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1 Copyright 2012 by ASME

Proceedings of the ASME 2012 International Design Engineering Technical Conferences &Computers and Information in Engineering Conference

IDETC/CIE 2012August 12-15, 2012, Chicago, IL, USA

DETC 2012-71247

ADAPTIVE HELIOSTAT SOLAR ARRAYS USING SHAPE-OPTIMIZED COMPLIANT

MIRRORS

Li Meng Department of Mechanical EngineeringMassachusetts Institute of Technology

Cambridge, MA [email protected]

Amy M. BiltonDepartment of Aeronautics and Astronautics

Massachusetts Institute of TechnologyCambridge, MA 02139

[email protected]

Zheng YouDepartment of Precision Instruments and

Mechanology,Tsinghua University

10086 Beijing, [email protected]

Steven Dubowsky*Department of Mechanical EngineeringMassachusetts Institute of Technology

Cambridge, MA [email protected]

Li Meng is a visiting-student of Department of Mechanical Engineering at MIT.* Corresponding author, Email: [email protected].

ABSTRACTIn a Solar Power Tower (SPT) system, the ideal shape of a

heliostat concentrator is a section of paraboloid which is a

function of the location in the array and the incidence sunangle. This shape is difficult to achieve and limits the system

efficiency. A shape-optimized compliant (SOC) design of

parabolic heliostats is presented here to solve this problem. An

approximation of the ideal shape is suggested to use an

optimized stationary paraboloid shape which only varies with

heliostat location in the array. A compliant structure design is

proposed that to use a simple flat mirror with a two-

dimensional tailored stiffness profile to form the required

parabolic surface using adjustment mechanisms at each

corner. This design is validated by numerical simulations

including FEA tools, ray tracing, and classical nonlinear

optimization. The annual performance shows that the SOC

heliostat will substantially improve the efficiency and benefit

the SPT system.

1 INTRODUCTION

1.1 BackgroundHeliostats are mirrors used to collect solar energy for a

Solar Power Tower (SPT), a thermal power system producing

clean and renewable electricity from solar radiation [1]. Bydeploying thousands of heliostats in a field, which redirect andconcentrate sunlight to a single receiver mounted on a hightower, SPTs can achieve high temperature and powerconversion efficiency in thermal cycle. Due to their large scalethey are potentially economically advantageous. Substantiaresearch has been done on heliostats and SPTs. The mostsuccessful systems for both technology demonstration andcommercial use are Solar One/Two in the US [2] and PS10/20in Spain (See Fig. 1) [3]. New concepts and technologies arealso being developed in other countries including GermanyIsrael, and China [4-5]. The current state-of-art SPTs produceon the order of 10MW of power but unfortunately have lowefficiency and high lifetime cost. This cost is dominated bytheir initial capital cost and land cost. Significantimprovements in system performance and cost reduction (by50%) are required for SPTs to reach the price of conventionalfossil-fuel power [1]. Recently, volumetric receiversbiomimetic heliostat field layouts, and new system concepts

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like beam-down/hillside SPTs are being researched to makeSPTs more practical [4, 6-7].

Figure 1. SOLAR POWER TOWER SYSTEMPS10 AND PS20, SPAIN

1.

A heliostat consists of one or more reflector surfaces, asupport structure, and a two-axis tracking device. Ideally, theshape of the reflector surface should be a paraboloid section tofocus the majority of reflected light on the fixed receiver,reducing sunlight falling on the mirror but not reaching thereceiver which is called optical spillage loss. This paraboloidshape is a function of the mirror position in the field and thesun angle that changes with time, season, and geographiclocation. So, developing efficient heliostats is challenging inboth designing and manufacturing. Current heliostats areexpensive and constitute approximately 50% of the cost of atypical SPT design [8-9].

Figure 2. FLAT HELIOSTAT (LEFT2) AND

STRETCHED MEMBRANE HELIOSTAT (RIGHT3).

