimization) DBO, to calculate structural moment coefficients, with comparative analysis conducted on their regression performance. The Grey Wolf Optimizer (GWO), a swarm intelligence optimization algorithm proposed by Mirjalili et al. from Griffith University, Australia in 2014, simulates the leadership hierarchy and hunting mechanisms observed in grey wolf populations in natural ecosystems²⁴. Particle Swarm Optimization (PSO), an algorithmic model developed by James Kennedy and Russell Eberhart in 1995, draws inspiration from the collective foraging behavior of bird flocks, its fundamental principle lies in simulating how avian populations achieve optimal destination identification through collective information sharing²⁵. The Back Propagation (BP) neural network, first conceptualized by Rumelhart, McClelland and their research team in 1986, represents a multilayer feedforward neural network trained via error backpropagation algorithm, this architecture has gained extensive application across various engineering domains due to its robust learning capabilities²⁶.

Employing operational condition parameters (elevation, azimuth and wind velocity) as independent predictors alongside force coefficients as response variables, the dataset necessitates comprehensive preprocessing before model training. This procedure involves three critical stages: identifying and rectifying anomalous samples deviating from established statistical distribution patterns; rescaling all features to a standardized bounded interval [0,1]; and implementing a 9:1 stratified partition between training and validation subsets. Comparative performance metrics for DBO and alternative optimization algorithms across different force coefficients predictions are subsequently visualized in Fig. 16.

Fig. 16

Training and testing sets for each coefficient.

Following predictive modeling, output data undergoes denormalization to reconstruct physical-scale values. Model fidelity is quantified via the coefficient of determination (R²), where values approaching unity indicate superior fit. Notably, DBO regression models consistently achieve R² > 0.8, outperforming benchmark algorithms and demonstrating robust generalization capacity. Complementary error metrics, specifically Root Mean Square Error (RMSE) and Mean Absolute Error (MAE), further validate predictive performance., with lower values signifying enhanced accuracy. As tabulated in Table 9, implementations using Grey Wolf Optimizer (GWO), Particle Swarm Optimization (PSO), DBO, and Back Propagation (BP) all yield RMSE/MAE values below 0.2 threshold. Across 13 validation samples, DBO-derived predictions exhibit exceptional concordance with experimental measurements, maintaining pointwise proximity while accurately replicating value trajectories. Though minor deviations occur in axial force and bending moment skewness predictions, DBO maintains statistically significant superiority across parametric predictions, establishing its efficacy for operational forecasting applications.

Although the test matrix consists of discrete points, the regression model established by the DBO inherently possesses continuous predictive capability. By utilizing the 130 operational conditions as training samples, the DBO learns the continuous functional relationship between the force coefficients (skewness, Csk, and kurtosis, Cku) and the variation of elevation and azimuth angles. For instance, for a point not directly tested, such as elevation=35° and azimuth=52.5°, DBO performs predictive interpolation based on surrounding tested conditions (e.g., the grid formed by elevations of 30°/40° and azimuths of 45°/60°), rather than simply assigning it to the nearest discrete point. Furthermore, DBO can capture the underlying physical essence through various mechanisms. For example, the transition in axial force skewness near 35° elevation reflects the gradual process of backflow transition from attached to separated flow. Through training on extensive surrounding conditions, DBO can infer the behavioral patterns within this transition zone. These strategies effectively mitigate the limitations of binary classification in regions of abrupt data variation.

Unlike general-purpose optimization methods such as Particle Swarm Optimization (PSO) and Grey Wolf Optimizer (GWO), which primarily focus on optimization efficiency, the DBO offers a distinct advantage by uniquely integrating algorithmic robustness with statistical rigor. This synergy enables DBO to derive Gaussian/non-Gaussian discrimination criteria that are both computationally efficient and grounded in clear physical meaning. DBO demonstrates marked superiority across several key performance metrics. First, in terms of predictive accuracy, DBO achieves higher coefficients of determination (R² > 0.8) and lower error rates (RMSE < 0.1) when predicting aerodynamic force coefficients such as skewness and kurtosis. For instance, in predicting the kurtosis of the bending moment, DBO attains an R² of 0.923, outperforming both PSO (0.940) and GWO (0.760). This enhanced accuracy ensures more reliable classification of load distributions, thereby substantially reducing uncertainty in the design process. Second, regarding the handling of nonlinearity, DBO’s bio-inspired mechanisms provide superior capability in exploring complex parameter spaces. This allows DBO to identify critical pattern transitions, such as the shift from Gaussian to non-Gaussian load distributions beyond specific elevation angles, that may be overlooked by PSO and GWO. Most importantly, in classification robustness, the DBO-based model delivers consistently superior performance across all 130 operational conditions, whereas PSO and GWO exhibit higher outcome variability. This robustness translates into clearer and more reliable guidance for designers. It effectively mitigates the risks of over-design or under-design caused by misclassifying load distribution characteristics. Consequently, DBO provides precise guidance for the design of heliostat support structures by clearly delineating when simplified Gaussian load models are sufficient and when more complex higher-order non-Gaussian models are necessary, thereby optimizing the balance between structural safety and economic efficiency.

