Applied Energy 145 (2015) 211–222

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Applied Energy

journal homepage: www.elsevier .com/locate /apenergy

Enhancing the efficiency of solar concentrators by controlled opticalaberrations: Method and photovoltaic application

http://dx.doi.org/10.1016/j.apenergy.2015.01.0850306-2619/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: alessandra.giannuzzi@unibo.it (A. Giannuzzi).

Alessandra Giannuzzi a,⇑, Emiliano Diolaiti a, Matteo Lombini a, Adriano De Rosa b, Bruno Marano c,Giovanni Bregoli a, Giuseppe Cosentino c, Italo Foppiani a, Laura Schreiber a

a INAF-Osservatorio Astronomico di Bologna, Via Ranzani 1, I-40127 Bologna, Italyb INAF-Istituto di Astrofisica Spaziale e Fisica Cosmica, Bologna, Via P. Gobetti 101, I-40129 Bologna, Italyc DIFA-Dipartimento di Fisica e Astronomia, Alma Mater Studiorum Università di Bologna, Via Ranzani 1, I-40127 Bologna, Italy

h i g h l i g h t s

�We developed a new optical designmethod for high performance solarconcentrators.� The method is based on optimizing

the optical shapes to match thereceiver features.� A dense array PV concentrator made

by few monolithic mirrors wasmodeled.� The optimization led to free-form

optics focusing high uniformirradiance spots.� The optimal optics/receiver coupling

increases the system conversionefficiency.

g r a p h i c a l a b s t r a c t

a r t i c l e i n f o

Article history:Received 21 June 2014Received in revised form 12 January 2015Accepted 23 January 2015Available online 2 March 2015

Keywords:Photovoltaic concentrator (CPV)Dense-array receiverNumerical optimizationOptical designZernike polynomials

a b s t r a c t

We present a general method, based on controlled static aberrations induced in the reflectors, to boostreceiver performances in solar concentrators. Imaging mirrors coupled with dense arrays suffer from sev-ere performance degradation since the solar irradiance distribution is bell-shaped: mismatch losses occurin particular when the cells are series connected. The method consists in computing static deformationsof the reflecting surfaces that can produce, for an adopted concentration ratio, a light spot matching thereceiver features better than conventional reflectors. The surfaces and the deformations have beenanalytically described employing the Zernike polynomials formalism. The concept here described canbe applied to a variety of optical configurations and collecting areas. As an example, we extensively inves-tigated a dense array photovoltaic concentrator, dimensioned for a nominal power of about 10 kWe. The‘‘flat’’ distribution of light we obtain can exploit the PV device cells close to their efficiency limit. A sig-nificant gain is thus obtained, with no need of secondary optics or complex dish segmentation and of spe-cial features in the receiver electrical scheme. In the design, based on seven 2.6 m mirrors, we addressedalso non-optical aspects as the receiver and the supporting mechanics. Optical and mechanical tolerancesare demonstrated not to exceed accurate, but conventional, industrial standards.

� 2015 Elsevier Ltd. All rights reserved.

212 A. Giannuzzi et al. / Applied Energy 145 (2015) 211–222

1. Introduction

Concentrating Photovoltaics technology (CPV) is experiencing agrowing interest thanks to the development of solar cells withcontinuously improved efficiency. At present, the best reported cellis a 0.165 cm2 multi-junction (MJ) cell having a new record of44.4% confirmed efficiency at direct irradiance concentration of302 suns (1 sun = 1000 W/m2) [1]. For both high concentration(HCPV) and low concentration (LCPV) systems the yearly installedcapacity increased significantly during the last five years [2].A simple advantage induced by this technology is that, given thecollected energy, the concentration performed by optical devicessuch as lenses or mirrors allows us to replace the area of photo-voltaic material with cheaper optical surfaces. Moreover, high effi-ciency cells are too expensive to be used in non-concentratingapplications. Despite most of the installed systems are point focuslens based as Fresnel [3–6] or micro-dish [7–9] systems, densearray systems have been recently investigated as profitable solu-tions for lowering the cost per watt-peak supplied [10,11]. In thistechnology the light is focused using one large reflective elementcalled dish, onto an array of photovoltaic MJ cells densely packedto form a single detector. If compared with lenses, mirrors havethe main advantage to not suffer from chromatic aberrations.These systems track the sun in two-axis during its daily motionand usually operate in high concentration mode, i.e. with solar fluxup to hundreds times the ambient value. Reflective dish concentra-tors with diameters ranging from few meters to few tens of metershave been already proposed and are at the beginning of their com-mercial development working at typical concentrations of 500�[12–14].

Traditional dish concentrators have paraboloidal shapes.Theoretically, their diameters could reach several tens of metersas the heliostats in central tower plants, the construction of mono-lithic mirrors being difficult at these scales. The size generallyimposes to approximate the profiles with cheap flat reflectingfacets mounted on a common frame and reproducing globallythe paraboloidal surface. As for the receivers, standard cells haverectangular shapes and the arrays are groups of cells denselypacked together mostly in series and parallels connections. Thearrays do consequently resemble rectangular shapes too. When astandard imaging mirror that produces a sun image intrinsicallycircular is coupled with a rectangular detector problems arise. Inthis condition some cells could be obscured if the spot is smallerthan the receiver, or part of the light could be lost if the detectoris smaller than the spot, these two effects contributing to a sub-stantial loss in efficiency. Moreover, the given irradiance distribu-tion is bell-shaped in contrast with the requirement of having allthe cells under the same illumination. In fact, interconnected cellshaving identical electrical characteristics and experiencing thesame irradiance/temperature conditions produce the same amountof output current and voltage. Mismatch losses occur instead wheninterconnected cells experience different conditions, in particularfor series connections. Still few investigations have been specifical-ly performed on current mismatches in dense array receiversexposed to high concentrations [15–17]. The issue of spatial lightuniformity is instead widely known for single cell devices [18–21] and the problem is commonly approached by the introductionof secondary optics (SO) [22–24] working as homogenizers. Thepresence of an extra secondary optics is rather useful to increasethe acceptance angle leading to a relaxation of tracking and align-ment tolerances. However, this solution has the disadvantage toincrease the system complexity and to add reflection losses, chro-matic aberration (if refractive) and mechanical problems as align-ment, stability or mounting. A useful review on the state of the artof the nonuniformity problem for single cell receivers has beenrecently published [25]. Few commercial systems and technical

data are available on secondary optics embedded in dense arrays.Some researches faced the uniformity problem from the receiverpoint of view, developing new electrical connections [26],embedding different cells in the same array [27] or designingnew receivers with radial symmetry [28].

