Tracking integration in concentrating photovoltaics using laterally moving optics

In this work the concept of tracking-integrated concentrat ing photovoltaics is studied and its capabilities are quantita tively analyzed. The design strategy desists from ideal concentration performa nce to reduce the external mechanical solar tracking effort in favor of a comp act installation, possibly resulting in lower overall cost. The proposed opti cal design is based on an extended Simultaneous Multiple Surface (SMS) al gorithm and uses two laterally moving plano-convex lenses to achieve hi gh concentration over a wide angular range of ±24. It achieves 500 × concentration, outperforming its conventional concentrating photovolta ic counterparts on a polar aligned single axis tracker. © 2011 Optical Society of America OCIS codes: (080.2740) Geometric optical design; (080.3620) Lens syste m d sign; (220.1770) Concentrators; (220.4298) Nonimaging optics; (350.6050) S olar energy. References and links 1. A. Luque and V. Andreev, Concentrator photovoltaics (Springer, 2007). 2. J. Chaves, Introduction to nonimaging optics (CRC Press, 2008). 3. S. Kurtz, “Opportunities and Challenges for Development o f a Mature Concentrating Photovoltaic Power Industry (Revision),” Tech. rep., National Renewable Energy Lab or tory (NREL), Golden, CO. (2010). 4. A. Bett, F. Ferrazza, J. Herzog, A. Marti, and W. Wettling, “Concentration Photovoltaics (CPV),” Tech. rep., EU Photovoltaic Technology Platform, The Strategic Research A genda (2006). 5. J. Karp, E. Tremblay, and J. Ford, “Planar micro-optic solar concentrator,” Opt. Express 18, 1122–1133 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18 -2-1122. 6. J. Hallas, J. Karp, E. Tremblay, and J. Ford, “Lateral trans lation micro-tracking of planar micro-optic solar concentrator,” Proc. SPIE 7769, 776904 (2010). 7. P. Kotsidas, E. Chatzi, and V. Modi, “Stationary nonimagin g lenses for solar concentration,” Appl. Opt. 49, 5183–5191 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=ao-49 -27-5183. 8. P. Kotsidas, V. Modi, and J. M. Gordon, “Nominally stationa ry high-concentration solar optics by gradient-index lenses,” Opt. Express 19, 2325–2334 (2011), http://www.opticsinfobase.org/abstract.cfm?URI=oe-19 -3-2325. 9. W. Sweatt, B. H. Jared, G. N. Nielson, M. Okandan, A. Filato v, M. B. Sinclair, J. L. Cruz-Campa, and A. L. Lentine, “Micro-optics for high-efficiency optical pe rformance and simplified tracking for concentrated photovoltaics (CPV),” International Optical Design Confe rence, Optical Society of America (2010), paper ITuC4. 10. E. Watson, “Analysis of beam steering with decentered mic rolens arrays,” Opt. Eng. 32, 2665–2670 (1993). 11. J. Gibson, B. Duncan, E. Watson, and J. Loomis, “Wide-angl e decentered lens beam steering for infrared countermeasures applications,” Opt. Eng. 43, 2312 (2004). 12. J. C. Mĩ nano and J. C. Gonz ález, “New method of design of nonimaging concentrators,” App l. 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Cebecauer, M.̌ Súri, and E. Dunlop, “Analysis of one-axis tracking strategi es for PV systems in Europe,” Prog. Photovolt: Res. Appl. 18, 183–194 (2010). 19. R. Winston and W. Zhang, “Pushing concentration of stati on ry solar concentrators to the limit,” Opt. Express 18, A64–A72 (2010),http://www.opticsinfobase.org/abstract.cfm?URI=oe-18 -S1-A64. 20. P. Benıtez and J. C. Mi ñano, “Ultrahigh-numerical-aperture imaging concentrator ,” J. Opt. Soc. Am. A14, 1988– 1997 (1997),http://www.opticsinfobase.org/abstract.cfm?URI=josaa -14-8-1988. 21. J. C. Mĩ nano, P. Beń ıtez, W. Lin, J. Infante, F. Mũ noz, and A. Santamar ı́a, “An application of the SMS method for imaging designs,” Opt. Express 17, 24036–24044 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17 -26-24036. 22. F. Mũnoz, J. M. I. Herrero, P. Ben ı́tez, J. C. Mĩ nano, W. Lin, J. F. Vilaplana, G. Biot, and C. Marta, “Novel fa st catadioptric objective with wide field of view,” Proc. SPIE 7780, 778704 (2010). 23. P. 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Introduction
