On the analysis of an elliptical Gaussian flux image and its equivalent circular Gaussian flux images

For a continuous elliptical Gaussian flux image over the infinite X-Y plane, the parameter relationships between the elliptical Gaussian flux image and the equivalent circular Gaussian flux images are clearly discussed in both mathematical and graphical ways in this paper with respect to the radial power distribution around the image centre (peak flux location in image plane). This paper presents six typical methods (SIGMA-2Mean, SIGMA-1Mean, SIGMA-0Mean in Class One, SIGMA-RPeak, SIGMA-RMean, SIGMA-SqrRMean in Class Two) to give the equivalent circular Gaussian flux images to the elliptical Gaussian flux image, tries to link these circular Gaussian fitting methods to the relevant solar engineering applications, and makes some assessment comments on the elliptical/circular Gaussian modelling in solar mirror optics. By comparing the approximation performance among these six typical fitting methods, it reveals the reason for the 90% intercept over-estimation phenomenon of Francisco J. Collado’s one-point circular Gaussian fitting practice relative to the experimental flux image. The detailed algorithm for automatically finding out the major/minor axes and the image centre of the digital elliptical flux image is also provided in this paper. SIGMA-2Mean and SIGMA-SqrRMean are equivalent for an elliptical flux image, but they are applied in different ways to figure out either the reasonable intercept factor of the experimental flux image on the physical target plane respecting the aperture region of interest, or the power spillage over the limited experimental target plane. At last, this paper introduces the interpolation reconstruction of an elliptical Gaussian flux image over a rectangular domain just based on the boundary pixel values, so it is quite useful for solar engineering, such as fast simulation of a flux image concentrated by a mirror, and also instant approximation of the flux density over the receiver aperture by the linear array of radiometers around the receiver aperture, when the central receiver system is in the normal working state.     

Measurements of solar flux density distribution on a plane receiver due to a flat heliostat

An experimental facility is designed and manufactured to measure the solar flux density distribution on a central flat receiver due to a single flat heliostat. The tracking mechanism of the heliostat is controlled by two stepping motors, one for tilt angle control and the other for azimuth angle control. A x-y traversing mechanism is also designed and mounted on a vertical central receiver plane, where the solar flux density is to be measured. A miniature solar sensor is mounted on the platform of the traversing mechanism, where it is used to measure the solar flux density distribution on the receiver surface. The sensor is connected to a data acquisition card in a host computer. The two stepping motors of the heliostat tracking mechanism and the two stepping motors of the traversing mechanism are all connected to a controller card in the same host computer. A software “TOWER” is prepared to let the heliostat track the sun, move the platform of the traversing mechanism to the points of a preselected grid, and to measure the solar flux density distribution on the receiver plane. Measurements are carried out using rectangular flat mirrors of different dimensions at several distances from the central receiver. Two types of images were identified on the receiver plane—namely, apparent (or visible) and mirror-reflected radiation images. Comparison between measurements and a mathematical model validates the mathematical model.     

Preliminary design of surrounding heliostat fields

Recently, the author has shown elsewhere a simplified model that allows quick evaluations of the annual overall energy collected by a surrounding heliostat field. This model is the combination of an analytical flux density function produced by a heliostat, developed by the own author, and an optimized mirror density distribution developed by University of Houston for the Solar One Project. As main conclusion of this previous work, it was recognized that such pseudo-continuous simplified model should not substitute much more accurate discrete evaluations, which manage thousands of individual heliostat coordinates. Here in this work, the difficulty of generating a preliminary discrete layout of a large number of heliostats is addressed. The main novelty is the direct definition of thousands of heliostat coordinates through basically two parameters i.e. a simplified blocking factor and an additional security distance. Such procedure, which was formerly theoretically suggested by the author, is put into practice here, showing examples and commenting their problems and advantages. Getting a previous set of thousands of heliostat coordinates would be a major first step in the complex process of designing solar power tower (SPT).     

Flux distribution of solar furnace using non-imaging focusing heliostat

This paper discusses on the flux distribution of a quasi-single stage solar furnace which consists of a non-imaging focusing heliostat as the primary stage and a much smaller spherical concentrator as a secondary. As the optics of the primary stage heliostat is of non-imaging nature, the analytical method for studying the flux distribution of the hot spot of this type of solar furnace would be complicated. Therefore, a digital simulation approach has been employed. Flux distributions of the hot spot for several different incident angles, which have covered all the extreme cases of operating conditions have been simulated. Simulation result shows that a solar furnace using an 8 × 8 m non-imaging focusing heliostat with 289 mirrors coupled with a spherical concentrator with 0.7 m aperture and 27 cm focal length is theoretically capable of achieving flux concentration of 25,000 suns. Concentration contours of flux distribution for several interesting cases are presented, the different working areas of high flux footage from 5000 to 15,000 suns have been compiled.     