Alternative ways of redesigning the primary collector have beenpoorly investigated but some good results has been obtained byChong et al. [29]. The proposed planar faceted concentratorcoupled to a dense array has been optimized to give a large uni-form illumination over the target area with a peak intensity of391 suns. However, such a concentrator is made by several mirrorsto be mounted and aligned before being orientated with the use ofline-tilting driving mechanism. Moreover, since the final spot is theoverlap of the multiple facets reflections, the size and the uniformi-ty of the final spot is influenced by projection and blocking effectswhich increase with the distance of the facets from the center ofthe whole assembly. For this reason, such a mosaic system is notable to both have big collecting area and high concentration ratiowithout embedding a high number of facets and high focaldistances, as reported in similar works [30–32]. In [32] the eco-nomical viability is however claimed for a specific configurationof faceted dense array system since a cost for the output powerbelow 2 euro/W has been calculated.

The strategy we suggest in this paper is to boost the spot unifor-mity by only acting on the primary reflector but using monolithicbig surfaces and avoiding the dish faceting into numerous smallerelements. In the proposed method, the shape of the mirrors isanalytically described by the Zernike polynomials and its optimiza-tion is numerically obtained to give a non-imaging optics able toproduce a quasi-square spot, spatially uniform and with prescribedconcentration. The free-form primary optics, optimized in this wayand validated by a ray tracing software, showed a substantial gainin efficiency without the employ of secondary optics. At the sametime, simple electrical schemes for the receiver are required. Theconcept has been investigated theoretically modeling a CPV appli-cation including a conceptual development of non-optical aspectsas the design of the receiver and of the supporting mechanics.For the proposed method and the specific CPV system developed,a patent application has been filed in Italy. A preliminary analyticalstudy, considering a residential utility, has been also performed inorder to understand the energetic and economic performance ofthe system [33]. The analysis indicates that the maximum sustain-able capital cost of the system ranges between 30,000 euros and45,000 euros depending on the years which are considered forthe return of the investment (10 or 20 years respectively). Furthermore detailed economical evaluations will be performed duringthe future constructive phases of the project.

2. Optical concept

From an optical point of view there is no need for an accurateimage at the receiver of a solar concentrator. The optical design cri-teria rather concern with the optimal transfer of light between thesource and the target chosen. To solve matching issues in concen-trators we thought to reinterpret optical concepts largely used inastronomy, where an accurate image formation is an essential pre-mise for efficient observations. In telescopes, controlled mirrorsdeformations are introduced by actuators to balance the opticalaberrations that degrade the wavefront coming from an observedsource [34–36]. What we developed instead is a sort of ‘‘reverse’’approach of the astrophysical method: the guideline is to applydeformations (active or static) to the mirrors of the solar collectorsto introduce aberrations in the wavefront, thus degrading the solarimage and, in the case of a CPV dense array system, focusing asquared spot with a prescribed irradiance. The result would be a

Fig. 1. Effects introduced on the Sun image by Zernike polynomials 4th, 11th and14th.

Table 1Principal Zernike modes involved in this study.

Zernike mode

4th 11th 14th

A. Giannuzzi et al. / Applied Energy 145 (2015) 211–222 213

better match between the irradiance features and the receiverperformance.

The technical feasibility of our concept is supported by indepen-dent studies and projects involving technology transfer processesfrom the astronomical instrumentation knowledge. Single mono-lithic reflectors suitable for concentrators (3.1 m wide) have beenalready realized in a customized furnace at the Steward Observato-ry Mirror Lab, at the University of Arizona [37]. A novel mirrorconcept based on an active laminate consisting of an ultra-thin(less than 1 mm) and ultra-light carbon-fiber shell bonded to apiezo-ceramic active layer have been recently investigated andmanufactured with the aim of reducing the cost of active mirrorsboth in telescopes and concentrators [38–40].

To describe the mirrors shape and to perform the optimizationfor a CPV dish, we used the Zernike polynomials, an analytical toollargely employed, especially in optics, to characterize functionsand data on a circular domain. They form an orthogonal basis onthe unit circle and real surfaces can be represented by linearcombinations of them. Every Zernike polynomial consists of threecomponents: a normalization factor, a radial component and anazimuthal component. The radial components are polynomialsderived from the Jacobi polynomials, whereas the azimuthalcomponent is sinusoidal. As in the Noll formalism [41], the Zernikepolynomials can be defined in polar coordinates ðq; hÞ:

Zjeven¼

ffiffiffiffiffiffiffiffiffiffiffiffinþ 1

pRm

n qffiffiffi2p

cos mh ð1Þ

Zjodd¼

ffiffiffiffiffiffiffiffiffiffiffiffinþ 1

pRm

n qffiffiffi2p

sin mh ð2Þ

Zj ¼ffiffiffiffiffiffiffiffiffiffiffiffinþ 1

pR0

nðqÞ ð3Þ

where q is the normalized radial coordinate ranging from 0 to 1 andh is the azimuthal angle ranging from 0 to 2p. In the formulas, mrepresents the azimuthal frequency and n the radial degree, bothare integer and the condition m 6 n; n� mj j ¼ even must be satis-fied. The index j is a mode ordering number and is a function of nand m. Eqs. (1) and (2) exist for m – 0 while Eq. (3) for m ¼ 0.The double indexing scheme is useful for unambiguously describingthe functions. In the formulas, Rm

n ðqÞ indicates polynomials withradial dependence.