Various innovative methods have been developed for solar

concentrators like heliostats to reduce the cost and complexityof fabricating precision shapes. In one solution, designersdivide the single surface into smaller flat or concave facets andmount them onto a honeycomb-like support structure and aligneach of them to the target (See Fig. 2 Left) [8]. Nonetheless, aninevitable moderate focusing error still exists while the multi-1 http://sine.ni.com/cs/app/doc/p/id/cs-115502 http://en.wikipedia.org/wiki/File:Heliostat.jpg3 http://www.pveng.com/FEA/FEAGeneral/LargeDisplace/LargeDisplace.php

facet design introduces complexity considerably as the facetnumber increases.

In another solution, researchers have developed methodsto form the mirror shape using compliant structures. Stretchedmembrane heliostats (See Fig. 2 Right) [8-9] use controlledpressure to adjust the reflector shape, but are complex and

imprecise. Other smart substructures with embedded actuatorsand sensors are also developed to control compliant mirrorshape [10]; however for large heliostat arrays this approachwould require many more actuators than would be practicalOther related research has studied forming parabolic troughcollectors and parabolic dishes by bending compliant structureswith optimized shapes and thicknesses [11-12]. However, theseworks only consider simple one dimensional bending to formparabolas using cable mechanisms. These works have notsolved the problem of bending plates simultaneously in twodirections to form paraboloids and develop a mechanism toperform this bending.

Consequently, the limit on heliostat performance becomes

an important issue for SPT design. First of all, because of thelimited concentration ratio, C(the mirror area divided by theprojected receiver aperture area), of individual heliostats, moremirrors are required to achieve higher temperature to improvethe thermal cycle efficiency. Second, in practical SPT systemsthe central receiver has a larger aperture area than theindividual heliostat surface or facet (e.g., C=0.047 for thefacets in Solar Two, [2]; C=0.73 for the heliostats in PS10[3]), which results in a dilemma for both mirror and receiverdesign. To reduce the receiver aperture, smaller mirrors haveto be chosen, which increases the number of tracking devicesTo increase mirror area, larger receiver aperture has to bedesigned, which causes both higher receiver cost and thermalloss [4].

1.2 Objective and ApproachThis research develops a new, low-cost, high efficiency

compliant heliostat concept, which will permit fewer mirrorsand a smaller central receiver. This concept could greatlyreduce costs and enable SPTs to be a more practical source ofclean energy.

In this approach, shape-optimized heliostats are formedfrom flat sheets with individually tailored compliances, whichcan be easily shipped to the site and assembled with

adjustment mechanisms to adapt in two dimensions asrequired. The mirrors are deformed into a fixed surface of aparaboloid section that optimizes the average heliostaperformance over the year. This shape is a function of heliostatlocation in the field.

To evaluate the effectiveness of this approach, an opticalefficiency metric is defined given the concentration ratio andapplied to a conventional representative system with flamirrors. Then the compliant heliostat mirror is designed withtailored surface stiffness in two dimensions that allows the

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surface to deform into a paraboloid section by simple manualmechanisms, such as screws with some guidance, on thecorners. The proposed simplification (statically optimizedshape) is shown to be effective by comparing to thetheoretically ideal adapting mirror surface, and a flat heliostatmirror using numerical simulation including the Finite

Element Analysis (FEA) method, optical ray tracing andclassical optimization.

2 PERFORMANCE METRIC AND DEVELOPMENTSOF HELIOSTAT MIRROR MOTION

2.1 Reference System AssumptionsA virtual heliostat field is established here as a reference

case. The system is assumed to be located at latitude of 35N,the approximate latitude of Los Angeles, California. A north-facing cavity receiver is used to collect energy from theheliostat field. The heliostat surface area is 1m 2 and the tower

height is 10m. The distance between the Sample Mirror andreceiver is 100m, as shown in Fig. 3.

Figure 3. REFERENCE HELIOSTAT FIELD.

2.2 Performance Metric

The performance metric or efficiency, , of a heliostatmirror is defined as the ratio of solar energy that falls on themirror to the reflected energy that reaches the receiver,

r

i

E

E (1)

where:Ei is the incidence solar radiation on the surface of theheliostat.