An analytic function for the flux density due to sunlight reflected from a heliostat

This paper presents an analytical model for the flux density due to a focused heliostat over the receiver plane of a tower solar plant. The main assumptions are: spherical and continuous surface of the mirror, linear conformal transformation in the complex plane equivalent to the reflection mapping between an on-axis aligned heliostat and the objective located on the receiver at the slant range necessary to produce the minimum circle of confusion, circular Gaussian distribution of the effective sunshape and the concentration function constant on the receiver or the image plane. Under the hypotheses presented earlier an exact convolution is obtained. The result, an analytic flux density function, relatively simple and very flexible, is confronted with experimental measurements taken from four heliostat prototypes of second-generation placed at the Central Receiver Test Facility (CRTF), Albuquerque, New Mexico, and compared indirectly with the predictions of the Helios model for the same heliostats. The model is an essential tool in the problem of the determination of collector field parameters by optimization methods.     

Automated high resolution measurement of heliostat slope errors

A new optical measurement method that simplifies and optimizes the mounting and canting of heliostats and helps to assure their optical quality before commissioning of the solar field was developed. This method is based on the reflection of regular patterns in the mirror surface and their distortions due to mirror surface errors. The measurement has a resolution of about 1 million points per heliostat with a measurement uncertainty of less than 0.2 mrad and a measurement time of about 1 min per heliostat. The system is completely automated and allows the automatic measurement of an entire heliostat field during one night. It was extensively tested at the CESA-1 heliostat field at the Plataforma Solar de Almería. Comparisons of flux simulations based on the measurement results with real flux density measurements were performed. They showed an excellent agreement and demonstrated in a striking manner the high measurement accuracy and high grade of detail in the simulation achieved by this technique.     

An analytical method for reflected flux density calculations

Conception, evaluation and real time control of solar “power tower” systems require the use of fast and accurate computer programs for calculating the flux density distributions on the receiver. Since the classical methods of “cone optics” and “hermite polynomial expansion” have some limitations of speed and accuracy, we have built an analytical model for calculating the convolution of the solar brightness distribution with the principal image of a heliostat (i.e. the fictive image for a “point sun”). We first characterize a principal image of a focusing heliostat by its shape and its geometrical concentration factor. Then this image is projected back onto the central plane (which passes through the center of the mirror), and considered as a flat reflecting surface. And the problem is reduced to density calculation for a flat heliostat. For each point of the receiver, the density of flux reflected by a heliostat is obtained by direct resolution of a convolution integral. The different formulations used to express the density function correspond to the various types of intersections between the image of the solar disk for the considered point and the principal image of the heliostat. Confrontation of this method with a program based on “cone optics” shows a good concordance of results and a strong decrease of computation time. We want to apply this method to the existing “THEMIS” solar plant built in France and to compare our results with real observations. Our density calculation programs will help conceiving fields of focusing heliostats for a new generation of power systems (gaz turbine systems).     

A numerical approach to the flux density integral for reflected sunlight

A computer model of the central receiver system must evaluate the flux density on the receiver due to sunlight reflected by the heliostats in the collector field. Several approaches are available but each has its limitations. The Monte-Carlo approach represents all of the heliostat behavior but is relatively slow in terms of CPU time and is not suitable for optimization purposes. FLASH is an analytically exact approach for flat polygonal heliostats but is slow and not applicable to dished heliostats or aureole effects. Cone optics programs evaluate the flux density by a direct numerical integration of the double integral, but this method is very slow if accuracy is required. HCOEF is a two dimensional Hermite polynomial method which is relatively fast and can be extended to include canting, focusing, solar limb, and guidance error effects. However, the polynomial approximation breaks down for near heliostats, small guidance errors, and aureole effects. The new image generators based on KGEN overcome this limitation, but running times compare to FLASH and are 3 or 4 slower than HCOEF.The new approach proposed in this study assumes isotropic gaussian guidance errors. Hence, the flux density integral reduces to several iterated single integrals which can be precalculated and stored in a table for interpolation as needed. The LBL solar telescope data are fed into a convolution integral which represents the guidance errors. Aureole effects can be switched on or off at this point. A vector of convoluted solar data is input to another integration which gives the table of normalized flux contributions. The tabular values depend on the position of the flux point with respect to an edge of the heliostat as seen in the image plane. The image map of the heliostat is linear unless ripples or irregularities occur; hence, effects due to canting and dishing can be included by a ray trace of the heliostat vertices.The use of tabular interpolation is not as fast as expected because of the time required to calculate the distance between the flux point and the image of the vertices. The accuracy of this method is limited by interpolation errors, and better results can be obtained with the same CPU time if more core is used for a larger table. It is possible to eliminate the table by introducing a Romberg type of integrator which bisects the interval until sufficient accuracy is achieved; however, this approach is inefficient unless the images are relatively small compared to the receiver.The convolution process in KGEN is fast and can be used to calculate moments for HCOEF and coefficients for FLASH which utilize the LBL data.     

