optical and structural optimization by linking fem and ...

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CIRCE is able to evaluate any combination of concentrator and ... Furthermore, CIRCE takes into account the so-called sun shape and existing mirror errors.
OPTICAL AND STRUCTURAL OPTIMIZATION BY LINKING FEM AND RAY TRACING ANALYSIS Christof Husenbeth1, Bernd Zwingmann2, Wolfgang Schiel3 1

Mathematician, Physicist, schlaich bergermann und partner (sbp sonne GmbH)

Address: Schwabstr. 43, 70197 Stuttgart, Germany, Phone: +4971164871912, E-mail: [email protected] 2 3

Civil Engineer, Dipl.-Ing., schlaich bergermann und partner (sbp sonne GmbH)

Managing Director, Dipl.-Phys., schlaich bergermann und partner (sbp sonne GmbH)

Abstract Concentrating solar power (CSP) plants require continuous research and development in order to improve their efficiency and competitive ability with conventional power plants. In this paper, a new method of numerical optimization of solar concentrators is described. The automated optimization tool presented here is generated by linking commercially available structural, math, and ray tracing software with self-programmed interfaces and coupling software. This development enables both structural and optical optimization of the concentrator during system design process. Keywords: Solar concentrator, solar engineering, finite element analysis, ray tracing analysis, numerical optimization 1. Introduction The development and the continuous improvement of solar concentrators (see Fig. 1) are two primary focuses of the scope of work of schlaich bergermann und partner. To design most efficient solar concentrators, over 30 years of experience, expert knowledge in various disciplines and modern methods of engineering are applied. One pillar of current concentrator advancement is the method of numerical optimization. In order to enable commercial engineering software to meet the requirements of solar concentrator optimization, additional interfaces and coupling software needed to be programmed. With this new concentrator optimization tool, a fully automated optimization process of solar concentrators has been developed, including ray tracing and structural analysis.

Fig. 1. Infinia PowerDish with gore facets 2. Basics and methods 2.1. Finite element method (FEM) In structural engineering, finite element analysis is commonly used to predict a structure’s behavior. Realistic material properties and geometry are used to obtain a numerical model of the structure, known as the stiffness

matrix. This matrix contains all essential properties for the structural calculations. Every matrix element relates to the stiffness properties of one (finite) part of the continuous structure. Due to the finite discretisation of the real structure, the stiffness matrix has to be seen as an approximation whose quality increases with the level of detailing. To simulate the structure’s behavior, realistic assumptions about the environmental boundary conditions are required. The loads acting on the structure, which cause movements and deformations, can be taken from appropriate codes or wind tunnel tests. The main loads for solar concentrators are self-weight and wind load. Various load scenarios within the concentrator’s operation modes need to be defined and have to be added to the finite element program input. To calculate the structure's deformation, the stiffness matrix is combined with every single load vector of the defined load scenario. The resulting vector of the analysis is a vector of displacements for all finite elements of the structure. The final structural response due to each load scenario is obtained by adding the deformation to the initial geometry. For every modification of the structure’s properties, the stiffness matrix has to be updated. Loads may also change due to changes of mass and geometry. The entire calculation has to be started from beginning again, resulting in an iterative design process. 2.2. Ray tracing analysis of optical performance For the optical analysis ray tracing software is used to precisely calculate the optical performance of a solar concentrator. With a suitable discretization of the concentrator and receiver surface, ray tracing software is able to determine the optical flux density for each point in the receiver area. The flux density is given in W/cm² or any equivalent unit. The data set of flux densities at all points of the discretized receiver area is referred to as flux distribution or flux matrix. All important properties of the analyzed concentrator can be derived from this flux distribution, as there are: intercept factor, total power to the received flux maximum, flux gradient etc. The ray tracing software used is CIRCE [2]. CIRCE is able to evaluate any combination of concentrator and target (receiver) geometry. Concentrator geometry can be based on parametric input or on discretized collector surfaces, such as the output of other software. Furthermore, CIRCE takes into account the so-called sun shape and existing mirror errors. The sun shape describes the brightness distribution of the sun, given as power per area and solid angle. This distribution gives information about the brightness of the sun surface as well as the circumsolar radiation, as a function of the angular distance from the center of the sun disk. Mirror errors result from unavoidable imperfections during mirror fabrication, assembly and tracking. They are treated in a statistical way, as a Gaussian distribution of the surface normal vectors which result in a Gaussian distribution of the reflected rays. Mirror errors and sun shape yield an angular brightness distribution which is called effective sun shape. Mathematically, it is a convolution of the error distribution and the sun shape.