Concentrating photovoltaic (CPV) systems employ optics to concentrate direct sunlight onto solar cells.The ratio of input and output aperture areas defines the concentration ratio of the concentrating system.The significant decrease of the required solar cell area provides a pathway to lower cost, as expensive semiconductor material is replaced with inexpensive mirrors or lenses.Furthermore, high-efficiency multi-junction solar cells can be used to boost the conversion efficiency of CPV modules [1].The angular acceptance of the optics and the achieved level of concentration are two measures that specify the overall performance of the optical design.The angular acceptance determines the maximum incident angle for which the light rays entering the optics still reach the solar cell.The second law of thermodynamics determines the maximum theoretical concentration limit for a given angular acceptance of the optics [2].Available CPV systems vary widely according to the desired concentration ratio, the kind of used optics (reflective and/or refractive) and the type of deployed solar cells.
A possible concept to classify these different systems is the concentration ratio.Highconcentrating photovoltaics (HCPV) typically perform at concentration ratios beyond 400× and makes use of multi-junction solar cells.This level of concentration normally demands an accurate dual-axis tracking of the sun's diurnal and seasonal movement.Medium-concentrating photovoltaics (MCPV) with typical concentration ratios around 10× to 20× are installed on single-or dual-axis trackers using silicon or other solar cells.Low-concentrating photovoltaics (LCPV) with concentration ratios around 2× to 4×, typically enhance conventional silicon solar panels and may be installed on a tracker [3].
Despite the vast number of these differing concepts, almost all have something essential in common: concentrating optics plus deployed solar cells are treated as an inseparable static unit which is then -usually within an array -packaged as a concentrating photovoltaics module.This module can then be installed on an external solar tracker.In contrast with this clear separation between (stationary) CPV modules and external solar trackers, the here introduced general concept of a tracking-integrated concentrating photovoltaic system is to transfer part of this external solar tracking functionality to the concentrating photovoltaic module.This inherent difference is schematically illustrated in Fig. 1.For conventional CPV modules, the acceptance angle α of the deployed optics is a single measure that determines the demands and tolerances that apply to the used external solar tracker.In case of a tracking-integrated CPV module, two parameters are necessary for a full description.The aperture angle α A of the optical system and the acceptance angle α of the deployed optics.The aperture angle of the optical system defines the angular range that can be covered by the integrated-tracking and therefore determines the demands for an additional external solar tracker, if needed.The acceptance angle determines the angular tolerances for each particular direction within the aperture angle of the optical system.This tracking integration in CPV can only be achieved through the introduction of additional degrees of freedom, such as relative movement between optics and solar cells.An example for a tracking-integrated CPV module, consisting of two moving optics arrays and a solar cell array is shown in Fig. 1