An optimization tool to design the field of a solar power tower plant allowing heliostats of different sizes

The design of a solar power tower plant involves the optimization of the heliostat field layout. Fields are usually designed to have all heliostats of identical size. Although the use of a single heliostat size has been questioned in the literature, there are no tools to design fields with heliostats of several sizes at the same time. In this paper, the problem of optimizing the heliostat field layout of a system with heliostats of different sizes is addressed. We present an optimization tool to design solar plants allowing two heliostat sizes. The methodology is illustrated with a particular example considering different heliostat costs. Copyright © 2017 John Wiley & Sons, Ltd.     

Performance enhancement of parabolic trough collectors by solar flux measurement in the focal region

Characterization of the optical performance and detection of optical losses of parabolic trough collectors are very important issues in order to improve the optical efficiency of these systems and to ensure the desired quality in solar power plants. Therefore two methods of measuring the solar flux in the focal region were developed: PARASCAN (PARAbolic Trough Flux SCANner) is a solar flux density measurement instrument which can be moved along the receiver axis. The sensor registers the flux distribution in front and behind the receiver with high resolution. The resulting flux maps allow to calculate the intercept factor and to analyse the optical properties of the collector at the finally interesting location, i.e. around the receiver. The camera-target-method (CTM) uses a diffuse reflecting Lambertian target and a calibrated camera which takes pictures of it. The target is held perpendicular to the focal line surrounding the receiver. With the resulting images of this fast and easy method it is possible to visualize the paths of the reflected rays close to the receiver and to detect local optical errors. Both methods are described in detail. Latest measurement results gained at the Eurotrough-II prototype collector built on the Plataforma Solar de Almería (PSA) in Spain are presented and consequences are discussed.     

Energy and exergy analyses and optimization study of an integrated solar heliostat field system for hydrogen production

In this paper, we propose an integrated system, consisting of a heliostat field, a steam cycle, an organic Rankine cycle (ORC) and an electrolyzer for hydrogen production. Some parameters, such as the heliostat field area and the solar flux are varied to investigate their effect on the power output, the rate of hydrogen produced, and energy and exergy efficiencies of the individual systems and the overall system. An optimization study using direct search method is also carried out to obtain the highest energy and exergy efficiencies and rate of hydrogen produced by choosing several independent variables. The results show that the power and rate of hydrogen produced increase with increase in the heliostat field area and the solar flux. The rate of hydrogen produced increases from 0.006 kg/s to 0.063 kg/s with increase in the heliostat field area from 8000 m² to 50,000 m². Moreover, when the solar flux is increased from 400 W/m² to 1200 W/m², the rate of hydrogen produced increases from 0.005 kg/s to 0.018 kg/s. The optimization study yields maximum energy and exergy efficiencies and the rate of hydrogen produced of 18.74%, 39.55% and 1571 L/s, respectively.     

Rotary-gradient fitting algorithm for polarization curves of Proton Exchange Membrane Fuel Cells (PEMFCs)

Experimental data obtained in electrochemistry traditionally have been fitted to models in order to obtain relevant parameters of the underlying processes. Polarization curves are among the most important representations of fuel cell performance, as they are useful tools for studying the performance of these cells in operation. On this basis, we developed an algorithm to fit the experimental polarization curves employing two different models. This algorithm combines different optimization techniques, such as rotary optimization and gradient optimization. Seven experimental polarization curves, obtained with different experimental proton exchange membrane fuel cell (PEMFC) membrane electrode assemblies (MEAs), are used to evaluate the proposed fitting techniques and theoretical models. The average quadratic errors of the models fitted by the algorithm are below 9 mV in all the curves, much less than the 750–1000 mV voltage variation in them. Therefore, we propose that the algorithm and the models are good options for use in fitting these data.     