Fig. 2. From left to right: gore dish, mirror facet and sub facet with surface normal

CIRCE discretizes each mirror facet into a large number of small mirror areas, so-called sub facets. Each sub facet is represented by its center point (x,y,z-coordinate) and its surface normal (see Fig. 2). An incoming solar ray is reflected at the center point of these sub facets. The result is the central reflected ray from each sub facet. The reflected power of a sub facet is then received by the target points that surround the central reflected ray. The distribution of the power to these receiver points is calculated by means of the effective sun shape. The flux distribution on the receiver surface is calculated by superimposing the flux distributions generated by all sub facets. The calculated flux densitiy values of all receiver points are then given as an ASCII text file, which can be further processed by post processing software. The ray tracing software itself needs a few seconds up to a few minutes for a complete ray tracing calculation. A typical calculation time takes approximately 10 seconds. 2.3. Optimization Mathematical optimization in general is the calculation of an extreme value of a so-called objective function [4]. The objective function describes the magnitude of a value depending on a number of input parameters. Using an optimization method, the parameters which result in an extreme value (generally a minimum value) of the objective function are obtained. Nonlinear optimization problems with various input parameters are complex challenges. The numerical effort increases exponentially with the number of parameters. The numerical stability of the optimization decreases with the number of parameters and the degree of nonlinearity. Although the shape of the objective function is normally unknown, structural optimization problems can be considered as highly nonlinear. To solve these problems, iterative numerical methods like the downhill simplex method or the gradient descent method were developed. Other possible methods are the Monte Carlo Method or evolution strategies. The Nelder-Mead downhill simplex method is a method for the optimization of nonlinear functions. Its main advantage consists in its stability. It is implemented as a predefined MATLAB[3] function called fminsearch. 3. Optimization of solar concentrators 3.1. Motivation Optimization of CSP plants aims to increase system efficiency and decrease system cost. Highly efficient plants with optimized operation performance will be able to compete with traditional systems that have been developed over a period of more than 100 years. Today, with the help of modern computational tools, the design of solar concentrators can approach their optimum in a short period of time. Optimized solar systems promise a more efficient use of resources. Optimized concentrators require less material and, due to their higher performance, the same amount of energy is produced on a smaller area of land. In addition to the important development of new innovative solar technologies, optimization of existing systems supports the evolution of solar technology. 3.2. Areas of optimization The optimization of solar concentrators can be classified into two areas, i.e. optical and structural optimization. Furthermore, there are some other possible areas of optimization, e.g. assembly process, system maintenance, material recycling and operation control, among others which are not discussed in this article. These not discussed areas of optimization are less applicable for numerical optimization because they are not continuous mathematical problems. The main objective of a solar concentrator optimization process is the improvement of its optical performance. The optical performance can be improved by changing of structural parameters of the concentrator. This procedure is typical for a well defined structure where small changes in the structure may have a significant impact on concentrator performance. One example is the optimization of the number and