(b).
It is important to differentiate between a tracking-integrated CPV and a micro-tracking CPV module, where in the latter case separate concentrating optics plus solar cell units are tracking the sun's movement individually.Even though a tracking-integrated CPV module might lack a perfect concentration performance over the entire aperture angle of the optical system, this shortcoming is deliberately chosen because there is a strong reduction of the external solar tracking effort, possibly resulting in a lower overall cost.Furthermore, external tracking constructions can be optimized with respect to size, stability and material consumption [4].This is especially important regarding possible roof top installations.On the other hand, additional degrees of freedom, such as internal movement raise the complexity of the CPV module and more optics arrays will reduce the optical efficiency on a system level.A detailed benefit-cost analysis will be necessary to identify those tracking-integrated CPV systems that show great promise for all demands.
First systems following this approach were already discussed in literature.One proposed micro-optic solar concentrator focuses sunlight into a planar slab waveguide with edgemounted photovoltaic cells.The laterally translating waveguide with respect to the lens array enables an integrated solar tracking [5,6].Another proposed design utilizes stationary meniscus lenses and longitudinally and laterally moving solar cells (or vice versa), which achieved about 25× line concentration and about 300× point concentration for an aperture angle α A = 30 • and an acceptance angle for sunlight α = 5 mrad [7].A recently published paper analyses highconcentration solar optics using spherical gradient-index lenses moving in three dimensions which are capable of daylong averaged flux concentration levels around 1000× [8].
A different proposed approach is to apply integrated solar tracking to fine-tune the total tracking functionality allowing coarse external solar tracking.This system has a measured point concentration of about 44× and a measured aperture angle of approximately α A = 10 • [9].However, except for the last example, these systems make only use of a relative movement between the optics and the receiver (waveguide or solar cells), respectively.
One main objective of this work is to establish the general concept of tracking-integrated CPV, specify the functionality and to highlight its capability.Contrary to most previously mentioned systems which make only use of a relative movement between optics and receiver, the investigation of two laterally moving optics layers to integrate tracking in CPV is subject of this work.The application of laterally moving optics is well-known in the domain of beam steering where moving lens arrays are used to redirect incident laser light [10,11].However, to integrate tracking in concentrating photovoltaics, the optics should combine both steering and concentration of the incident sun light.The objective of this investigation is to demonstrate the potential of laterally moving optics for tracking-integrated concentrating photovoltaics.
This article is structured as follows.In Section 2, some possible configurations of external solar module mountings are shortly discussed and their requirements for tracking integration are presented.For a comparison later on, meniscus lenses comprising a laterally moving receiver are introduced in Section 3 and their concentration performance for a specific aperture angle is investigated.These lenses are designed using the Simultaneous Multiple Surface design algorithm in two dimensions (SMS2D) [12].To make use of this design algorithm in case of laterally moving lenses, an extended algorithm is introduced in Section 4. This extended SMS2D algorithm is used in Section 5 to design (equivalent to the systems in Section 3) tracking-integrated optics and investigate their concentration performance.Finally, in Section 6, conclusions are drawn and an outlook is given.