3. Case of single on-axis mirror

An analysis we performed with the ray tracing software Zemax�

showed that, starting from a spherical mirror, very few deforma-tions described by specific Zernike polynomials (modes) canstrongly help in solving the uniformity and shape problem in densearray receivers. Considering an imaging mirror with deformations,its surface z (the so-called sag) can be approximated by the follow-ing formula:

z ¼ cr2

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ð1þ kÞc2r2

p þXN

i¼1

AiZiðq; hÞ ð4Þ

where N is the number of polynomials, Ai is the coefficient associat-ed to the ith polynomial, r is again the radial coordinate in the cho-sen units, q and h are the polar coordinates defined before. Eq. (4)depends on the curvature c (which equals the reciprocal of the cur-vature radius) and the conic constant k. The first term in the equa-tion represents an ideal conic surface (spherical if k ¼ 0) while thesecond term represents the deformations described by Zernikepolynomials. The number of terms needed for a good surface mod-eling grows together with the number of deformations occurring atdifferent scales.

For a single spherical mirror focusing on axis, we identifiedthree main polynomials: the 4th, the 11th and the 14th. Fig. 1shows how the solar spot produced at a fixed distance by aspherical mirror can be modified by introducing controlled

deformations related to the three modes here mentioned. Thismodel can be also extended to mirrors with an off-axis focus: inthat case the number of Zernike modes involved in the spotshaping is higher.

The identified modes are shown in 2D and 3D in Table 1. Thedeformation associated with the 4th mode (defocus) basicallyenlarges the image and contributes to spread the light quitesimilarly to the effect of shifting the receiver plane. The 11th mode(third order spherical) contributes to redistributing the rays main-taining an image radial symmetry and changing the image irradi-ance profile. These two polynomials do not have any impact onthe spot shape since they have no azimuthal dependence. A defor-mation corresponding to the 14th polynomial (vertical quadrafoil)contributes to make a circular spot square along two preferentialdirections rotated 45�, depending on the coefficient sign. The effectof this specific deformation is less evident if the mirror is in focusmode: that is the reason for a combined use of the modes 14th and4th. Alternatively, the same effect of this combination can beobtained by positioning the receiver slightly behind or above thecorrect focal plane and avoiding (partially or completely) thedeformations related to the 4th mode. Since it is easier for a singlemirror to produce a square uniform image when the defocus is big-ger, this means that the lower the concentration factor the betterthe method works. The size of the spot to obtain depends on thedesired concentration factor.

A prescribed irradiance could be also obtained by employingthis concept to design concentrators with several optimized mir-rors focusing at the same receiver. In this case, the final illumina-tion pattern impinging on the receiver would result in the sumof the incoherent illumination patterns produced by each singlemirror, as we are going to show in the next sections.

4. Case of a CPV dense array system: design choices

A multi-mirror configuration can be useful to solve the issue ofbuilding a single huge mirror. In order to avoid a mosaic of hun-dreds reflective elements [15], we choose to design a CPV dishmade by few monolithic mirrors mounted close together on the

Fig. 2. Optical layouts: (a) 3D, (b) x–y plane, and (c) y–z plane.

214 A. Giannuzzi et al. / Applied Energy 145 (2015) 211–222

same structure. The selected configuration is the hexapolar gridand it has been already used in Stirling applications as well as insome ground based optical telescopes. In the hexapolar configura-tion the elements are placed on rings so that the (n + 1)th ring con-tains six elements more than the nth ring, the central ring havingonly one element. We decided to consider only the central mirrorand a ring of six mirrors arranged around it. Fig. 2 presents theoptical layouts of the proposed system. The mirrors of the secondring have been labeled from 2 to 7 counter-clockwise. The z-axishas been set as the direction of the incoming rays and it is perpen-dicular to the central mirror vertex. This optical condition of align-ment with the solar direction should be the system nominalworking state.

Considerations about the concentration ratio to be investigatedand the mechanical compactness have been made also in compar-ison with similar existing prototypes and plants. Since thisresearch activity has been carried out with the specific goal of find-ing new solutions in the field of clean micro-generated distributedelectricity, our dish has been conceived as a power system suitablefor the market of medium residential contexts or small farms. Wedecided the mirror diameter to be around 2–3 m, to avoid con-struction difficulties. The diameter of the single mirror has beenset to D ¼ 2600 mm, for a total system size of about 7800 mmand a resulting total optical area slightly bigger than 35 m2. Sup-posing an irradiance at the collecting aperture of 1000 W/mm2,

the entry power would be around 35 kW: with a receiver workingalmost at the efficiency of the best presently available cells (be-tween 30% and 40%), such a system would be able to deliver morethan 10 kWe. Utility scale applications could be anyway consid-ered, together with the scaling of the single elements for higherenergy outputs.

The detector distance has been set to h ¼ 4800 mm in order tohave a low ratio of detector distance to total diameter. Consideringthis ratio similar to the focal ratio in imaging systems, a value f=0:5should be approached to maximize the concentration but also toallow a more compact structure.

We investigated two concentration levels, 500� and 800�. Toobtain these concentrations, we applied a defocus to the mirrorswhich is the common method to modulate the concentration deliv-ered at the receiver. A paraboloid in focus mode would have a col-lected flux too high for the cells working range (up to fewthousands of suns at present). In our case, another reason to avoidextreme concentrations is that the deformations introduced by theZernike modes are more efficient in reproducing the image fea-tures required when a defocus occurs.

The concentrator has been initially designed putting mirrorswith the same diameter D on the same plane. The reference systemhas been chosen so that incoming rays are parallel to the z-axis,while the mirrors vertexes lay in the x–y plane. Each mirror hasbeen placed at d ¼ 2680 mm (in the x–y plane) from the central

Table 2Positions, tilt angles and curvatures of the seven mirrors.

Mirr1 Mirr2 Mirr3 Mirr4 Mirr5 Mirr6 Mirr7

X pos (mm) 0.00 0.00 2320.88 2320.88 0.00 �2320.88 �2320.88Y pos (mm) 0.00 2680.00 1340.00 �1340.00 �2680.00 �1340.00 1340.00ax (�) 0.00 �14.59 �7.41 7.41 14.59 7.41 �7.41ay (�) 0.00 0.00 12.60 12.60 0.00 �12.60 �12.60Radius of curv. (mm) 10101.00 11480.10 11480.10 11480.10 11480.10 11480.10 11480.10

A. Giannuzzi et al. / Applied Energy 145 (2015) 211–222 215

mirror vertex to prevent shading effects. The mirrors of the exter-nal ring have been tilted respect to the central one in order to focusall the chief rays from the Sun center at the center of the receiverplane having coordinates ð0;0;hÞ. This optical restriction is option-al, but we aimed at simplifying the mechanical structure. Thegeometrical laws fulfilling this optical condition are easily deriv-able and once fixed the distance d in the hexapolar grid the posi-tional/tilting parameters of the mirrors can be immediatelycalculated. The tilt of the external mirrors reduce by 5% the collect-ing projected area of the whole system from 37.17 m2 to about35.25 m2. Positions, tilts and curvatures of the seven mirrors arelisted in Table 2. The generic mirror surface sag has been describedby Eq. (4).