position of mirror support points to reduce mirror deflection under dead and wind load (improving performance) by acceptable mirror stress levels (ensure mechanical stability). A fixed number of supports are moved along the mirror area to obtain the location that generates the best optical performance during operation. Structural boundary conditions like mirror stress are checked before and after the optimization process. The aim of structural optimization is to reduce redundant structural parts or evaluate new structural ideas on the background of optical requirements. Consequently, the desired optical quality is defined as a boundary condition in structural optimization process. The impact of structural changes on the optical quality is repeatedly compared to this reference value. Structural design criteria are checked for every change of geometry. One example for this procedure is the length-to-height relationship of solar concentrating elements (SCE) for parabolic troughs. 3.3. Objective functions The objective function of the optimization process and its parameters are defined dependent on the area of optimization. For the optical evaluation of a concentrator, the deformation of the reflecting surface is the main criteria. Within the deformation, the rotation of a surface element around its axes perpendicular to the sun ray direction is especially sensitive. To generate an optical objective function, the deformations calculated by FEM are evaluated. This can be done by a simple average value or root mean square (RMS) calculation or by using ray tracing software. Ray tracing software directly calculates the flux distribution on the receiver and enables the evaluation of the optical performance. Each type of concentrator should be optimized using the most appropriate kind of objective function. For parabolic troughs, the intercept value is usually used. For dish-stirling and solar tower systems, high intercept and limited maximum flux values are required. For concentrating photovoltaics (CPV), the constant flux distribution on the entire receiver surface is used as objective function. The main objective function used for structural optimization is the low mass criteria. The mass of the structure is minimized by using fewer elements, reducing the mass of elements and changing connection details. Parameters that influence the mass, e.g. the number of support points and the thickness of the reflecting mirrors, are included in this function. A simplified method of structural optimization is estimating the optical performance based on structural behavior, i.e. beam deflection or rotation of the entire structure. These criteria cannot accurately describe the optical performance. They are, however, used as a fast comparison value between different systems, especially at the beginning of the concentrator design process. The global structural behavior is not generally applicable as an objective function for optical optimization, because it only provides relative information about optical performance. Only ray tracing analysis can give absolute results about flux distribution and optical performance. 3.4. Parameters and boundary conditions The parameters of the objective function are the varying values in an optimization problem. The parameters of optical and structural optimization are structural properties, or simply the concentrator’s geometry. For the optimization process, parameters with high influence on the objective function are used. The fewer parameters are chosen, the more probable a successful application becomes. Typical examples for optimization parameters are material thickness, beam span, cross section size and juncture location. The most important boundary condition for all optimization processes is the structural design verification. All valid concentrator geometries have to fulfill the design requirements in ultimate limit state. This means that the structure will not fail in survival situations, independent of optical performance. The survival load case does not change during the optimization process. In addition to this condition, the parameters themselves are limited to a maximum value. The material thickness cannot increase endlessly. The number of connection points is also limited as well as the crosssection size is restricted by connection details.

For structural optimization, minimal optical boundary conditions are defined which have to be reached by every valid geometrical solution during the optimization process. One example is the minimum intercept or maximum peak flux on the solar receiver. 4. Linking FEM, ray tracing and optimization tool 4.1. Optimization cycle The automatized numerical optimization requires an exchange of information between the three main parts of the calculation. In the FEM part, the geometry of the concentrator is defined by supplied input parameters. Deformations and structural design information of the concentrator is obtained by calculations based on environmental conditions. Finally, information about the optical performance changes due to these deformations is generated by ray tracing software. During each cycle of the optimization process, the objective function is calculated with the input parameters, the obtained performance and the structural design considering all boundary conditions. By interpretation of the objective function solution, new input parameters are obtained using the optimization method and a new optimization cycle (see Fig. 3) starts.