Mounting of CPV modules (with or without integrated tracking)
The conservation of étendue links the acceptance angle of an optical system to the maximum achievable concentration ratio C for line and point concentration systems [13].Due to the earth's obliquity of the ecliptic, the sun's relative diurnal and seasonal movement imposes certain demands on the used external solar trackers according to the acceptance angle of the concentrating optics.In case of HCPV including external dual axis solar tracking, the CPV module follows the sun's movement throughout the day and the year as accurately as possible.This means that the acceptance angle is only necessary to meet the sun's divergence angle (0.28 • ) and additional mechanical tracking tolerances.
Different Nonimaging Optics designs ensure high acceptance angles and high concentration performance that approach the theoretical limits [14,15].For example, an acceptance angle α = ±1.1 • at 1000× concentration was demonstrated for a Fresnel-based photovoltaic concentrator [14].A positive side effect of dual axis tracking is that the CPV module is always aligned perpendicular towards the sun.The maximized insolation (incident solar radiation) thereby increases the annual energy yield of the installation.For this reason solar trackers are also used for classic solar panel installations.The tracking-integrated CPV approach takes over part of the external tracking requirements.Therefore, it is possible to use only external single axis trackers or even fixed CPV modules.

CPV module mounting on single axis trackers
The possible alignment of a single axis tracker can reduce the necessary aperture angle of the CPV module and thus the demands on the tracking integration.The aperture angle will be minimal for a polar aligned single axis tracker, installed towards the South.In this particular configuration, the tilt angle is equal to the latitude of the installation and the rotational axis of the single axis tracker equals the earth's axis of rotation.The aperture angle of the system then reduces to approximately ±24 • (axial tilt of the earth) in North-South direction and the sun's half divergence angle in East-West direction which is shown in Fig. 2(a).The design and the theoretical concentration limit of a point concentration system on a polar aligned single axis tracker has been discussed in the past and achieved about 300× concentration [16,17].For deviations from the polar aligned rotational axis in North-South direction, the angular range becomes asymmetric regarding the origin and the necessary aperture angle for tracking integration increases.The angular range for a horizontally aligned single axis tracker for latitude of 30 • (Seville, Spain) is shown in Fig. 2

(b).
Of course, in contrast to dual axis tracking, single axis tracking suffers from off-axis cosine losses.However, the comparison of the potential annual energy yield for different single axis tracker with dual axis tracker installations shows moderate differences, e.g. for miscellaneous places in Europe [18].It is obvious that the application of dual axis trackers in HCPV mainly arises from the theoretical point concentration limit for single axis trackers than a maximized insolation -it is rather an appreciated necessity.Therefore, the tracking-integrated approach can be very useful to further increase the point concentration ratio for single axis tracker installations.Within the upcoming sections, the tracking integration for point concentration systems on a polar aligned single axis tracker will be investigated and discussed.This specific configuration is defined by the aperture angle α A = 24 • and the acceptance angle α = 0.28 • .

