Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “error testing”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 1,009 records · Page 56
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Linearised Fokker–Planck collision model for gyrokinetic simulations

We introduce a gyrokinetic, linearised Fokker–Planck collision model that satisfies conservation laws and is accurate at arbitrary collisionalities. The differential test-particle component of the operator is exact; the integral field-particle component is approximated using a spherical harmonic and a modified Laguerre polynomial expansion developed by Hirshman and Sigmar (1976 Phys. Fluids 19 1532). The numerical methods of the implementation in the δf-gyrokinetic code stella (Barnes et al 2019 J. Comput. Phys. 391 365–80) are discussed, and conservation properties of the operator are demonstrated. The collision model is then benchmarked against the collision model of the gyrokinetic solver GS2 in the limiting cases of a reduced test-particle collision operator and energy- and momentum-conserving operator. The accuracy of the full collision model is investigated by solving the parallel Spitzer-Härm problem for the transport coefficients. It is shown that retaining collisional energy flux and higher-order terms in the field-particle operator reduces errors in the transport coefficients from 10%–25% for a simple momentum- and energy-conserving model to under 1%.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

Deep Learning-enhanced Block-Diagram Modeling of Solar Power Systems

Data-driven models of power system inverter-based resources are desired to run simulations faster than with detailed electromagnetic transient models, to hide proprietary design details, to support control system design applications, and to aggregate the effects of distributed energy resources. This paper applies a customized Hammerstein Wiener framework to train block diagram models from thousands of electromagnetic transient simulations or experimental test records. The block diagram models integrate with larger grid simulations as voltagecontrolled current sources or current-controlled voltage sources for several simulators. Guidelines for block architecture and training are presented. Three-phase balanced, three-phase unbalanced, and single-phase examples all achieve an acceptable root mean square error of no more than 0.05 per-unit.

Mcdermott, Thomas E. [Private consulting company]↗
DOE OSTI2022

Modeling land use and land cover change: using a hindcast to estimate economic parameters in gcamland v2.0

Abstract. Future changes in land use and cover have important implications for agriculture, energy, water use, and climate. Estimates of future land use and land cover differ significantly across economic models as a result of differences in drivers, model structure, and model parameters; however, these models often rely on heuristics to determine model parameters. In this study, we demonstrate a more systematic and empirically based approach to estimating a few key parameters for an economic model of land use and land cover change, gcamland. Specifically, we generate a large set of model parameter perturbations for the selected parameters and run gcamland simulations with these parameter sets over the historical period in the United States to quantify land use and land cover, determine how well the model reproduces observations, and identify parameter combinations that best replicate observations, assuming other model parameters are fixed. We also test alternate methods for forming expectations about uncertain crop yields and prices, including adaptive, perfect, linear, and hybrid approaches. In particular, we estimate parameters for six parameters used in the formation of expectations and three of seven logit exponents for the USA only. We find that an adaptive expectation approach minimizes the error between simulated outputs and observations, with parameters that suggest that for most crops, landowners put a significant weight on previous information. Interestingly, for corn, where ethanol policies have led to a rapid growth in demand, the resulting parameters show that a larger weight is placed on more recent information. We examine the change in model parameters as the metric of model error changes, finding that the measure of model fitness affects the choice of parameter sets. Finally, we discuss how the methodology and results used in this study could be used for other regions or economic models to improve projections of future land use and land cover change.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Measurement of the muon anomalous precession frequency at the Muon g − 2 Experiment at Fermilab