Fig. 3. Optimization cycle The entire cycle is controlled by the optimization tool. In every cycle of the optimization process, the input parameters are obtained by the function of the optimization method. The tool also starts the subsequent FEM program and ray tracing software and ends the optimization process when the defined end conditions are achieved, including the maximum process size. The optimization tool documents the history of the process and executes the commands. 4.2. Linking elements Linking elements have to be defined to provide the interfaces for communication between the programs. General information about the optimization process has to be entered into the optimization tool, including the objective function, the input parameters that have to be optimized, optimization options, and boundary conditions. This initial input is only done once at the beginning of the process. The first interface is the entry of the input parameters to the FEM software in each cycle. At the second interface, the geometry, deformation and structural design information is returned from the FEM software to the optimization tool. After preprocessing, the concentrator data is given to the ray tracing software by the third interface. The fourth and last interface is the output of the ray tracing solution back to the optimization tool. 4.3. Proposal for realization of an optimization tool In the following the development of a software tool providing all the required linking elements and preprocessing needed to integrate commercial software into an optimization process is described. The main program of the tool is a MATLAB based program with an input window that controls the subsequent programs. The FEM calculation is done by SOFiSTiK [1] and the ray tracing analysis by CIRCE. These programs have input and output interfaces based on ASCII text files, which make program-to-program communication possible. In the SOFiSTiK input file geometry and loads of the solar concentrator are defined in various modules using CADINP language. Executing the sps.exe from the optimization tool interprets the file and calculates the concentrator for the defined scenarios. The optimization parameters that are defined in the head of the SOFiSTiK input file are the determining values for the geometry generation. The parameters are changed by

overwriting the files head caused by the optimization tool in every cycle of the optimization process. The solution of the FEM calculation is stored in a binary database, which includes all inputs and results. The initial SOFiSTiK input file is programmed by a structural engineer as the starting point for the optimization process. For the model, shell and beam elements are used to simulate the properties of the structure. The reflecting surface is made out of plane elements that are defined by their four corners. The required information about geometry, deformation, forces, stress and utilization level of the concentrator is exported from the database by programmed routines and written out in an ASCII file which is the basis for the following calculations. With the FEM output file, the preprocessing of the optical ray tracing calculation is started. The preprocessing is done by the optimization tool. CIRCE requires a specially formatted ASCII file which includes the center point and normal direction of the curved reflecting mirror surface. This information is obtained by superimposing the initial geometry on the FEM calculated edge displacements. Additionally, information about sun shape, optical surface errors and receiver geometry are needed. The input data is provided by the user at the beginning and stays constant during the optimization process. In the end of the ray tracing analysis, a flux matrix is stored as an ASCII data set. The flux matrix describes the flux distribution over the entire receiver. It is imported into the optical post processing of the optimization tool and the relevant values for the optimization process are calculated. Relevant values are, for example, receiver intercept, maximum flux and flux gradient. In the last step of the optimization cycle, the structural and the optical solution is compared to the requirements of the optimization problem and the solutions of previous cycles. The validity of the obtained solution is checked by comparing the results to the given boundary conditions. The optimization method obtains the input parameters for the next cycle. If the minimum requirements for an optimum solution are achieved, the optimization process ends successfully. If the maximum number of loop cycles is reached without an adequate solution, the optimization ends without success. In the case of an unsuccessful optimization, the optimization process can be started again with fewer requirements. The most effective of the incomplete optimization outcomes can be used, or the optimization problem can be changed to improve the chance of a successful optimization outcome. 5. Example 5.1. Description The optimization method is presented in the following example. A gore mirror with four support points is to be optimized to produce maximum power on the receiver. Structural input parameters were manipulated to maximize power under load conditions. The mirror is supported by support bearings, whose location on the back surface of the mirror can be varied. The locations of the support bearings are referred to as support points. Their location can be defined by the following four parameters utilizing the mirror’s symmetry (see Fig. 4): -

Inner radial distance x1 r, i.e. the distance between the inner support points and the inner edge of the facet

-

Inner edge distance x2 edge of the facet

-

Outer radial distance x3 r, i.e. the distance between the outer support points and the outer edge of the facet

-

Outer edge distance x4 edge of the facet

, i.e. the distance between the inner support points and the left (resp. right)