Stationary CPV module mounting
With the elimination of all external tracking, the aperture angle of a stationary trackingintegrated CPV module should encompass the full angular range of the sun's movement throughout the year.The geometrical considerations for stationary module mounting are similar to the discussed single axis tracker case.The necessary aperture angle in North-South direction meets its minimum for a polar plane alignment, where the panel surface faces the equator with a tilt angle equal to the local latitude.This can be used to push the concentration of stationary solar concentrators to its limits (4× concentration, eight hours per day year around) [19].
For this particular configuration, a tracking-integrated line concentrator in East-West direction results in the same aperture angle as in the polar aligned single axis tracker case (see Fig. 2(a)).The same applies to deviations from the polar plane alignment.The angular range becomes asymmetric regarding the origin and the necessary aperture angle increases.For a horizontally aligned module at latitude of 30 • (Seville, Spain) the angular range coincides with Fig. 2(b).Within the upcoming sections, the tracking integration for line concentration systems in a polar plane alignment will be investigated and discussed.This specific configuration is characterized by the aperture angle α A = 24 • and the acceptance angle α = 0.28 • .

Stationary lenses comprising a laterally moving receiver
In case of stationary single lenses, only moving solar cells remain as a degree of freedom including longitudinal and lateral movement.However, from a manufacturing point of view it is easier to use moving optics and thus stationary solar cells.To reduce the complexity of the integrated tracking as well as the thickness of the overall CPV module the optics are restricted to lateral movement only.Classical (imaging) optics design is based on minimizing a chosen merit function which quantifies the image quality for defined sets of rays.This common design concept can easily be adopted to enable the design of solar concentrators.The optics could be described as aspherical surfaces and instead of the image quality, the concentration performance would serve as a merit function.The optimization is then done by parametrically varying the optical surfaces.By using multi-parametric optimization (a common tool of any optical design software), these algorithms normally start from an initial set of parameters to a final optimized set of parameters.Since it is a non-convex optimization, it can not be guaranteed that these algorithms find the global minimum.
For the lens design in this publication, the Simultaneous Multiple Surface design method in two dimensions (SMS2D) is used which was developed as a nonimaging optics design method.The SMS design method sets up the problem in a different way.In contrast to multi-parametric optimization, SMS surfaces are piecewise curves made of several portions of Cartesian ovals that map initial ray sets to final ray sets.It involves the simultaneous calculation of N optical surfaces using N one-parameter ray sets for which specific conditions connect the initial with final ray sets [2].
The imaging properties of the SMS design method were first investigated for an ultrahighnumerical-aperture imaging concentrator [20].A more general formulation of the SMS2D method for imaging systems comprises perfect imaging of N ray sets at the correspondent N image points.The SMS method offers the flexibility to choose the ray sets and their associated imaging points.The obtained image quality over the entire field of view showed superior results [21].Recently, the SMS2D design method was used to design a catadioptric objective with a wide field of view [22].These examples clearly show the potential of the SMS2D design algorithm for imaging optical designs and strengthen the possibility it can be used for tracking-integrated designs as well.A further advantage of the SMS design is the existence of a three-dimensional algorithm to implement free-form optics [23].Especially for trackingintegrated point concentration optics on a single axis tracker mounting (see Sec. 2.1), free-form optics are capable to exceed the performance of a rotational symmetric optical system.However, the three-dimensional implementation is beyond the scope of this work and therefore only rotational symmetric systems will be investigated.
The used SMS2D design algorithm of this section can be found in [2,21] and will therefore not be discussed in detail here.The system's design is described by the f-number of the lens, its refraction index n (from now on, n=1.5), two design angles of opposite sign and their focus positions in the receiver plane.Fig. 3 shows such a calculated exemplary meniscus lens for design angles θ = ±10 • and the two correspondent parallel ray sets.
Several lenses with different design angles are calculated for a constant f-number f/1.875, which is also a parameter that can be optimized.Their line and point concentration ratios are investigated using a ray tracing software and 50.000 rays each simulation.(All ray tracing sim-θ = ± 10°L aterally moving receiver Fig. 3. Example of a calculated meniscus lens for two parallel ray sets and design angles θ = ±10 • .The laterally moving receiver is indicated. ulations in this paper are only monochromatic and performed with Advanced Systems Analysis Program (ASAP) version 2009 V2R2 from Breault Research Organization.)Both, positions and directions of the particular rays are randomly chosen within the aperture of the lens and the divergence angle of the source (±0.28 • ) using uniform distributions.The actual receiver size is defined in such a way that 95% of the energy entering the lens' aperture is collected and the optimal receiver position is obtained by the ray tracer.The mapping of the lateral central receiver position r is close to a law r(θ ) = c • tan(θ ).The ratio of entrance aperture and receiver area is then determining the line or point concentration ratio, respectively.It has to be mentioned that refractive line concentrators (even for monochromatic light) perform differently for rays normal to the direction of symmetry than for non-normal rays.This effect causes that the apparent refractive index of the lenses is increased and the rays will get defocused.
Fig. 4(a) shows the line concentration ratio for rays normal to the direction of symmetry for different design angles θ for a system's half aperture angle of 24 • (polar plane alignment).All curves are characterized by a peak concentration at the incident design angle.With an increasing design angle the peak concentration shifts and the left slope is decreasing while the right slope is increasing.The result for θ = 18 • shows the rudimentary optimized (using MATLAB's fminsearch unconstrained nonlinear optimization) case where the concentration ratio values C(0 • ) and C( 24• ) meet at approximately 10× concentration.This criterion ensures maximum concentration over the entire aperture angle of ±24 • .However, it does not necessarily guarantee maximum energy yield over the entire year.This will also depend on the insolation and thus on the geographical location and the site-specific climatic conditions [24].
A similar analysis is also presented for point concentration performance and a system's half aperture angle of 24 • (polar aligned single axis tracker).All SMS2D designs are now rotated around the optical axis and the ray tracing results in spots in the receiver plane.Again, a 95% transmission efficiency of the energy entering the entrance aperture and reaching a circular receiver area determines the particular point concentration ratio.The interpretation can be done analogous to the line concentration approach.By increasing the design angle, the peak point concentration ratio is shifted until the concentration for C(0 • ) and C(24 • ) meet slightly above 100× concentration over the entire angular range for θ = 18 • .
The investigated stationary lenses comprising a laterally moving receiver demonstrate the procedural method and capabilities of the SMS2D design algorithm for tracking-integrated line and point concentration systems with an aperture angle of 24 • .However, a 100× point concentration is not sufficient to make use of high-efficient solar cells.As mentioned in the introduction, the main objective of this work is the investigation of two laterally moving lens arrays.To properly evaluate the benefit from an additional moving lens array, the number of two curved optical surfaces should remain constant.