Er is the energy reflected into the receiver aperture.This metric is a function of the sun angle, , concentration

ratio, C, and the mirror design. The atmospheric attenuationand shading effects are neglected here, so

, ,mirror designC (2)

The sun angle is defined as the angle between incidenceand reflected sunlight on the mirror surface (See Fig. 4), whichcan be assumed to be constant over the surface since the mirror

size (1m) is rather small in comparison to the receiver distance(100m). The ratio of the projected heliostat area in thedirection of the sun to the surface area should be high forpractical SPTs [6]. This ratio, which is cos /2, is known as thecosine efficiency. The mathematical development presentedhere assumes a cosine efficiency greater than 75%.

Figure 4. THE LIGHT PATH OF HELIOSTAT.

Recall that the concentration ratio C is defined as themirror area divided by the projected receiver aperture areagiven by:

secm

r

AC

A (3)

where:Am is the mirror area.Ar is the receiver aperture area. is the angle between normal of receiver plane and

reflected sunlight (See Fig. 4).For a cavity receiver which has a plane aperture, the

term should be considered because the projected aperture sizewill decrease from the mirror perspective. For a cylindricalreceiver, this term reduces because of the structure symmetry

Therefore, C is a function of mirror location in the field asshown in Fig. 5. This shows that the optimal mirror shaperequirements will vary throughout the field. For a given andC, is only a result of mirror design.

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Figure 5. CONCENTRATION RATIO C (WITHAM/AR=1) VARIATION OVER THE FIELD.

2.3 Flat Mirror PerformanceFor a flat mirror, the efficiency can be obtained as,

min ,1

co2

1

sC

(4)

The detailed derivation can be found in the Appendix A.With a cosine efficiency above 75%, this reveals theperformance of flat heliostat will decrease significantly withincreasing concentration ratio C.

2.4 Ideal Mirror Surface Determination and

PerformanceTo concentrate parallel sunlight to a small target, the

mirror surface has to be a section of paraboloid whose focalpoint is the receiver and symmetry axis is parallel to thesunlight as shown in Fig. 4. This surface is a function ofmirror location and sun angle. A temporary coordinate system,XYZ, is introduced to deduce the surface in the body-fixedmirror coordinate system XYZ.

Figure 6. COORDINATE SYSTEM.

Figure 6 shows the origin of XYZ placed at the center othe heliostat; Z axis is parallel to sunlight; Y axis isperpendicular to Z and within the plane of receiver-mirror-sunlight; X axis constitute right-hand coordinate system withY and Z. In XYZ, the ideal parabolic surface is,

2 2

tan4 2

x yz y

f

(5)

(1 cos )

2

Rf

(6)

where:fis focal length.R is the receiver-mirror distance.The body-fixed mirror coordinate system XYZ, in which Z

axis is the normal of mirror surface center, is obtained byrotation from XYZ by /2 around the X axis.

By coordinate transformation the surface governingequation in XYZ is,

2 2 2 2 2cos sin sin 4 sec 02 2 2

x y z yz fz

(7)

Since the parabolic surface can theoretically concentratethe sunlight into an infinitely small focal point, the efficiencyof ideal surface will be always 100% for any sun angle andconcentration ratio C.

3 COMPLIANT MECHANISM DESIGN

The ideal mirror shape is a function of mirror location andsun angle. It varies continuously, but implementing shapevariation in real-time is difficult and costly. Here, anapproximate design called a Shape-Optimized Complian(SOC) heliostat is proposed. A SOC heliostat has a fixedparabolic shape that is optimized for maximum annual averageperformance, and thus is only a function of location. Since therequired surface deviation in Z-direction is relatively smalcompared to the mirror size, it is proposed to implement theshape by two-dimensional bending of a compliant surfaceWith appropriate stiffness design, only two pairs of smalladjustment mechanisms are applied to the four cut corners o

the square sheet. Hence, the compliant heliostats can be easilyadjusted to the optimized shape during installation.