The anomalous magnetic moment of the muon, $a_\mu = \frac{g-2}{2}$ is the fractional deviation of the muon $g$-factor from the value of 2. It emerges as the cumulative effect of the virtual particles participating in the muon interaction with a magnetic field via quantum loop corrections. Its value encodes all the possible interactions between the virtual particles and, for this reason, represents an important test of the Standard Model (SM). In particular, any deviation from the SM theoretical evaluation could be due to new physics contributions. The new Muon $g-2$ (E989) Experiment at Fermilab is currently operating to repeat and improve the previous E821 experiment at Brookhaven National Laboratory (BNL), aiming to reduce the experimental error by a factor of 4 to the final accuracy of 140 parts per billion (ppb). On April 7th, 2021, the E989 collaboration published the first result based on the first year of data taking (Run-1), measuring $a_\mu = 0.001~165~920~40(54)$ with a precision of 460 ppb. The measured value is consistent with the BNL measurement and strengthens the long-standing tension with the data-driven SM prediction to a combined discrepancy of 4.2$\sigma$. On the theory side, however, new efforts involving lattice-QCD techniques are starting to question the current consensus on the theoretical prediction, demanding new improvements on both the experimental and theoretical sides. The E989 collaboration is now finalizing the analysis of Run-2 and Run-3 data and a new publication is expected in the first half of 2023 with a combined statistical uncertainty of 200 ppb. The anomalous magnetic moment $a_\mu$ is measured as the ratio between the muon spin anomalous precession frequency, $\omega_a$, and the average magnetic field experienced by the muons as they circulate in the storage ring. This thesis presents a precession frequency analysis of the Run-1 data and an evaluation of the related systematic uncertainties. A new positron reconstruction developed for the analysis of the subsequent data-taking periods, aiming to reduce some of the major systematic uncertainties of the $\omega_a$ measurement, is presented. The author's involvement in the production of the Run-2/5 data and in the precise calibration of the detectors is discussed. Finally, the complete Run-1 $a_\mu$ result is presented.

43 PARTICLE ACCELERATORS↗
DOE OSTI2022

Angular systematics-free cosmological analysis of galaxy clustering in configuration space

ABSTRACT Galaxy redshift surveys are subject to incompleteness and inhomogeneous sampling due to the various constraints inherent to spectroscopic observations. This can introduce systematic errors on the summary statistics of interest, which need to be mitigated in cosmological analysis to achieve high accuracy. Standard practices involve applying weighting schemes based on completeness estimates across the survey footprint, possibly supplemented with additional weighting schemes accounting for density-dependent effects. In this work, we concentrate on pure angular systematics and describe an alternative approach consisting in analysing the galaxy two-point correlation function where angular modes are nulled. By construction, this procedure removes all possible known and unknown sources of angular observational systematics, but also part of the cosmological signal. We use a modified Landy–Szalay estimator for the two-point correlation function that relies on an additional random catalogue where angular positions are randomly drawn from the galaxy catalogue, and provide an analytical model to describe this modified statistic. We test the model by performing an analysis of the full anisotropic clustering in mock catalogues of luminous red and emission-line galaxies at 0.43 < z < 1.1. We find that the model fully accounts for the modified correlation function in redshift space, without introducing new nuisance parameters. The derived cosmological parameters from the analysis of baryon acoustic oscillations and redshift-space distortions display slightly larger statistical uncertainties, mostly for the growth rate of structure parameter fσ8 that exhibits a $50{{\ \rm per\ cent}}$ statistical error increase, but free from angular systematic error.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

High-dimensional isotomics, part 2: Observations of over 100 constraints on methionine's isotome

The abundances of different isotopic forms of a compound, or isotopologues, will vary based on its physical and chemical history. The number of isotopologues increases combinatorically with the size of a molecule, and even small molecules such as amino acids have thousands of potentially observable isotopic variants. However, due to the analytical challenges of separating and observing isotopologues, only a few dimensions of isotopic diversity are routinely measured. Overcoming these challenges requires both an experimental method to observe many isotopic properties and a theoretical framework for interpreting these experiments. In Part 1, we presented such a theoretical framework; here, we demonstrate an experimental method, which we apply to methionine. Our approach uses a Q Exactive HF Orbitrap to perform several “M + N experiments”, where a sample is ionized, a subset of its isotopologues with cardinal mass N daltons greater than the unsubstituted isotopologue is selected and fragmented, and the proportions of all detectable isotopic forms of those fragment ions are quantified. We perform M + 1, M + 2, M + 3, and M + 4 experiments of a methionine sample and standard where the sample has a 100 ‰ enrichment of 13C at the methyl carbon relative to the natural 13C abundance at that position in the standard, and is otherwise identical to the standard. We observe isotopic forms of 8 fragment ion species for each version of the M + N experiment. With the assistance of a forward model of expected mass spectra, we identify isotopic peaks for each fragment ion based on observed mass and abundance, screen these for data quality, and quantify abundances for 146 unique isotopic peaks at precisions of ≈ 0.3–3 ‰. We present our direct observations and use them to reconstruct the concentrations of 19 individual singly, doubly, and triply-substituted isotopologues; doing so gives fewer constraints and broader error bars than working with the direct observations, but may be more interpretable for some applications. We also examine possibilities for measuring additional peaks, which are primarily limited by the detection limit of the Orbitrap-IRMS method. We then suggest some possible uses of our direct measurements for chemical forensics and hypothesis testing. Furthermore, our results demonstrate the diversity of isotopic constraints currently observable and interpretable for organic molecules.