5. Design method

To optically model our system, an end-to-end IDL� code hasbeen written on purpose. Each step of the procedure and theresults have been verified with the optical design software Zemax�

as reference. The code includes four main subgroups of routines:the first for individually modeling the optical part; the second forthe receiver implementation; the third for optimizing the optics;the last one for calculating tolerances of optical/mechanicalparameters.

5.1. Optical modeling

The initial optical parameters, which are the initial conditions ofthe simulations, have been set by a ray tracing analysis performedby Zemax�. The Sun has been modeled as a finite source with anangular diameter of 0:53�, neglecting its shape variations causedby the altitude changing during the day. The curvatures have beenset so that the mirrors could produce a spot with a size compatiblewith the mean geometrical concentration chosen. The concentra-tion ratio has been defined as the total mirrors area perpendicularto the axis of the central mirror divided by the total area of thereceiver, supposing a receiver and a spot ideally with the same size.We ignored the obscuration introduced by the receiver itself.

The Zernike modes corresponding to deformations useful to ful-fill our requirements of shape and uniformity have been selectedafter fixing the curvature. The deformations needed for the centralmirror are the three described in paragraph Section 2, but othermodes (from 5th to 8th) are necessary for the six off-axis mirrors.

Table 3Correlation between the Zernike coefficients of the seven mirrors.

Mirr1 Mirr2 Mirr3 Mirr4

Z4 Z4(1) Z4(2) Z4(2) Z4(2)Z5 0.00 0.00 �Z6(2)� cos 30� Z6(2)�Z6 0.00 Z6(2) �Z6(2)� sin 30� �Z6(2Z7 0.00 Z7(2) Z7(2)� sin 30� �Z7(2Z8 0.00 0.00 Z7(2)� cos 30� Z7(2)�Z11 Z11(1) Z11(2) Z11(2) Z11(2)Z14 Z14(1) Z14(2) Z14(2) Z14(2)

The selection criteria is that the superimposition of all the generat-ed spots could produce an irradiance distribution with the desiredfeatures. Symmetry properties have been imposed for the six mir-rors in the external ring to reduce the degrees of freedom of ourproblem. For example, these mirrors have been chosen with thesame curvature radius and the same values of the 4th, 11th and14th Zernike coefficients. As consequence, the non-zero coeffi-cients are linked between mirrors by the geometrical relationsshown in Table 3. In this way, opposite mirrors are equal but rotat-ed by p and the final optical model results to be made of only fourdifferent types of surfaces. It could be certainly possible to identifymore coefficients to improve the performance however increasingthe complexity of the system. This condition would be more suit-able both on construction and calibration stages. The independentmodes identified for our system are eight, three for the central mir-ror (Z4(1), Z11(1) and Z14(1)) and five for the external ones, allderived from the modes of the mirror number 2 (Z4(2), Z6(2),Z7(2), Z11(2), Z14(2)) according to the relations shown in Table 3.The mirrors of the ring can not have all the same shapes even if thiswould be the best constructive condition. The 14th Zernike modein fact corresponds to a deformation able to modify the circularsymmetry of the ray bundle into a square and it has an azimuthaldependence. The simple rotation of a given surface would lead to adifferent analytical description in terms of its Zernike coefficients,except for the coefficients with pure radial dependence. Thismeans that a ring generated by replicating mirror number 2 andsimply rotating the replicas according to the position in the ring,would give a series of identical spot rotated as in Fig. 3a. The super-imposition of these spots would certainly not lead to a final squareshape. On the contrary, fixing the 14th coefficient to the same val-ue for all the surfaces, the features in Fig. 3b are obtained. The phy-sical size of the figure is 4 � 105 lm.

The optical scheme described is simulated by the ray-tracingcode written on purpose. The code output is the final spot pro-duced by the concentrator. In the algorithm, the continuous opticalsurfaces of the mirrors have been discretized by a fixed number ofsub-apertures. The rays striking every sub-aperture are reflectedtoward the receiver according to the classic reflection law. TheSun has been modeled as an homogeneous circular source with adiameter of 0.53�, thus applying a realistic divergence model. Thenumber of rays traced from the Sun has been set in order to mini-mize sampling errors. To calculate the nominal mirrors shape, wesupposed an ideal tracking condition in which the central solarray strikes the central mirror vertex parallel to the optical axis.

Mirr5 Mirr6 Mirr7

Z4(2) Z4(2) Z4(2)cos 30� 0.00 �Z6(2)� cos 30� Z6(2)� cos 30�

)� sin 30� Z6(2) �Z6(2)� sin 30� �Z6(2)� sin 30�

)� sin 30� �Z7(2) �Z6(2)� sin 30� Z7(2)� sin 30�

cos 30� �Z7(2) �Z6(2)� cos 30� �Z7(2)� cos 30�

Z11(2) Z11(2) Z11(2)Z14(2) Z14(2) Z14(2)

Fig. 3. Effect introduced in the spot generated by each mirror by the introduction of (a) a Z14 value rotated according to the mirror location and (b) a common Z14 value.

Table 5Electrical parameters of the AZUR SPACE 3C40 cell at 500� and 1000�.