, i.e. the distance between the outer support points and the left (resp. right)

Fig. 4. Geometry and parameters of the investigated mirror If these edge distances are very small, the center of the mirror will sag because of its dead load, which will lead to relatively large rotations of the mirror surface. On the other hand, very large edge distances will cause sagging of the edge of the facet, which will also lead to large surface rotations. An optimization of the edge distances shall yield the locations of the support points that minimize optical losses that are caused by sagging of the mirror surfaces. 5.2. Optimization process The variable that has to be optimized is the vector x = [x1 x2 x3 x4], which contains the variables that determine the locations of the support points. The values r1, r2 and are constants. The objective criterion of the optimization process is to maximize the received power Prec on a circular receiver. Under load conditions, it depends on the locations of the support points, so that Prec can be written as Prec(x). As optimization algorithms commonly look for minimum values, the objective function f must be defined in such a manner that a maximum value of the power on the receiver leads to a minimum of the objective function, e.g. f (x) = 1/Prec(x) or f (x) = - Prec(x). The evaluation of the objective function f for a certain value of the variable x consists of the following steps: 1. The values x1, x2, x3 and x4 contained in the vector x are input in the corresponding variable declaration in the SOFiSTiK input file that controls the FEM calculation. 2. The FEM calculation is carried out under dead load conditions. The result is the geometry of the concentrator, which is given as the undeformed geometry and the deformations due to considered load conditions. This information is stored as matrices in ASCII files that are created after the FEM analysis by a self programmed interface routine inside the FEM software. 3. These text files are read in by the coupling software, which serves as preprocessing for the ray tracing software. The coupling software calculates the information that is needed by the ray tracing software. This information consists of the coordinates of the center points of the discretized mirror sub facets, their surface normals and their projected areas. After calculating this information, the coupling software saves it in files that can be read in by the ray tracing software. 4. The ray tracing algorithm is carried out and the flux distribution on the receiver surface is obtained. 5. The coupling software integrates the flux distribution and the power on the receiver Prec is obtained. These steps are carried out until the minimum value of the objective function is found. The MATLAB routine fminsearch is used for this purpose.

5.3. Summary of results The developed method was carried out in order to optimize the locations of the pad positions. With nonoptimized locations, the flux distribution of the deformed concentrator was wider spread than with optimized positions of the pad locations, as shown in Fig. 5. For reference, the flux distribution of the undeformed concentrator is also shown. It is demonstrated that structural deformation has a great impact on the flux distribution. By optimization, the power on the receiver increases and thus the efficiency of the concentrator is improved.

Fig. 5. From left to right: finite element model of the example, flux distribution for undeformed shape, flux distribution under dead load without and with optimized support location. 6. Conclusion The combination of different software tools (optical, mechanical, mathematical etc.) is used to generate a powerful optimization tool. This optimization tool enables an in-depth investigation of complex optimization challenges, e.g. structural and optical optimization of solar concentrators. Using numerical optimization, the efficiency and competitive ability of CSP plants can be improved, which supports the development of this technology. The optimization process includes the definition of sensitive structural parameters, the FEM supported calculation of the structure’s behavior under realistic environmental conditions and the interpretation of influence on optical performance by the deformed shape. Boundary conditions for structural properties, structural design and flux distribution can be added to the optimization process. Acknowledgements We thank David Wollin and Prateek Ropia for their support in programming and proofreading. References [1] SOFiSTiK AG Oberschleissheim, (2010). SOFiSTiK: Basics [2] V.J. Romero, (1994). CIRCE2/DEKGEN2: A Software Package for Facilittated Optical Analysis of 3-D Distributed Solar Energy Concentrators, Sandia Report SAND91-2238 [3] The MathWorks, Inc., (2010). MATLAB & Simulink: Releas notes for R2010a [4] J. A. Nelder, R. Mead, A simplex method for function minimization. Computer Journal, 7 (1965) 308-313