58 GEOSCIENCES↗
DOE OSTI2025

A proposed high-intensity radiometer calibration method using concentrated solar radiation

Accurate calibration of irradiance measurement devices, or radiometers, is essential for ensuring the reliability of measurements in high heat applications such as concentrating solar power (CSP), aerospace, defense, and pulsed power systems. Despite the critical need, existing calibration standards and service providers are limited to irradiance levels below 100 kW/m 2 and specific radiation sources, which is insufficient for many applications. For instance, CSP technologies, particularly those under the Department of Energy’s Solar Energy Technologies Office (SETO) Gen 3 program, require accurate measurements of broadband irradiance at levels exceeding 2000 kW/m 2 . In even more extreme scenarios, such as re-entry vehicles, heat levels can surpass 10000 kW/m 2 . Current ISO standards, specifically ISO 14934–2 and ISO 14934–3, are constrained to lower irradiance levels and dependent on black body heat sources, limiting their applicability for high-intensity broadband irradiance measurements, particularly in concentrated solar applications. Here, to address this shortfall, the National Solar Thermal Test Facility (NSTTF) at Sandia National Laboratories (SNL) proposes a calibration method and facility capable of characterizing radiometers up to 2750 kW/m 2 using concentrated solar irradiance. Calibrating with concentrated sunlight is important for solar applications as it aligns the calibration process with the solar spectrum. This alignment is crucial for minimizing systematic errors and avoiding the need for additional corrections that may arise when radiometers designed for solar applications are calibrated using black-body or electrical sources. This paper presents the present day NSTTF characterization facility and procedure, detailing the proposed calibration method and uncertainty quantification. The presented method builds upon 1980′s NSTTF methodology and involves both theoretical and empirical methods to establish a robust relationship between gauge voltage output and irradiance intensity, quantifying both measurement and fitting errors. By addressing the limitations of existing standards and extending the characterization range, this work provides an advancement in the field of high-intensity irradiance measurement and instrumentation characterization.

Gardon gauge↗
DOE OSTI2020

DropSynth 2.0: high-fidelity multiplexed gene synthesis in emulsions

Abstract Multiplexed assays allow functional testing of large synthetic libraries of genetic elements, but are limited by the designability, length, fidelity and scale of the input DNA. Here, we improve DropSynth, a low-cost, multiplexed method that builds gene libraries by compartmentalizing and assembling microarray-derived oligonucleotides in vortexed emulsions. By optimizing enzyme choice, adding enzymatic error correction and increasing scale, we show that DropSynth can build thousands of gene-length fragments at >20% fidelity.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Verification of a Reduced-Order Extension Spring Model

Adopting reduced order models (ROMs) of springs lowers the computational cost of stronglink simulations. However, ROMs introduce currently unquantified error to such analyses. This study addresses that lack of data by comparing a hexahedral mesh to a commonly used ROM beam mesh. Two types of analyses were performed, a quasi-static displacement-controlled pull and a haversine shock, examining basic spring properties as well as dynamics and stress/strain data. Both tests showed good similarities between the hexahedral and beam meshes, especially when comparing reaction force and stress trends and maximums. Equivalent plastic strain results were not quite as favorable, indicating that the beam model may be less likely to correctly predict spring failure. Despite reducing computation times by over 48 hours in all shock cases, appropriate use of the ROM should carefully balance this advantage with its reduction in accuracy, especially when examining spring failure and outputting variables such as equivalent plastic strain.