Isc (A) Voc (V) Imax (A) Vmax (V) Pmax (W) FF (%) g (%)

500� 2.151 3.144 2.102 2.842 5.98 88.0 39.01000� 4.239 3.170 4.135 2.762 11.42 85.0 37.8

216 A. Giannuzzi et al. / Applied Energy 145 (2015) 211–222

5.2. Receiver implementation

To simulate the performance of a dense array receiver, we con-sidered an electrical model for the PV cells. Neglecting any tem-perature or spectral variation, the physical behavior of a cell canbe in first approximation summarized by the following set of equa-tions uniquely depending on the concentration factor �:

Iscð�Þ ¼ � � Iscð1Þ ð5Þ

Vocð�Þ ¼ Vocð1Þ þ ndKT lnð�Þ

qð6Þ

Pmaxð�Þ ¼ Imaxð�Þ � Vmaxð�Þ ð7Þ

FFð�Þ ¼ Pmaxð�ÞIscð�Þ � Vocð�Þ

ð8Þ

gmaxð�Þ ¼Pmaxð�ÞPinð�Þ

¼ Iscð�Þ � Vocð�Þ �FFð�ÞPinð�Þ

ð9Þ

where Pin is the total power received by the cell and Iscð�Þ; Vocð�Þare short circuit current and open circuit voltage at a given concen-tration, gmax is the nominal conversion efficiency, nd is the diodeideality factor, T is the absolute temperature of the cell, K is theBoltzmann constant and q is the electron charge. A more exhaustivemodel involving dependencies on T and spectral variations can befound in [42]. Eq. (8) defines the Fill Factor FF as the ratio betweenthe power at the maximum power point Pmax and the product of theopen circuit voltage and short circuit current. It is typically betterthan 75% for good quality MJ solar cells. It is also an index of theperformance of a solar cell in terms of generated power and itshould be as close as possible to 100%: graphically, the FF is a mea-sure of the squareness of the solar cell I—V curve and is also the areaof the largest rectangle which would fit in the curve.

Our receiver has been analytically designed and numericallysimulated using a datasheet of commercially available high con-centration cells 3C40 produced by AZUR SPACE [44] with a nom-inal efficiency of 39% at 500� (around 38% at 1000�) at ambienttemperature. The reference cell has main features described inTable 4.

In addition to efficiency, the cell datasheet gives other outputparameters (Table 5) necessary in the simulations to predict thecells power output at different illuminations. Moreover, since wedeal exclusively with reflective elements, no chromatic aberrationare introduced. The temperature can also be considered reasonably

Table 4Main features of the AZUR SPACE 3C40 cell implemented in the simulations.

Base material GaInP/GaAs/Ge on Ge substrateAR coating TiOx/Al2OxChip size 5.59 � 6.39 mm2 = 35.25 mm2

Active cell area 5.5 � 5.5 mm2 = 30.25 mm2

constant as efficient cooling systems have been shown inliterature.

The receiver electrical design has been chosen in order to mini-mize the power matching problem even maintaining high degree oflinearity and easiness of construction: attention has been paid toseries connected cells since the output current in this case corre-sponds to the current produced by the worst illuminated cells ofthe series.

The choice of the number of cells to connect has been madestarting from the concept that a receiver should have a certain areato perform at a certain mean concentration. The array design has toresemble, with the right connections, an irradiance distributionwhich is mostly square and uniform and probably degradingtoward the borders. To simplify the scheme, we decided to simu-late different receivers starting from the same base unit, which isa string of series connected cells. A scheme with many parallelswould lead to a lower dependence from irradiance gradients, butit has the inconvenience to give high current and small voltagesin output. High voltages are instead more suitable for the standardrange of inverters while small currents limit the resistive losses.We thus chose to conceptually design different receivers type toperform at different output voltages. Fig. 4a and b shows the thirdof the array implemented for which we will show also the toler-ance results. It is a detector made by 56 strings of 36 cells. Thestrings spatial positioning is shown in Fig. 4a where each stringis represented by a narrow rectangle. There are 32 strings in thecentral square zone, which corresponds roughly to the maximumuniform area obtainable by the optimization, and 4 lateral zonesmade by 6 additional modules each. The total number of cells is2016. This scheme allows cells in series to be irradiated with simi-lar fluxes and at the same time, the strings and the groups containthe same number of elements thus ensuring small parallel mis-matches. This scheme does not have cells at the corners, sincethe spillage losses in case of 500� have been evaluated in the orderof 5%. The electrical connections are arranged as follows (Fig. 4b):cells in each strings are series connected as well as strings with thesame color. The central zone is then made by 8 blocks of cells eachcontaining 4 adjacent substrings (the subdivision of each coloredareas have been omitted), while the lateral strings are series con-nected in concentric frames. The 14 resulting blocks are finally par-allel connected.

The same electrical scheme has been also used for simulatingthe concentration 800�. In this case the cells of the base string

-200 -100 0 100 200

X (mm)

-200

-100

0

100

200

Y (

mm

)

(a)

-200 -100 0 100 200

X (mm)

-200

-100

0

100

200

Y (

mm

)

(b)

Fig. 4. Type-3 receiver design at 500�. The (a) panel shows the subdivision in strings. The (b) panel shows which strings are series connected (zones with the same color). The14 resulting blocks are parallel connected. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

A. Giannuzzi et al. / Applied Energy 145 (2015) 211–222 217

are only 27 and the central zone is made by 24 strings since thehigher concentration results in a smaller irradiated area. The paral-lel connected blocks are 12. Spillage losses at the corners arearound 8–10% but again we preferred to preserve the array sym-metry avoiding cells in these areas.

To analytically calculate the electrical performance, we devel-oped a routine implementing the Eqs. (5)–(9) modeling the celloutput current and voltage as functions of concentration, neglect-ing resistive effects. As for the electrical scheme, the routine imple-ments the classical equations for calculating voltages and currentsin series and parallel connections. Only these connections areinvolved while no model has been implemented for the bypassdiodes. A temperature of T ¼ 298 K has been considered and a rea-sonable value for the ideality factor nd ¼ 3:3 has been assumed totreat the junctions as real. The other initial parameters used are inTable 5. Being FF only dependent on Voc , it has been calculatedusing a classical empirical formula [43] approximated for zeroresistivity:

FFð�Þ ¼ vocð�Þ � lnðvocð�Þ þ 0:72Þ1þ vocð�Þ

ð10Þ

where vocð�Þ is the open circuit voltage normalized by the factorndKT=q.

5.3. Optimization procedure

The optimization procedure employs a downhill simplexmethod. We decided to minimize a merit function related to con-version efficiency. In particular it has been defined as the negativeefficiency of the receiver �g as defined in Eq. (9): each evaluationof this function requires the calculation of the efficiency by the raytracing procedure and the receiver model previously explained. Wesummarize the optimization steps as follows.