3.1 Analytical Design

3.1.1 Surface Curvature RequirementsThe principal curvatures of a general paraboloid surface

can be obtained in the cylindrical coordinate system (,, z) as

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shown in Fig. 7. For a heliostat, they are functions of themirror location R and sun angle . The detailed mathematicalderivation is presented in Appendix B.

Figure 7. PARABOLOID AND ITS PRINCIPALCURVATURES ON A CYLINDRICAL COORDINATE

SYSTEM.

32

22

cos1 2

22 (1 )

4

Rf

f

(8)

2

2

1 1

2 cos2 1 24

Rff

(9)

Since R is much larger than the heliostat size, thesecurvatures can be approximated as constants over the surface.The body-fixed Y-axis aligns with the principal direction withcurvature of while X-axis aligns with the other principaldirection of curvature. Thus, the deflections in Z-directionare given by,

22

16 cos

1

8

2

x

lw l

R

(10)

2

2

cos2

1

1

8 6y

l

wR

l

(11)

where: l is the length of the heliostat surface.With R>>l and a cosine efficiency above 75%, the

maximum surface deflection is much smaller than the mirrorsize,

max ,x yw w l (12)

(e.g., for the sample 1m2 mirror in the reference case themaximum deflection is about 1mm). This permits thefeasibility of mirror shaping using two-dimensional bending.

3.1.2 Mechanism DesignThe mechanism for the compliant heliostat is shown in

Fig. 8. The mirror surface is a square and its diagonal linesalign with the principal directions in which the moments areapplied to form the curvatures. The corners of the surface areequipped with adjustment mechanisms applying momentstoward the center. Because the adjustments are much smallerthan the mirror they can be regarded as concentrated momentsapplied on the corners. To ensure the external loads are puremoments, the Madjustment mechanisms are free to move inX-direction, while the M adjustment mechanisms are free to

move in both the Y and Z directions. The entire structure isfixed to its two-axis tracking system due to restrictions on Yaxis and Z-axis by the X-track and the restriction on X-axis bythe Y-track.

Figure 8. OVERVIEW OF COMPLIANTMECHANISM.

With this mechanism framework, the internal bendingmoments of the mirror elements in XYZ coordinate system aregiven by,

( )xm D (13)

( )ym D (14)

0xym (15)

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3

212 1

EhD

(16)

where:D is the stiffness of the mirror element.Eis the Youngs modulus.

is the Poissons ratio of the material.To achieve the paraboloid curvatures given in (8) (9), thestiffness is tailored to a specific profile. It should be pointedout that the calculation here neglects the singularity thatresults from applying concentrated moment loads on thecorners. This will be considered later in this paper.

3.1.3 Tracking DemandsSince and are different, the two tracking axes for the

proposed heliostat are unique. Axis 1 is parallel to the linebetween the receiver and the mirror center (See Fig. 9); Axis 2is the body-fixed X-axis which is orthonormal to the plane

determined by the sun, the receiver, and the mirror center. Thistracking method, called spinning-elevation, is different fromthe conventional azimuth-elevation heliostat tracking [13].

Figure 9. TRACKING AXIS DEMANDS.

3.1.4 Variation of StiffnessAs mentioned before, the stiffness of the compliant

heliostat surface has to be designed for the simple appliedmoments to shape the surface into a paraboloid to reach highoptical performance. The equilibrium diagram is shown in Fig.10. All the calculations use the small displacementassumption.

Figure 10. FREE BODY DIAGRAM.

The applied moments are related to the internal momentsby,

2

( )2

ly

l yy

M m dx

(17)

2

( )2

lx

l xx

M m dy

(18)

In classical plate bending theory, the internal moments aredifficult to solve because of the singularity caused by theasymmetrical concentrated load [14]. The calculations neglecthe challenges of the detailed corners modeling. However, herethey yield an approximate solution for the required stiffnessBy substituting (13) (14) into (17) (18), a set of integraequations for the multi-variable stiffness function, D(x,y), isobtained as,

2

2

ly

ly

MDdx

(19)

2

2

lx

lx

MDdy

(20)

The continuous solution for (19) (20) is difficult to findbut a discontinuous approximate solution can be constructed,