42 ENGINEERING↗
DOE OSTI2020

Trans-Neptunian Objects Found in the First Four Years of the Dark Energy Survey

In this paper, we present a catalog of 316 trans-Neptunian bodies (TNOs) detected from the first four seasons ("Y4" data) of the Dark Energy Survey (DES). The survey covers a contiguous 5000 deg 2 of the southern sky in the grizY optical/NIR filter set, with a typical TNO in this part of the sky being targeted by 25–30 Y4 exposures. This paper focuses on the methods used to detect these objects from the ≈60,000 Y4 exposures, a process made challenging by the absence of the few-hour repeat observations employed by TNO-optimized surveys. Newly developed techniques include: transient/moving object detection by comparison of single-epoch catalogs to catalogs of "stacked" images; quantified astrometric error from atmospheric turbulence; new software for detecting TNO linkages in a temporally sparse transient catalog, and for estimating the rate of spurious linkages; use of faint stars to determine the detection efficiency versus magnitude in all exposures. Final validation of the reality of linked orbits uses a new "sub-threshold confirmation" test, wherein we demand the object be detectable in a stack of the exposures in which the orbit indicates an object should be present, but was not individually detected. This catalog contains all validated TNOs which were detected on ≥6 unique nights in the Y4 data, and is complete to r 23.3 mag with virtually no dependence on orbital properties for bound TNOs at distance 30 au < d < 2500 au. The catalog includes 245 discoveries by DES, 139 not previously published. The final DES TNO catalog is expected to yield >0.3 mag more depth, and arcs of >4 yr for nearly all detections.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2025

Field validation of dynamic mechanical torque measurements using fiber-optic strain sensors for geared wind turbines

Abstract Accurate knowledge of the mechanical loads of wind turbine gearboxes has become essential in modern, highly loaded gearbox designs, as maintaining or even improving gearbox reliability with increasing torque density demands is proving to be challenging. Unfortunately, the traditional method of measuring dynamic mechanical torque using strain gauges placed on the outer surface of a rotating shaft and transmitting the resulting signal is unsuitable for serial deployment due to technical and economic constraints. An alternative method based on fiber-optic strain sensors placed on the stationary outer surface of the gearbox ring gear has been proposed. Like shaft torsion, the radial deformation of the ring gear is proportionate to the rotor torque. Placing the sensors on a stationary component is a cost-effective alternative for serial implementation because the need for complex and expensive data transfer via wireless transmission or a slip ring is eliminated. In this paper, we present the results of an extensive field experiment conducted to evaluate the torque measurement accuracy of this novel sensing solution installed on the gearbox of a Gamesa G97 2-MW wind turbine at the National Renewable Energy Laboratory’s Flatirons Campus. Torque measurements derived from fiber-optic strain sensors placed on the ring gear of the planetary stage are compared to conventional torque measurements from strain gauges placed on the main shaft. Two different torque estimation data processing methods were evaluated, with the method based on operational deflection shapes providing the most accurate results with an average normalized root mean square error below 0.7% for a load revolution distribution analysis. The effect of operating conditions on the torque estimate was also investigated, and the third planet-passing operational deflection shape was found to be the least sensitive to nontorque load-related effects. The fiber-optic strain sensors’ successful operation during the complete test campaign has demonstrated a robust and accurate solution for fleet-wide enhanced gearbox remaining useful life estimation.

17 WIND ENERGY↗
DOE OSTI2023

Evaluation of supply air temperature control performance with different control strategies at air handling units

Due to its nonlinear feature, supply air temperature control systems at air handling units using single-loop fixed gain control frequently suffer unstable response under dynamic conditions. This paper is aimed at identifying a more effective control strategy to stabilize the valve control and achieve desirable supply air temperature response under various operation conditions by evaluating three control strategies, cascade, gain scheduling and cascade plus gain scheduling controls. Firstly, a control-oriented model is developed and calibrated through open-loop tests. Then, the control issues with single-loop fixed gain control are evaluated through simulations and field tests, which indicates that the system has more aggressive response or becomes unstable under lower load conditions when the chilled-water flow rate is less than 0.1 L/s. Finally, three control strategies are proposed and evaluated. Further, the results reveal that all three proposed controls can achieve better robustness with the integral square error of supply air temperature decreased to 16%, 47% and 5%, respectively, under lower load condition. However, since high-frequency oscillation of the control valve occurs with cascade and cascade plus gain scheduling control, the gain scheduling control is recommended considering a trade-off among the supply air temperature response, valve lifetime and implementation complexity.