Quick evaluation of the annual heliostat field efficiency

The recent world wide interest in solar power tower justifies the presentation of a simplified model that allows quick evaluations of the annual overall energy collected by a surrounding heliostat field, which is sent to the electric power generating system (EPGS). The model is the combination of an analytical model of the flux density produced by a heliostat from Zaragoza University, an optimized mirror density distribution developed by University of Houston for the Solar One project and molten salt receiver efficiencies measured during the Solar Two project. The abilities of the model are successfully compared against the scarce open data about the next Solar Tres demonstration plant-a 15 MWe solar tower with molten salts storage. This simplified model could be valid for rather preliminary optimizations, although it should not substitute much more accurate discrete evaluations that manage thousands of individual heliostats with their actual shadowing and blockings, performed every few minutes using actual meteorological data.     

Off-axis aberration correction surface in solar energy application

A new fixed aberration correction surface for applications in solar energy is proposed. The surface equation was derived based on the theory of focusing heliostat and the technique of canting the mirror array at a particular incident angle. The performance of the optimized new surface is simulated and compared with that of a spherical surface for two different ranges of incident angles, i.e., 0°–33° and 33°–57°. The comparison result shows that there are considerable improvements: not only is the size and concentration of the sun’s image improved, but it is also possible to shift the hot spot from noon time to early morning and late afternoon.     

Wind power statistics and an evaluation of wind energy density

In this paper the statistical data of fifty days' wind speed measurements at the MERC-solar site are used to find out the wind energy density and other wind characteristics with the help of the Weibull probability distribution function. It is emphasized that the Weibull and Rayleigh probability functions are useful tools for wind energy density estimation but are not quite appropriate for properly fitting the actual wind data of low mean speed, short-time records. One has to use either the actual wind data (histogram) or look for a better fit by other models of the probability function.     

A new code for the design and analysis of the heliostat field layout for power tower system

A new code for the design and analysis of the heliostat field layout for power tower system is developed. In the new code, a new method for the heliostat field layout is proposed based on the edge ray principle of nonimaging optics. The heliostat field boundary is constrained by the tower height, the receiver tilt angle and size and the heliostat efficiency factor which is the product of the annual cosine efficiency and the annual atmospheric transmission efficiency. With the new method, the heliostat can be placed with a higher efficiency and a faster response speed of the design and optimization can be obtained. A new module for the analysis of the aspherical heliostat is created in the new code. A new toroidal heliostat field is designed and analyzed by using the new code. Compared with the spherical heliostat, the solar image radius of the field is reduced by about 30% by using the toroidal heliostat if the mirror shape and the tracking are ideal. In addition, to maximize the utilization of land, suitable crops can be considered to be planted under heliostats. To evaluate the feasibility of the crop growth, a method for calculating the annual distribution of sunshine duration on the land surface is developed as well.     

Optimal time-invariant operation of a power and water cogeneration solar-thermal plant

Conceptual design, system-level models, and optimization of operation are presented for a cogeneration solar-thermal plant. The solar-thermal energy collected and concentrated in a salt pond is used in a regenerative Rankine steam cycle with an extraction turbine to produce electricity and process steam. The desalination system is based on reverse osmosis (RO) and multi-effect distillation (MED). An equation-oriented modeling environment is used for the development of time-dependent system-level models required for optimization of the plant. A meteorological radiation model is used to estimate the hourly distribution of beam radiation as a function of time (day and hour), location, and local weather (mainly visibility and humidity). A recently developed model is used to estimate the field efficiency, including projection losses and shading/blocking for a given heliostat layout. Time-invariant optimal operating conditions are presented for a summer day, considering Cyprus as a case study. Seawater desalination processes, RO and MED, are modeled by adapting and extending models from the literature. A control-volume model is developed for the steam cycle based on the first and second law, with given isentropic efficiencies, turbine leaks, and a detailed model for thermodynamic properties of steam/water. This model is validated and allows for optimization over a wide range of operating conditions, e.g., various extraction pressures. The optimization problem is formulated as a nonlinear program (NLP) with dynamics embedded and a heuristic global optimization approach is used. The sequential method of optimization is used, decoupling the simulation from the optimization. The results show that for the plant size considered (4 MW_e equivalent nominal capacity) and the MED design chosen based on the literature and industry practice, RO is preferred over MED from an energy point of view. In addition, under the current feed-in tariff (FiT) and water prices in Cyprus, extracting steam for MED is not recommended. In contrast, if current market prices for electricity and water in Cyprus are used, i.e., FiT is neglected, with a typical steam cycle design, extracting steam for MED at low pressures yields maximum income. A new process configuration is presented based on the findings from the case studies, resulting in significantly higher income and exergetic efficiencies.     