The initial values chosen for the parameters to be optimized areinserted in the optimization routine. The routine operates perform-ing a multidimensional minimization of a function funcðxÞ where xis an n-dimensional vector of parameters, using a downhill simplexmethod requiring only function evaluations and not derivatives.Additional input for the routine are the fractional tolerance to beachieved in the function value as well as the range of the para-meters variation.

The optimization procedure transfers the parameters value tothe ray-tracing procedures which gives the image as output, thenthe block simulating the receiver performance gets in input theimage focused by the optics. The image is represented by a matrixcontaining the local concentration impinging on each receiver cell.

The receiver model distinguishes between cells series and parallelconnected, imposing the current of a series cells as the current pro-duced by the worst illuminated cell. Subsequently, the current andvoltage output for each series/parallel are summed to give the totaloutput and the efficiency. After calculating the efficiency of theoptics coupled with that receiver, the procedure changes the para-meters value iteratively in the range specified, modifying theoptics and calculating a new image, a corresponding new efficiencyand comparing the values of the simplex obtained. When the mini-mum is found within the threshold, the routine returns an n-ele-ment vector corresponding to the function minimum value. Thiskind of method could be applied to other type of receivers and itcould be improved by extending the variables (for example thecurvatures that here we considered fixed).

5.4. Tolerance calculation

After obtaining the nominal image produced by the optimizedoptics, a tolerance calculation has been implemented to assessthe feasibility of the results. Tolerances have been obtained forboth optical and geometrical parameters. We considered 25 para-meters for each of the 4 different mirrors. Additional parametersare the two tracking angles and the receiver position along the z-axis, for overall 178 parameters. The parameters include tilts andpositions of the mirrors, their curvatures and the Zernike coeffi-cients up to the 6th radial order (from 4th to 21th). The reasonfor considering up to this order lays in the connection betweenthe radial degree of the polynomials and the spatial scale of thedeformations: the degree of a polynomial on a certain surface(which has a diameter of 2.6 m in the proposed design) roughlydefines the spatial scale (period) of the associated deformation sothat, for example, a 6th degree deformation on 2.6 m diameterwould be roughly half meter (2:6=6 m ¼ 0:43 m). It has beenevaluated that higher degree deformations, i.e. occurring on spatialscales smaller than the considered scale, can be reasonably con-trolled by surface polishing of candidate materials (aluminum,molded plastics, etc.). The tolerances have been also calculatedfor polynomials with nominal null coefficients since all the polyno-mials up to a certain degree are necessary to model the irregulari-ties down to a given scale.

The nominal image produced by the optics with the optimizedparameters and the corresponding receiver efficiency have beencalculated and stored as terms of comparison. We chose a rangeof variation for each parameter and a minimum tolerable efficien-cy. The tolerated efficiency degradation was equally split among allthe parameters, assuming their effects as uncorrelated. Degradedefficiency has been calculated for the minimum and maximum

Table 6Values in mm of the Zernike coefficients optimized at the two considered concen-trations considering type-3 receivers.

Z4(1) Z11(1) Z14(1) Z4(2) Z6(2) Z7(2) Z11(2) Z14(2)

500� 1.124 0.137 0.098 1.486 �0.616 0.223 0.003 �0.217800� 1.103 0.070 �0.108 1.053 �0.714 0.280 0.019 �0.144

218 A. Giannuzzi et al. / Applied Energy 145 (2015) 211–222

values of a given parameter, keeping nominal values for all theother parameters: if the degraded efficiency is acceptable, theminimum and maximum values of the given parameter are adopt-ed as tolerances for that parameter, otherwise the variation rangeof the parameter is reduced and the process is repeated until con-vergence. After computing the tolerances for each parameterseparately, the global effect has been evaluated by perturbing allthe parameters simultaneously in a random fashion according tothe computed tolerances and evaluating the correspondingefficiency.

6. Results: the SOLARIS concentrator

The results shown in Table 6 have been obtained by optimizingour optics at two concentrations (500� and 800�) with type-3receivers. The values of the Zernike coefficients not shown can bederived from the relations in Table 3.

The bi-dimensional and the x-cross section irradiance producedby the optimized optics have been simulated by Zemax� for thetwo concentration ratios and they are shown in Figs. 5 and 6.The x-cross section irradiance is evaluated on the central row par-allel to the x-axis of the bi-dimensional irradiance pattern. All thesimulations have been performed supposing 1 sun irradiance at theconcentrator aperture, which is the common value in Standard TestConditions (STC).

The performance obtained for other receivers types described inSection 5.2 are listed in Table 7. The efficiency g is the output pow-er of the receiver divided by the total power collected by the optics.The optimized systems show a conversion efficiency of about 30%in all the cases with 500� and of 28% in the only analyzed casewith 800�. The case with higher concentration is interesting forthe development of new generation cells because it shows thatthe proposed method gives good results also at higher concentra-tions. Moreover, the higher the concentration the smaller the num-ber of cells employed in the receiver. The case with concentration800� in fact includes only 1152 cells, almost half of the cells need-ed for the concentration 500� (2016 elements).

The relative efficiency grel in Table 7 has been defined consider-ing the only effective power impinging on the array, i.e. accounting

(a)

Fig. 5. (a) 2D and (b) x-cross section irradiance produced by the optics coupled to the typbar are W/mm2. (For interpretation of the references to colour in this figure legend, the

for spillage losses at the corners/edges. This parameter is useful toevaluate the average cells performance in the array. In three of thefour cases, its value is above 31% and it must be compared with themaximum theoretical efficiency reported in Section 5.2 for theactive part of the cell considered, i.e. 33% for concentration 500�and 32% for 1000�. This means that the cells in the arrays workreally close to their nominal performance under the irradiance pro-duced by the optimized optics.

Looking at the results in Table 7 with concentration 500�, themain difference between the three receivers analyzed lays in theoutput parameters values. Even if the total power produced is quitesimilar in all the cases (slightly higher than 10 kWe), the outputcurrent and voltage are very different. The third receiver has beendesigned specifically with a high number of series connections toobtain a high voltage value (409.2 V) suitable for the availableinverters and with small current (25.3 A) to limit the resistive loss-es. This condition is convenient from an electrical point of view,but it leads to tighter tolerances, as shown below.