42 ENGINEERING↗
DOE OSTI2020

Quantitative assessment of eddy viscosity rans models for turbulent mixed convection in a differentially heated plane channel

Turbulent mixed convection between two vertical, infinite parallel plates at different temperatures is studied using various two-equation turbulence models. The numerical simulations are performed at a turbulent Reynolds number of Re τ = 150 and a Grashof number of Gr = 9.6 × 10 5 . Comparisons are made against the highly trusted DNS results. Consistent with the DNS approach, the current simulations are performed using constant properties and the Boussinesq approximation to predict the influence of buoyancy. Previous studies have provided assessments of two-equation turbulence models for various scenarios, but often rely on a qualitative “eye” test in order to determine the most appropriate model to predict a given flow. This study aims to provide a new form of quantitative assessment that accounts for both the physics captured by the turbulence model as well as the magnitude of the system response quantities (SRQ) using a modified symmetric mean absolute percent error (SMAPE) method. This method is designed to be approachable to researchers at any level and can be applied to system response quantities from multiple research fields. Uncertainty quantification is also performed to determine the discretization error for each turbulence model. Recommendations are made as to which turbulence models best capture the physics – hydrodynamically and thermally – using both local and global validation metrics. Lastly, a sensitivity analysis is performed on the damping functions used in the most accurate models. This underpins the potential of model developments and adjustments most worth pursuing for buoyant flows. Finally, this framework provides a more physics-based comparative analysis of the selected turbulence models.

42 ENGINEERING↗
DOE OSTI2024

Evaluation of the Planetary Boundary Layer Height From ERA5 Reanalysis With MOSAiC Observations Over the Arctic Ocean

The planetary boundary layer height (PBLH) is a crucial indicator reflecting the region of the atmosphere characterized by continuous turbulence. Here, we use radiosonde and surface meteorological observations (4–7 times per day, year-round measurements) during the Multidisciplinary drifting Observatory for the Study of Arctic Climate (MOSAiC) expedition to derive the PBLH (PBLH MOSAiC ), and further evaluate the PBLH from the ERA5 reanalysis (PBLH ERA5 ). Comparisons between PBLH MOSAiC and PBLH ERA5 from different perspectives reveal that: (a) The overestimation of PBLH ERA5 when the sea ice concentration is >90% is significant with the centered root mean squared error reaching up to 201 m; (b) The difference between the two products is notably pronounced in cold seasons, while it is comparatively diminished in warm seasons; (c) In neutral boundary layers, differences in PBLH ERA5 are larger compared with stable and convective boundary layers. In addition, the analysis of error sources indicates that the bias of PBLH ERA5 is sensitive to the bias of vertical thermal structure and wind speed profiles in ERA5 data sets in all conditions. Finally, we find a Random Forest model effectively reduces the bias of PBLH ERA5 with the index of agreement reaching up to 0.71 in the test data set, while a multiple linear regression demonstrates comparable performance to the Random Forest model.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2026

Crowdsourcing the Frontier: Advancing Hybrid Physics‐ML Climate Simulation via a $\$$50,000 Kaggle Competition

Subgrid machine-learning (machine learning [ML]) parameterizations have the potential to introduce a new generation of climate models that incorporate the effects of higher-resolution physics without incurring the prohibitive computational cost associated with more explicit physics-based simulations. However, important issues, ranging from online instability to inconsistent online performance, have limited their operational use for long-term climate projections. To more rapidly drive progress in solving these issues, domain scientists and ML researchers opened up the offline aspect of this problem to the broader ML and data science community with the release of ClimSim, a NeurIPS Data sets and Benchmarks publication, and an associated Kaggle competition. This paper reports on the downstream results of the Kaggle competition by coupling emulators inspired by the winning teams' architectures to an interactive climate model (including full cloud microphysics, a regime historically prone to online instability) and systematically evaluating their online performance. Our results demonstrate that online stability in the low-resolution real-geography setting is reproducible across multiple diverse architectures, which we consider a key milestone. All tested architectures exhibit strikingly similar offline and online biases, though their responses to architecture-agnostic design choices (e.g., expanding the list of input variables) can differ significantly. Multiple Kaggle-inspired architectures achieve state-of-the-art results on certain metrics such as zonal mean bias patterns and global Root Mean Squared Error, indicating that crowdsourcing the essence of the offline problem is one path to improving online performance in hybrid physics-AI climate simulation.