A solar furnace using a horizontal heliostat array

A two-component solar furnace, condenser-heliostat combination, is described in which the condenser faces downward at 30° towards a heliostat comprised of numerous rows of plane mirrors mounted on a horizontal turntable. It is shown that for a south-facing condenser, with the angle of the final flux beam limited to 30° below the horizontal, the rows of heliostat mirrors may be mounted so they overlap, resulting in a reduction of the edge losses occurring when the heliostat mirrors are all held in a single plane. The over-all size of the heliostat turntable is calculated for a 6-hour workday throughout the year, and a suggestion is made for using the heliostat control mechanism to provide shutter action. The saving in flux possible by the elimination of an independent shutter is estimated at about eight per cent.     

High resolution dual-plane stereo-PIV for validation of subgrid scale models in large-eddy simulations of turbulent premixed flames

A measurement strategy for the experimental validation of subgrid scale (SGS) models for large-eddy simulations (LES) of turbulent premixed flames is presented. The approach is based on a dual-plane stereo-PIV technique. The measurement of the flow field is performed in two parallel planes which allows the determination of velocity gradients in all three directions. The flame front position in the PIV images is deduced from the clearly observable step in the particle number density between burnt and unburnt gases. This facilitates the single-shot based evaluation of important quantities for reacting flows, e.g., the density weighted rate-of-strain tensor. Also filtered quantities like the SGS scalar flux of the reaction progress variable can be directly determined by spatial averaging over several regions of interest which reproduces the application of the filter function in LES. Moreover, the measured data allows the direct interpretation of SGS model formulations since besides the filtered values, also resolved data are generated. Thus, statistical a-priori tests of SGS models are possible. The measurement strategy is explained, a statistical evaluation of the density weighted rate-of-strain tensor is given, and exemplarily an instantaneous distribution of the measured radial SGS scalar flux is compared with predictions of two models, the gradient diffusion model and the Clark model. Starting from a reference operation point of a turbulent V-shaped flame the following three parameters - Reynolds number, fuel-air ratio and fuel type - have been varied independently. First results show that the gradient diffusion model fails completely, while the Clark model predictions show a high degree of correlation to the directly determined flux components, especially in the reactant zone. More advanced modeling, however, may be needed, to incorporate for instance heat-release effects more closely.     

Heliostat field optimization: A new computationally efficient model and biomimetic layout

In this article, a new model and a biomimetic pattern for heliostat field layout optimization are introduced. The model, described and validated herein, includes a detailed calculation of the annual average optical efficiency accounting for cosine losses, shading and blocking, aberration and atmospheric attenuation. The model is based on a discretization of the heliostats and can be viewed as ray tracing with a carefully selected distribution of rays. The prototype implementation is sufficiently fast to allow for field optimization. Parameters are introduced for the radially staggered layout and are optimized with the objective of maximizing the annual insolation weighted heliostat field efficiency. In addition, inspired by the spirals of the phyllotaxis disc pattern, a new biomimetic placement heuristic is described and evaluated, which generates layouts of both higher insolation-weighted efficiency and higher ground coverage than radially staggered designs. Specifically, this new heuristic is shown to improve the existing PS10 field by 0.36% points in efficiency while simultaneously reducing the land area by 15.8%. Moreover, the new pattern achieves a better trade-off between land area usage and efficiency, i.e., it can reduce the area requirement significantly for any desired efficiency. Finally, the improvement in area becomes more pronounced with an increased number of heliostats, when maximal efficiency is the objective. While minimizing the levelized cost of energy (LCOE) is typically a more practical objective, results of the case study presented show that it is possible to both reduce the land area (i.e. footprint) of the plant and number of heliostats for fixed energy collected. By reducing the capital cost of the plant at no additional costs, the effect is a reduction in LCOE.