The tolerance results are here shown only for the concentration500� with the type-3 receiver, giving some qualitative indicationsfor the other cases studied. The parameters which differ from mir-ror to mirror are summarized in Tables 8 and 9 while the commonparameters related to the receiver position are shown in Table 10.Five out of seven mirrors have been omitted from the list sincetheir tolerances are similar to those of the second mirror exceptfor discretization effects. The last row in Table 8 is the root squaresum (RSS) of the Zernike coefficients and it is one of the mostimportant tolerance indicators in our analysis since it representsthe tolerated surface sag deviation. For all the mirrors, this para-meter is in the order of tenths of a millimeter. The shape deviationtolerated is also compatible with the manufacturing irregularitiesof candidate materials (molded plastics or aluminum) for thedeformed/deformable mirrors. The tracking errors shown inTable 10 are quite small if compared to other CPV concentrators(normally in the order of 1 milliradian or more). In any case, thetracking accuracy can be achievable with standard tracking solu-tions commonly employed in telescopes since these systems canalso reach subarcseconds tolerances. Good pointing and activetracking systems are already developed also for solar concentrators[45], but their performances should be further improved to allowour tolerances.

The calculations have been performed setting a threshold of 3%on the efficiency, i.e. tolerating a degradation of the performancefrom 29.4% down to 26.4%. This value has been chosen as reason-able for this type of systems, but it can be varied depending onthe required performance. In general, for small perturbations, thetolerance on a parameter scales linearly with the threshold value.

-150 -100 -50 0 50 100 150

X position at Y=center (mm)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Irra

dian

ce (

Wat

ts/m

illim

eter

s sq

uare

d)

(b)

e-3 designed for 500�. The physical size of the figures is 350 mm. Units in the colorreader is referred to the web version of this article.)

(a)

-150 -100 -50 0 50 100 150

X position at Y=center (mm)

0.0

0.2

0.4

0.6

0.8

Irra

dian

ce (

Wat

ts/m

illim

eter

s sq

uare

d)

(b)

Fig. 6. (a) 2D and (b) x-cross section irradiance produced by the optics coupled to the type-3 designed for 800�. The physical size of the figures is 350 mm. Units in the colorbar are W/mm2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 7Electrical performance obtained after the optimization run with the three receiversimplemented.

Iout (A) Vout (V) Pout (W) g (%) grel (%)

Receiver 1 (500�) 98.7 105.2 10288.0 29.2 30.5Receiver 2 (500�) 50.5 204.6 10324.8 29.7 31.6Receiver 3 (500�) 25.3 409.2 10354.5 29.4 31.2Receiver 3 (800�) 32.6 302.6 9868.1 28.0 31.4

Table 8Zernike coefficients tolerances calculated for the system with 500� coupled with atype-3 receiver.

Units Parameter Mirr1 Mirr2

Nominalvalue

Tolerance Nominalvalue

Tolerance

mm Z4 1.124 0.063 1.486 0.063Z5 0.000 0.063 0.000 0.031Z6 0.000 0.250 �0.616 0.063Z7 0.000 0.031 0.223 0.016Z8 0.000 0.031 0.000 0.016Z9 0.000 0.031 0.000 0.031Z10 0.000 0.031 0.000 0.016Z11 0.137 0.008 0.003 0.016Z12 0.000 0.016 0.000 0.008Z13 0.000 0.016 0.000 0.008Z14 0.098 0.016 �0.217 0.031Z15 0.000 0.016 0.000 0.016Z16 0.000 0.016 0.000 0.004Z17 0.000 0.004 0.000 0.008Z18 0.000 0.016 0.000 0.008Z19 0.000 0.008 0.000 0.008Z20 0.000 0.016 0.000 0.016Z21 0.000 0.016 0.000 0.008

mmffiffiffiffiffiffiffiffiffiffiffiP

Z2q

0.2762 0.1122

Table 9Tolerances on other parameters calculated for the system with 500� coupled with atype-3 receiver.

Units Parameter Mirr1 Mirr2

Nominalvalue

Tolerance Nominalvalue

Tolerance

mm Radius ofcurv.

10101.0 25.0 11480.1 25.0

tilt x 0.0 0.4 �254.6 0.2mrad tilt y 0.0 0.9 0.0 0.1

tilt z 0.0 1.7 0.0 1.7

offset x 0.0 5.0 0.0 2.5mm offset y 0.0 2.5 2680.0 2.5

offset z 0.0 25.0 0.0 3.1

Table 10Tolerances calculated for to the common parameters.

Units Parameter All mirrors

Nominal value Tolerance

mrad Tracking error x 0.0 0.11Tracking error y 0.0 0.01

mm Receiver offset z 4800.0 2.5

A. Giannuzzi et al. / Applied Energy 145 (2015) 211–222 219

The tolerances are strictly related to the electrical scheme imple-mented in the receiver. For example we calculated that with thereceiver involving more parallels and with the same threshold,the tolerances would be three times more relaxed. In that casehigher output current would be produced, the output power beingapproximately the same.

The mechanical model is shown in Fig. 7. From the analysis ofthe Zernike polynomials, the desired deformations on the mirrorscan be applied by a restricted number of actuators positioned ona certain number of control points. For the system with the chosendimensions, these points are located radially on three circumfer-ences every 10�. A possible way to obtain the final surfaces is to

use spherical mirrors and to set the deformations by the actuators.Another approach is to freeform mirrors already shaped with thefinal form desired, the actuators being employed only to compen-sate the shape errors once the mirrors have been placed on theirown support. All these mirrors could be made by aluminum sheets,since this material is particular suitable for its lightness and itsductility. Molded plastic could be also a candidate substrate mate-rial (if compatible with the requested tolerances) after the deposi-tion of a high reflective layer. During the realization, the systemshould be aligned within tolerances. For this reason, we conceiveda 2-step procedure. The first phase consists in the mirrors position-ing on their own supports and the calibration of their nominalshape. This test can be performed in laboratory and it requires apoint light source, a beam splitter, a Shack–Hartmann (SH) wave-front sensor [46] with a camera. The camera acquires the imageof a point source reflected back by the mirror which can be usedto recognize the wavefront shape and the mirror surface map.The actuators are tuned iteratively until the measured surfacemap matches its nominal value (within tolerances, see Tables 9and 10). To accelerate the calibration procedure, an interactionmatrix records the SH sensor reaction to the specific movement

Fig. 7. Shaded models of the SOLARIS concentrator: (a) front side and (b) rear side.