The DBO facilitates precise load identification for heliostat support structures by generating Gaussian/non-Gaussian distribution maps, which enable differentiated design strategies based on load characteristics. In low-elevation Gaussian regions, simplified second-moment methods enhance computational efficiency, whereas higher-order statistics are employed in high-elevation non-Gaussian zones to accurately evaluate tail risks. The quantitative relationships established by DBO between force coefficients and azimuth/elevation angles provide a basis for optimizing heliostat positioning, effectively reducing non-Gaussian moments by avoiding specific operational conditions. Furthermore, the closed-loop correction mechanism integrating load modeling with reliability assessment refines dynamic response analysis, significantly improving wind resistance. This data-driven methodology represents a transition from traditional empirical approaches to predictive design, ensuring structural safety while optimizing construction costs.

Division of the structure forces follow Gaussian distribution

Leveraging the computational framework established previously, force coefficients across operational scenarios are derived parametrically from elevation angles, azimuth orientations, and wind velocities. Application of skewness (C_sk) and kurtosis (C_ku) metrics enables binary classification of Gaussian versus non-Gaussian force distributions among the 130 operational regimes for heliostat support structures. This methodology yields intuitive visualization of load distribution characteristics under varying operating parameters, as depicted in Fig. 17. Through systematic evaluation of shear, axial, and bending moment statistics against established Gaussian-distribution thresholds, operational conditions are partitioned accordingly. The resultant distribution map categorizes all 130 operating states by Gaussian compliance status, with wind direction angle β represented vertically and elevation angle α horizontally.

Fig. 17

Statistical analysis of the skewness and kurtosis coefficients. The coefficient value refers to the kurtosis coefficient or skewness coefficient

Figure 18 demonstrates that aerodynamic forces on heliostat support structures exhibit primarily Gaussian distribution characteristics under specific angular configurations: drag forces at α=0°~20°, lift forces during α=0°~20° or β=90°~180° orientations, and overturning moments within α=60°~90° combined with β=0°~60°. These observational patterns align with fundamental turbulence principles articulated by Wang Yingge²⁹, whose application of the Central Limit Theorem to non-Gaussian wind pressure distributions reveals that multiscale atmospheric vortices randomly distribute vortex cores throughout spatial domains. Each vortex core applies localized forces upon contacting structural surfaces while simultaneously transferring mechanical energy. Consequently, the resultant aerothermal forces acting on heliostat supports represent the cumulative effect of numerous superimposed point vortices interacting with the structure.

Fig. 18

Distribution of shear force, axial force, and bending moment.

For the F_x and F_z on the heliostat support structure, when the Elevation angle is small, the mirror is approximately parallel to the ground, and the incoming wind along the X-axis direction acting on the heliostat has little correlation, it can be approximately considered that the effect of each point vortex along the X-axis direction is independently and identically distributed. When the number of point vortices acting on the heliostat is sufficient, the sum of all point vortex effects has Gaussian characteristics. Increasing mirror elevation angles trigger organized vortex formation downstream of the panel’s peripheral edges as airflow separates. Under such conditions, X-axis-aligned point vortices lose statistical independence and identical distribution properties. These coherent vortical systems impart strongly correlated aerodynamic forces along the X-axis orientation, violating the statistical independence prerequisite essential for Central Limit Theorem applicability. Consequently, both drag (F_x) and lift (F_z) forces exhibit pronounced non-Gaussian characteristics. Conversely, at minimal elevation angles when the reflector approaches horizontal alignment, analogous vortex organization develops along the Y-axis following edge-flow separation. The resulting Y-direction vortical structures generate spatially correlated loading distributions that similarly contravene Central Limit Theorem requirements. This mechanistic framework explains the consistent non-Gaussian behavior observed in overturning moments (M_y) under such operational regimes.

This study establishes a comprehensive theoretical framework for assessing the propagation of experimental uncertainties, including measurement instrument errors, wind tunnel simulation deviations, and operational parameter control inaccuracies, through the DBO model for wind load characterization on heliostat supports. The analysis delineates three primary transmission pathways: the introduction of errors into force coefficients during data preprocessing, where wind speed uncertainties are quadratically amplified in dynamic pressure calculations; the distortion of the feature space due to azimuth and elevation positioning tolerances, particularly critical in flow transition zones; and the incorporation of label noise during model training, potentially affecting convergence and generalizability. Theoretical evidence indicates that while wind speed measurement errors exert the most significant influence due to their squared relationship with reference pressure, angular inaccuracies induce the highest classification uncertainty in critical regions, such as elevations between 30° and 50°, where flow separation patterns are highly sensitive. Despite these inherent uncertainties, the proposed Gaussian/Non-Gaussian discrimination thresholds (|C_sk|≤0.2, C_ku≤3.2) are demonstrated to incorporate a practical tolerance for measurement variability, while DBO’s bio-inspired mechanisms confer inherent robustness against noise, ensuring the reliability of its performance superiority over conventional algorithms like PSO and GWO. The study concludes by advocating for a future transition towards probabilistic output frameworks, which would quantify classification confidence levels derived from uncertainty propagation models, thereby providing engineers with risk-aware decision-making tools and bridging the gap between theoretical model performance and practical design application under uncertainty.