Extended SMS2D algorithm to design laterally moving optics
The SMS2D lenses in Sec. 3 showed promising properties to design tracking-integrated CPV optics for a given aperture angle.The flexibility to choose two design angles of opposite sign and their associated image points enables a viable concept to adjust the overall concentration ratio to the desired aperture angle.However, the question arises how to use this algorithm to design optics layers that move relatively to one another.The simplest possible arrangement of such a system consists of two moving plano-convex lenses and a receiver plane.This basic setup and all relevant parameters are shown in Fig. 5.For a symmetric angular range regarding the origin (e.g., aperture angle for polar plane alignment), both lenses are characterized by an axis of symmetry.
R Fig. 6.Extended SMS2D design procedure to include motion by the alternate addition of surface segments on the top and bottom surfaces of two plano-convex lenses.The calculation starts through the center of the bottom surface which determines the optical path length (a).It then proceeds through mirroring (b) to the edges (c) of the lenses.Finally, all chains add up to the final lenses that map the incident rays to the receiver point R (d).
The fundamental idea is to extend the design algorithm in order to shift the second lens already by an offset x relative to the first optical axis.The setup is then fully described by the distance between first lens and receiver W, the position of the second lens p, the offset x and the diameters of both lenses.The extended design algorithm works as follows.Instead of explicitly using two design angles of opposite sign, the algorithm uses only one design angle and the axes of symmetry.It can be easily shown that both approaches can be transformed into each other.The basic construction of the starting SMS chain is shown in Fig. 6(a).
The design starts with the definition of a receiver point R and the offset x of the second lens.Due to the axial symmetry, a point P 0 can be chosen on the bisector line of the second lens in a certain distance.Given the symmetry of that lens, the normal n 0 is pointing perpendicular towards the receiver plane.The ray coming from R is now refracted at P 0 , taking the refractive index of the lens' material into account.The refracted ray propagates to point F 0 which can be determined by the desired central thickness of the second lens.Since the upper surface of the second lens is fixed to be plane (normal perpendicular to receiver plane), the ray refracts in F 0 towards F 1 .The position of F 1 defines the distance between both planar lens surfaces.As the lower surface of the first lens is plane, the ray is now refracted towards P 1 , where the normal n 1 is chosen to refract the ray in a direction parallel to the designated direction of the design angle.Due to the axial symmetry of the first lens, the point P 1 and its normal are now mirrored to Q 1 .From the path of this first ray the optical path length S between R and the incident wave front is calculated.Next, a second order function is chosen that goes through P 1 and Q 1 and that is perpendicular to the normal vectors in these points.This function defines the continuously differentiable SMS starting segment and determines the central thickness of the first lens.Finally, a set of rays parallel to the incident direction is launched through these points and the corresponding points on the segment P 0 P 2 are calculated.Each ray's path is determined to an intersection point on the upper planar surface of the second lens.For all these rays the constant optical path length S is used to calculate the points and normal vectors on segment P 0 P 2 that refract those rays towards R.
For the next step in Fig. 6(b), the axial symmetry of the second lens is used to mirror the lens segment P 0 P 2 to the new segment P 0 Q 2 .A set of rays coming from R is launched through the points on P 0 Q 2 .These rays are refracted at the segment and then propagated through both planar lens surfaces.Again, the constant optical path length is used to calculate the points and normal vectors on a new segment P 1 P 3 to refract each ray in a direction parallel to the direction of the design angle.
Similar to this previous step, the axial symmetry of the first lens is used in Fig. 6(c) to mirror the lens segment P 1 P 3 to segment Q 1 Q 3 .The procedure resembles the final steps in Fig. 6(a).A set of rays parallel to the incident direction is launched through the segment Q 1 Q 3 which enables the calculation of a new segment P 2 P 4 .The steps in Fig. 6(b) and (c) can now be alternately repeated until a stop criterion is reached.Fig. 6(d) shows an exemplary result for this design algorithm.It should be stated that for x→0 the extended SMS2D algorithm approaches the classic SMS2D.