220 A. Giannuzzi et al. / Applied Energy 145 (2015) 211–222

of each single actuator. This matrix has to be inverted and used totransform the SH sensor signal into incremental corrections toapply to the actuators. The second stage is an alignment on Sunof each mirror on the whole frame. A mask dimensioned as thereceiver and realized in a material resistant to temperatures of afew hundreds degrees is needed. Concentric frames of pinholeson the mask transmit part of the light impinging on the receiverplane to diodes or other electronic light-sensitive devices. Such atool allows to sample the irradiance distribution produced by theoptics and to adjusted iteratively the position of each mirror onthe common frame until the desired irradiance is obtained.Another interaction matrix is used to record the diodes reactionto the parameters to align. This matrix is then inverted and usedto translate the measured signal into corrections for the mirrorpositioning.

The new concentrator resulting from the investigation carriedout has been called ‘‘SOLARIS (SOLAR Image Squaring) Concentra-tor’’ and it has been patented in Italy. The patent is owned by boththe University of Bologna and the National Institute of Astro-physics (INAF), the two research institutes involved in the project.Main subjects of the patent are both the innovative concentratingCPV application and the method for the numerical optimization ofreflective surfaces. The procedures to test/calibrate the reflectiveshapes and to align the mirrors on Sun, as well as the receiverand the mechanical design are all parts of the patent. The modeland the obtained results will be validated with the described pro-cedures during the forthcoming prototyping stage.

7. Summary and discussion

We developed a new optical designing method for solar concen-trators. In particular, dense array photovoltaic applications need anaccurate control on both shape and irradiance of the collected lightspot to perform at high efficiency. These systems are experiencingin the last years growing interest (from market and research) asfeasible solutions in the production of cost competitive electricityon demand, especially in very sunny environments and off-gridcommunities. The development of solar cells that can work at veryhigh irradiance imposes a technological jump also from an opticalpoint of view, to let these systems work at the same performanceof the employed cells. The proposed method is based on controllingthe optical shapes so that the spot produced by the mirrors canresemble the optimal features for the chosen receiver without

including secondary optics. The deformations to apply have beenanalytically modeled by the Zernike polynomials and the deformedmirrors have been simulated by ray tracing routines developed onpurpose. At the same time, different schemes of dense array recei-vers have been designed using reference cells with known featuresand simulated by implementing simple electrical models for pho-tovoltaic devices. The deformed optics have been numerically opti-mized to maximize the performance of the concentrator as afunction of the coupled receiver. The method has been fruitfullyemployed to solve the prescribed irradiance problem at high con-centration in CPV dishes. It has led to the design of a novel CPVoptics, the SOLARIS concentrator. Both the method implementedand the specific application developed have been patented in Italy.

The main advantage of using big monolithic mirrors is to havefew optics to manage respect to the complex segmented opticsproposed in other researches involving dense arrays. Despite thistechnology is quite recent and commercial plants are not as dif-fused as the refractive fresnel lens based systems, our method todesign dense array concentrators opens a new scenario for devel-oping PV systems that could perform at very high efficiency work-ing at high concentrations. This efficiency boosting up to nominallevels and, at the same time, the relaxation of the constraints onthe receiver design and the recent development of new materialsfor optical application suggest interesting perspectives of costreduction.

The concentrator developed is a single stage multi-mirror sys-tem made by 7 monolithic optics placed in an hexapolar arrange-ment and all focusing on the same receiver. The principalinvestigated design has a mean concentration ratio 500�. Thedeformations applied to the optics allow them to produce a solarspot resembling a square shape with smoothed corners. The irradi-ance pattern inside the spot obtained is highly uniform. At thisconcentration, the optimized optics can boost the conversion effi-ciency of the whole receiver up to 30%, almost the same theoreticalperformance of the single cell used in the calculations which isaround 33% (considering only the active areas). The receiver hasbeen designed as simple as possible, using exclusively strings ofidentical cells in series. The strings are then organized in parallelsor series connections, with a Cartesian configuration and notinvolving bypass diodes in the design.

From an optical point of view, different considerations can beenmade to extend the purposes and the applications of the methodconceived. Similar systems with different concentrations can be

A. Giannuzzi et al. / Applied Energy 145 (2015) 211–222 221

surely designed ever keeping in mind the optimization method hasbeen tested for the two concentration 500� and 800�, and that theresults are better in the first case considered thanks to the higherdefocus involved. Despite this, we demonstrated that our methodcan work efficiently also at many hundreds of concentration ratio.

Method improvements could be done by a further investigationof the convenient deformations to introduce, exploring for examplethe effects related to Zernike polynomials of higher degrees. Theselected deformations and the optical configuration used in thiswork are indeed only an example of the method proposed: otherconcentrators could be designed by adding deformations or chang-ing the geometrical/optical parameters as a function of the desiredspot features. Systems with single or multiple mirrors (different ornot) could be implemented and different geometrical configura-tions explored. Also the mirrors aperture could be varied in shapeand size depending on the amount of output power needed or onthe economical/constructive constraints. The final spot could resultfrom a superimposition of images not necessarily centered in thesame point, as in the studied cases. Another interesting applicationcould result from exploring the performance of deformable opticsincluding very simple reflective secondary optics to recover possi-ble light losses at the receiver borders or to relax the tolerances(thus enhancing the acceptance angle).

A great advantage of employing actively deformable opticscould be given by the tuning of the concentration ratio. Using con-venient deformable materials, flexible systems could be obtainedembedding different type of receivers but exploiting the sameoptics. Also from the receiver point of view, great improvementscould be obtained in terms of electric efficiency, involving opti-mized electrical schemes or thinking to future monolithic recei-vers. Finally, an extension of this method could be also helpful insolving thermal problems. Thermal concentrators do also need acertain uniformity in the light collected to optimally transfer theenergy to the exchanging fluid. The proposed technique could beimplemented to correct possible optical aberrations thus boostingthe concentration up to its limit.

Acknowledgements

This research activity has been financially supported by a MIURPhD Grant and a Fondazione CARISBO grant within the researchframework on renewable sources, energy spare and distributedmicro-generation.

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