Conclusion

This investigation systematically examines the stochastic properties of wind-induced loads acting upon heliostat support structures, establishing diagnostic criteria for identifying Gaussian-distributed force components. Concurrently, the research characterizes the probabilistic behavior of fluctuating wind pressure distributions across heliostat reflector surfaces. Primary findings from this comprehensive analysis reveal the following critical insights:

• (1) Through systematic evaluation of temporal waveforms, probability density distributions, and higher-order statistics (C_sk and C_ku) for heliostat structural loads across 130 operational regimes, this study establishes definitive Gaussian-distribution criteria for wind-induced forces as follows: data conform to Gaussian statistics when satisfying parametric thresholds of -0.2 ≤ C_sk ≤ 0.2 and C_ku ≤ 3.2; otherwise, distributions are classified as non-Gaussian. Validation demonstrated 96.92% diagnostic accuracy across all measured load components (F_x, F_z , M_y). This empirically-derived standard provides critical refinement to conventional wind engineering frameworks, where stationary Gaussian wind load assumptions are routinely employed for analytical simplification despite documented non-Gaussian phenomena in structural responses. However, real wind fields often exhibit non-stationary and non-Gaussian features, which significantly compromise structural safety assessments. Non-stationary wind loads, commonly observed during transient climatic events such as typhoons and thunderstorms, demonstrate time-varying intensity and spectral properties. These loads may induce transient resonance or abrupt dynamic responses. Neglecting the non-stationary and non-Gaussian nature of wind loads could lead to systematic overestimation of structural safety margins. While the hybrid criteria proposed in this study (integrating statistical indicators with graphical analysis) demonstrate efficacy in identifying non-Gaussian loads, their implementation requires further incorporation into dynamic response analysis frameworks. Such integration would enable comprehensive refinements spanning from load modeling to reliability evaluation, thereby establishing a closed-loop correction methodology.

• (2) Parametric force coefficients were derived through computational intelligence optimization, enabling a systematic analysis of the Gaussian distribution properties across operational regimes for heliostat support structures. The analysis reveals that aerodynamic drag and lift forces predominantly exhibit Gaussian characteristics at low elevation angles, transitioning to non-Gaussian behavior at elevated tilt positions. Conversely, overturning moments demonstrate an inverse pattern, showing non-Gaussian properties at minimal elevation angles while conforming to Gaussian statistics at steeper inclinations. The predictions generated by the DBO-optimized algorithm show a marked improvement over alternative methodologies, as evidenced by their close alignment with experimental measurements in both magnitude and temporal variation trends.

• (3) This study elucidates the physical mechanism behind the non-Gaussian characteristics of wind loads through an atmospheric vortex superposition model: When the elevation angle of the heliostat is small, the random vortex effects along the X/Z-axis directions satisfy the independent and identically distributed condition, resulting in Gaussian characteristics for shear force F_x, bending moment M_y, and axial force F_z under the central limit theorem. However, as the elevation angle increases or under specific azimuth angles, airflow forms spatially strongly correlated organized vortex structures at the mirror edges, violating the statistical independence prerequisite. This leads to pronounced non-Gaussian characteristics in F_x, M_y, and F_z. The essence of this mechanical transition lies in whether the spatial correlation of turbulent structures exceeds the applicability boundary of the central limit theorem, where the systematic spatial coupling of organized vortices constitutes the core physical mechanism driving non-Gaussian features. Through the analysis of the criterion’s accuracy, it can be concluded that adopting this evaluation standard to determine whether the stress on the heliostat support structure follows a Gaussian distribution is feasible.

• (4) The research findings can provide direct theoretical support for the wind-resistant design of heliostats. For operating conditions exhibiting Gaussian characteristics, it is recommended to adopt a probability-based design method utilizing second-order moments, which can simplify the structural reliability analysis process. For non-Gaussian conditions, the introduction of higher-order statistical measures is required to construct extreme value estimation models.In terms of algorithmic applications, the global search capability demonstrated by the DBO suggests its potential extension to multi-objective optimization problems in wind engineering, such as topology optimization of photovoltaic supports and flutter prediction of wind turbine blades. Future research should focus on conducting wind field reconstruction experiments in complex terrains including mountainous and gobi regions to establish the mapping relationship between turbulence integral scale and skewness coefficient.