Laterally moving lenses comprising a moving receiver
This extended SMS2D algorithm is now used to design different tracking-integrated optical systems comprising two plano-convex lenses and a receiver plane.Analog to Sec. 3, the line and point concentration ratio of such systems is investigated for monochromatic light, a half aperture angle of 24 • and an angular acceptance α = 0.28 • (polar plane/polar axis tracker alignment).
The second lens is positioned centered between first lens and receiver plane.All optical systems are designed to meet the same f-number f/1.875 as in Sec. 3. Furthermore, a criterion for exclusion ensures that the diameter of the second lens does not exceed the diameter of the first lens, such that it is possible to create an array of this basic lens structure.The concentration performance of different systems is again evaluated using the ray tracing analysis as in Sec. 3. A simple iteration method within the ray tracer makes sure that the second lens is always properly shifted for any incident direction.The optimal receiver position is again obtained by the ray tracer.An analysis of the mapping of the positions shows no strict law.For angles well below the actual design angle the mapping for both, second lens and central receiver position is close to a law r(θ ) = c • tan(θ ).However, by reaching and exceeding the design angle there are some derivations from these expressions.
Analog to Fig. 4, Fig. 7 shows the line concentration ratio for rays normal to the direction of symmetry and the point concentration ratio for different design angles against the incident angle from 0 • to 24 • .As mentioned previously, the line concentration system will perform differently for non-normal rays which will get defocused.Both, line and point concentration ratio show the same characteristics for different design angles as in the stationary SMS2D case.The peak concentration at the design angle follows a monotone decline of the concentration ratio to both sides.The results for θ = 19 • and C(0 • ) ≈ C(24 • ) were obtained by rudimentary optimization (using MATLAB's fminsearch unconstrained nonlinear optimization) for line and point concentration systems, respectively.The final results are approximately 20× line concentration and 500× point concentration over the entire angular range, both for the sun's half divergence angle of 0.28 • .Fig. 8 shows two single-frame excerpts from ray tracing animation videos to demonstrate the tracking-integrated concentration performance of the line concentration Fig. 8(a) (Media 1) and point concentration Fig. 8(b) (Media 2) over the entire aperture angle ±24 • and for α = 0.28 • .
Even though the number of curved optical surfaces and the f-number maintain the same as in Sec. 3, the additional moving optics helps to raise the line and point concentration performance considerably by a factor of about 2 and 5, respectively.However, the additional moving optics layer increases the complexity of the internal tracking.The tracking-integrated line concentration system clearly exceeds the concentration of its stationary solar concentrator counterparts for a polar plane alignment but will suffer from defocused non-normal incident directions.The tracking-integrated point concentration system exceeds the concentration ratio of its conventional CPV counterparts on a polar aligned single axis tracker.Additionally, it already enables the potential use of high-efficient multi-junction solar cells, even though a rotational symmetric system is clearly not the optimal solution given the asymmetry of the angular aperture space.A further enhancement is expected for an extended SMS algorithm in three dimensions.
This quantitative evaluation of the benefits from an additional moving lens array clearly demonstrates the potential of this concept and the extended design algorithm for tracking integration in concentrating photovoltaics.In addition to this design study, further desired attributes of concentrating photovoltaic systems are the uniformity of the flux, color mixing on the solar cell's surface and an increased acceptance angle which is directly related to misalignment tolerance.A well accepted solution to achieve this is a final stage concentrator on top of the solar cell [25].A future generalization to N surfaces of the extended SMS2D design algorithm presented in this work should also cover this component and possibly help to increase the aperture angle, ensure color mixing and increase the acceptance angle.Polychromatic simulations and tolerance analysis will then help to verify the full potential of this approach.

Conclusion
Within the scope of this work, the general concept of tracking-integrated concentrating photovoltaics has been clearly defined and possible configurations were discussed.One main objective was a quantitative analysis of the potential of laterally moving optics.An extended SMS2D design algorithm was presented and used to design tracking-integrated optical systems comprising two plano-convex lenses and thus two curved optical surfaces.In comparison to a single lens consisting of two curved optical surfaces comprising a laterally moving receiver, the additional moving optics helped to raise the concentration performance considerably.This design approach enables a strong reduction of the external solar tracking effort along with high concentration, exceeding the concentration of its conventional solar concentrator counterparts at the same time.Therefore, external tracking constructions can be optimized with respect to size, stability and material consumption, possibly resulting in a lower overall cost.A future generalization of the introduced extended algorithm to N optical surfaces in three dimensions will help to explore the full potential of this approach.

Fig. 1 .
Fig. 1.Inherent difference between a conventional (stationary) CPV module with its acceptance angle α (a) and a tracking-integrated CPV module with the aperture angle α A of the optical system and the acceptance angle α for a particular direction (b).

ig. 2 .
Angular range of incident sun light for CPV modules mounted on a single axis tracker that is (a) polar aligned or (b) horizontally aligned.(Both for Seville, Spain)

ig. 4 .
Line concentration (a) and point concentration (b) ratio against the incident angle for different SMS2D lenses comprising a laterally moving receiver.

Fig. 5 .
Fig.5.Schematic drawing of a basic optical system consisting of two laterally moving lenses and a receiver plane.The offset indicates the lateral shift of the second lens for an off-axis incident ray set.All relevant parameters describing the system are indicated.

ig. 7 .
Line concentration (a) and point concentration (b) ratio against the incident angle for different extended SMS2D lens designs comprising a lateral moveable receiver.

Fig. 8 .
Fig. 8. Exemplary single-frame excerpts from ray tracing animation of (a) line concentration (Media 1) and (b) point concentration (Media 2) over the entire angular range.The used designs are the two results obtained for θ = 19 • for line and point concentration.