Environmental sciences↗
DOE OSTI2025

Real-time elemental analysis of liquids for process monitoring using laser-induced breakdown spectroscopy with a liquid wheel sampling approach

This article presents an engineered sampling system that used a rotating wheel to form a thin liquid layer, permitting the use of laser-induced breakdown spectroscopy (LIBS) for in situ, real-time elemental impurity quantification during liquid processing. The sampling approach was demonstrated on eight elements from across the periodic table (Na, Al, K, Ca, Ti, Sr, Mo, and Yb). Univariate and multivariate calibrations were presented for each element. The average value for percent root mean square errors of cross-validation for the multivariate models was 3.64%, highlighting the method's strong prediction accuracy. Additionally, the limits of detection for each analyte were estimated from their univariate models: Na = 0.0532, Al = 18.5, K = 0.105, Ca = 0.273, Ti = 67.7, Sr = 0.640, Mo = 22.4, and Yb = 22.9 μg mL –1 . Finally, a test in which multivariate models were used to monitor a liquid system for 80 min was performed to investigate the real-time monitoring capabilities of this liquid LIBS sampling approach. Rigorous measurements were performed to effectively predict the absence and concentrations of multiple analytes as they were spiked and diluted. This demonstration showed the feasibility of using LIBS for real-time liquid quantification models with estimated precision ≤ 8.1%. Finally, the limitations of this approach and potential future improvements are discussed.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

J-PLUS: Support vector regression to measure stellar parameters

Stellar parameters are among the most important characteristics in studies of stars which, in traditional methods, are based on atmosphere models. However, time, cost, and brightness limits restrain the efficiency of spectral observations. The Javalambre Photometric Local Universe Survey (J-PLUS) is an observational campaign that aims to obtain photometry in 12 bands. Owing to its characteristics, J-PLUS data have become a valuable resource for studies of stars. Machine learning provides powerful tools for efficiently analyzing large data sets, such as the one from J-PLUS, and enables us to expand the research domain to stellar parameters. The main goal of this study is to construct a support vector regression (SVR) algorithm to estimate stellar parameters of the stars in the first data release of the J-PLUS observational campaign. The training data for the parameter's regressions are featured with 12-waveband photometry from J-PLUS and are crossidentified with spectrum-based catalogs. These catalogs are from the Large Sky Area Multi-Object Fiber Spectroscopic Telescope, the Apache Point Observatory Galactic Evolution Experiment, and the Sloan Extension for Galactic Understanding and Exploration. We then label them with the stellar effective temperature, the surface gravity, and the metallicity. Ten percent of the sample is held out to apply a blind test. We develop a new method, a multi-model approach, in order to fully take into account, the uncertainties of both the magnitudes and the stellar parameters. The method utilizes more than 200 models to apply the uncertainty analysis. We present a catalog of 2 493 424 stars with the root mean square error of 160 K in the effective temperature regression, 0.35 in the surface gravity regression, and 0.25 in the metallicity regression. We also discuss the advantages of this multi-model approach and compare it to other machine-learning methods.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Shaded fraction and backtracking in single-axis trackers on rolling terrain

A generalized closed-form equation for the shaded collector fraction in solar arrays on rolling or undulating terrain is provided for single-axis tracking and fixed-tilt systems. The equation accounts for different rotation angles between the shaded and shading trackers, cross-axis slope between the two trackers, and offset between the collector plane and axis of rotation. The validity of the equation is demonstrated through comparison with numerical ray-tracing simulations and remaining minor sources of error are quantified. Additionally, a simple procedure to determine backtracking rotations for each row in an array installed on the rolling terrain (varying in the direction perpendicular to the tracker axes) is provided. The backtracking equation accounts for a desired shaded fraction (including complete shade avoidance) as well as an axis-collector offset. Test cases are provided to facilitate implementation of these equations.

14 SOLAR ENERGY↗