, 20

lb x y

D x

c

othersy

(21)

where b, c are undetermined constants. So that,

2

2

2

2 ( )2 2

ly

ly

cb y b

Ddx l lc y b y

(22)

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2

2

2

2 ( )2 2

lx

lx

cb x b

Ddy l lc x b x

(23)

Substituting D to (17) (18), the external adjustmentmoments are obtained as follows,

3

2

cos122

26 1 2 cos2

EbhM cb

RR

(24)

3

2

cos1 22

26 1 2 cos2

EbhM cb

RR

(25)

In this solution, the Eqn. (19) (20) are satisfiedeverywhere except the corners. By definition of stiffness (16),this solution can be implemented by varying the thickness ofthe plate as shown in Fig. 11. The design consists of a squareframe with thickness of h, and a thin square center withthickness ofh/5. The adjustment mechanisms are located at thecorners of the frame. Note that since the thickness of the centersquare of mirror is nominally zero, as shown in (21), it is sethere to h/5 to be able to support the reflective surface, such as aMylar sheet. This area will have a negligible effect on thesurface stiffness profile. This constructive stiffness design onlyuses square sheets and two different thicknesses which can beeasily manufactured, for example, by lamination of three

layers.

Figure 11. VARIATION OF STIFFNESS (LEFT: TOPVIEW; RIGHT: CROSS-SECTION VIEW).

3.2 FEA Validation and Numerical OptimizationFor this study, a numerical simulation to verify the

heliostat mechanism design was developed using thecommercial Finite Element Analysis (FEA) software ADINA,and synthesized with the optical ray tracing and optimization

process written in Matlab. The integrated simulation toolincludes:

FEA: With the external moments as inputs, thedeformation and rotation of discrete elements aregenerated over the mirror surface which is meshed toresolve both thicknesses.

Ray-tracing: The inputs include the sun anglereceiver distance, and the surface information resultfrom FEA module. The normal direction for eachelement is calculated using the rotation data. Then thereflected points on focal plane located at receiver aresolved geometrically.

Optimization: The objective is to minimize the opticaspillage loss that counts the elements by which thereflected lights miss the receiver aperture. Thecorresponding design variables are the adjustmenmoments. This unconstrained nonlinear optimizationprocess is programmed using Newtons method usingthe finite difference for gradient calculations.

The parameters for simulation include: distanceR (100m)sun angle (60); thickness h (5mm); width b (0.2m)adjustment size (0.2m); Youngs modulus E (200GPa)Poissons ratio (0.3); The FEA model contains 790 nodes.

The optimized deformation result is shown in Fig. 12(Left) with the largest deviation wmax shown in Tab. 1. Theerror between optimized FEA deformation and the idealparaboloid surface is shown in Fig. 13 and in Tab. 1. Thesimulation resulted surface is close to the analytical paraboloidsection especially in the center. The large errors occur on theedges of the mirror and the transition points between differenthicknesses.

Figure 12. DEFORMATION PLOT.

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Figure 13. SURFACE DEFLECTION ERROR.

Table 1 DEFLECTION RESULTS

MaximumDeflection

Analytical (m) FEA+Optimization(m)

wmax 1.0640103 1.0814103

Deflection

Error

Mean (m) Standard Deviation

(m)

| wFEAwAnalytical | 1.8284105 1.0523105

Figure 14 shows the ray tracing points on a 10 cm receiverfor the numerically optimized mirror. It can be seen that themajority of the light falls in a center 5cm5cm region of thereceiver and the design provides an efficiency of 88.4% for areceiver radius of 10cm and concentration ratio of 32. Bycomparison, for the same conditions, the conventional flatmirror heliostat has an efficiency of 4.17%, as given by Eqn.

(4). Although the efficiency of such compliant heliostat isshown to be high, some small deviations exist between theideal shape for the SOC and the actual shape. First, the surfacedetermined by the optimization with the smallest deflectionerror only has an efficiency of 85%, which reflects theinaccuracy of predicting local curvatures. This can beimproved by integrating the ray tracing and efficiencycalculation into the optimization and setting thecomprehensive optimization objective, which howeverconsumes more computing time and increases theprogramming complexity. Second, the external momentspredicted by analytical model are larger than the optimizedvalues. This is the result of the substantial assumptions andapproximation made in the analytical modeling.

Figure 14. REFLECTED POINTS ON FOCALPLANE.

4 PERFORMANCE OF SHAPE-OPTIMIZEDCOMPLIANT HELIOSTATSIn this section, the annual average performance of the

shape-optimized compliant (SOC) heliostat is compared to theflat mirror, the ideal stationary parabolic shaped mirror, andthe ideal time-dependent adaptive mirror.

The sample heliostat in the reference system is located at99m (North) and 10m (East) with a distance of 100m to thereceiver as shown in Fig. 4. Neglecting any tracking errors, theupper bound of sun angle for the sample mirror is 95.74which occurs when the sun rises in the morning, while thelower bound is reached in the afternoon, varying by the seasonfrom summer to winter with the range of [25.66, 71.77]

Therefore, the annual performance of heliostat is defined bythe mean of the efficiency distribution over the range of inpusun angles.

This paper develops a stationary compliant heliostatwhose shape is optimized corresponding to the annual averageperformance. The optimization process used to determine theshape is similar to section 3.2 with the objective changed tomaximize the annual energy collection. Meanwhile, diverseconcentration ratios, meaning various receiver sizes, areinvestigated in this section to explore the feasibility of theheliostat design.

The results (See Fig. 15) show that the SOC heliostat canachieve much higher performance than the flat one, and

almost performs as well as the ideal case. Using 90% as astandard of efficiency, the SOC can function with aconcentration ratio of 28 which results in a receiver radius of10.7cm. With the same receiver, the flat mirror delivers lesthan 10% of the light into the aperture. For the ideal stationaryparabolic mirror, the receiver size can be further reduced to5.6cm and maintain a 90% efficiency. Such results suggest thaSOC heliostats are much more efficient than conventional flatmirrors. They perform nearly as well as the ideal stationary

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parabolic mirrors, and even the ideal time-dependent adaptingones, but are much simpler to implement.

Figure 15. PERFORMANCE RESULTS.

5 CONCLUSIONThis paper presents a low-cost, high efficiency, location-

based adaptive heliostat mirror. The design uses a tailoredcompliant surface with simple adjustment mechanisms toachieve the desired parabolic shape. The shape is optimized toachieve the maximum annual efficiency of transferring solarradiation to the receiver with a two-axis spinning-elevationmode tracking device. This is a stationary simplification of theideal time-dependent adapting shape. These adaptive mirrorscan be configured in the field during assembly and adjusted asrequired.

The results of numerical analysis show that the shape-

optimized compliant heliostats can be substantially moreeffective than flat mirrors. They also perform closely to theideal fixed parabolic mirrors. These low-cost adaptiveheliostats will permit smaller receivers and larger mirrors forSolar Power Towers to improve system efficiency andeconomic benefits.

ACKNOWLEDGMENTSThe authors would like to thank the King Fahd University

of Petroleum and Minerals in Dhahran, Saudi Arabia, forfunding the research reported in this paper through the Centerfor Clean Water and Clean Energy at MIT and KFUPM. Theauthors would also like to thank the Chinese Scholarship

Council for the support of Li Meng for his research in MIT.The authors would also like to thank Leah Kelley, ElizabethReed, and Aditya Bhujle for their generous help during thedevelopment of this work.

REFERENCES[1] Kalogirou, S.A., 2004. Solar Thermal Collectors and

Applications. Progress in Energy and Combustion Science30 (3), 231-295.

[2] Kelly, B., and Singh, M., 1995. Summary of the FinaDesign for the 10 MWe Solar Two Central ReceiverProject. Solar Engineering: 1995, ASME, 1, p. 575.

[3] Solcar, Inabensa, Fichtner, Ciemat, DLR, 2006. FinaTechnical Progress Report: 10 MW Solar Thermal PowerPlant for Southern Spain.

ec.europa.eu/energy/renewables/solar_electricity/doc/2006_ps10.pdf[4] vila-Marn, A.L., 2011. Volumetric receivers in Solar

Thermal Power Plants with Central Receiver Systemtechnology: A review. Solar Energy 85 (5), 891-910.

[5] Yao, Z., Wang, Z., Lu, Z., Wei, X., 2009. Modeling andsimulation of the Pioneer 1MW solar thermal centrareceiver system in China. Renewable Energy 34 (11)24372446.

[6] Noone, C.J., Torrilhon, M., Mitsos, A., 2012. Heliostatfield optimization: A new computationally efficient modeand biomimetic layout. Solar Energy 86 (2), 792803.

[7] Slocum, A.H., Codd, D.S., Buongiorno, J., Forsberg, C.

McKrell, T., Nave, J.-C., Papanicolas, C.N., Ghobeity, A.Noone, C.J., Passerini, S., Rojas, F., Mitsos, A., 2011Concentrated solar power on demand. Solar Energy 85(7), 15191529.

[8] Mancini, T., 2000. Catalog of heliostats. Solar PACESTechnical Report No. III-1/00.

[9] Vogel, W., Kalb, H., 2010. Large-scale solar thermalpower. Weinheim: Wiley-VCH.

[10]Irschik, H., 2002. A Review on Static and Dynamic ShapeControl of Structures by Piezoelectric ActuationEngineering Structures 24(1), 5-11.

[11]Li, L., Kecskemethy, A., Arif, A.F.M., and Dubowsky, S.2011. A Novel Approach for Designing Parabolic MirrorsUsing Optimized Compliant Bands. Proceedings of theASME 2011 International Design Engineering TechnicalConferences & Computers and Information inEngineering Conference, August 28-31, 2011Washington, DC, USA.

[12]Li, L. and Dubowsky, S., 2011. A New Design Approachfor Solar Concentrating Parabolic Dish Based onOptimized Flexible Petals. Journal of Mechanisms andMachine Theory, 46 (10), 1536-1548.

[13]Chen, Y. T., Kribus, A., Lim, B. H., Lim, C. S., Chong, KK., Karni, J., Buck, R., Pfahl, A., and Bligh, T. P., 2004Comparison of Two Sun Tracking Methods in the

Application of a Heliostat Field. ASME Journal of SolarEnergy Engineering, 126(1), pp. 638-644.[14]ukasiewicz, S.A., 1976. Introduction of Concentrated

Loads in Plates and Shells. Progress in Aerospace Science17 (2), 109-146.

APPENDIX A: OPTICAL EFFICIENCY OF FLATMIRRORS

For a flat mirror, the total solar energy captured is,

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cos2

i mE GA

(26)

where: G is the solar constant.The reflected energy flux on the receiver is,

cosr

D G (27)

So the energy radiated on the receiver is,

cos2

cos

cos2

2 cos

m

r r

r

m

m r

A

GA cos A

E

A

GA cos A

(28)

Thus the efficiency for flat mirror is,

min ,1co

2

1

sC

(29)

APPENDIX B: PRINCIPAL CURVATURES OF APARABOLOID

Geometrically, the paraboloidF (See Fig. 7) is a surface ofrevolution which can be parameterized as,

2

, ( cos , sin , )4f

F (30)

For any point PF, the tangent plane is spanned bytangent vectors Fand F:

(cos , sin , )2f

F (31)

( sin , cos , 0) F (32)

The shape operator is,

32

22

2

2

10

2 (1 )4

10

2 14

P

ff

S

ff

(33)

Since SP is a diagonal matrix, the principal curvatures ofF at any point are equal to those entries located on principaldiagonal with principal directions parallel to F and Frespectively.

32

22

1

2 (1 )4

ff

(34)

2

2

1

2 14

ff

(35)

For a given mirror location R and sun angle , can bedetermined as,

sinR (36)

with the expression (6) of focal length f the two curvatures oheliostat surface are,

322

2

cos1 2

22 (1 )4

Rff

(37)

2

2

1 1

2 cos2 1 24Rf

f

(38)