Abstract

Beam-steering lens arrays enable solar tracking using millimeter-scale relative translation between a set of lens arrays. This may represent a promising alternative to the mechanical bulk of conventional solar trackers, but until now a thorough exploration of possible configurations has not been carried out. We present an approach for designing beam-steering lens arrays based on multi-objective optimization, quantifying the trade-off between beam divergence and optical efficiency. Using this approach, we screen and optimize a large number of beam-steering lens array configurations, and identify new and promising configurations. We present a design capable of redirecting sunlight into a <2° divergence half-angle, with 73.4% average yearly efficiency, as well as a simplified design achieving 75.4% efficiency with a <3.5° divergence half-angle. These designs indicate the potential of beam-steering lens arrays for enabling low-cost solar tracking for stationary solar concentrators.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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C. S. Schuster, “The quest for the optimum angular-tilt of terrestrial solar panels or their angle-resolved annual insolation,” Renewable Energy 152, 1186–1191 (2020).
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J. Blank and K. Deb, “Pymoo: Multi-Objective Optimization in Python,” IEEE Access 8, 89497–89509 (2020).

P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and S. . Contributors, “SciPy 1.0: Fundamental algorithms for scientific computing in Python,” Nat. Methods 17(3), 261–272 (2020).
[Crossref]

2019 (3)

K. E. Moore, “Optimization for as-built performance,” Proc. SPIE 10925, 1 (2019).
[Crossref]

C. C. Olson, “Automated design of optical architectures using novel encoding methods and a multi-objective optimization framework,” Proc. SPIE 11105, 11 (2019).
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H. J. D. Johnsen, A. Aksnes, and J. Torgersen, “Pushing the limits of beam-steering lens arrays,” Proc. SPIE 11120, 10 (2019).
[Crossref]

2018 (2)

2016 (4)

Bráulio Fonseca Carneiro de Albuquerque, F. L. de Sousa and A. S. Montes, “Fonseca Carneiro de Albuquerque Multi-objective approach for the automatic design of optical systems,” Opt. Express 24(6), 6619–6643 (2016).
[Crossref]

A. J. Grede, J. S. Price, and N. C. Giebink, “Fundamental and practical limits of planar tracking solar concentrators,” Opt. Express 24(26), A1635–A1646 (2016).
[Crossref]

H. Apostoleris, M. Stefancich, and M. Chiesa, “Tracking-integrated systems for concentrating photovoltaics,” Nat. Energy 1(4), 16018 (2016).
[Crossref]

V. Narasimhan, D. Jiang, and S.-Y. Park, “Design and optical analyses of an arrayed microfluidic tunable prism panel for enhancing solar energy collection,” Appl. Energy 162, 450–459 (2016).
[Crossref]

2015 (2)

L. D. DiDomenico, “Towards doubling solar harvests using wide-angle, broad-band microfluidic beam steering arrays,” Opt. Express 23(24), A1398–A1417 (2015).
[Crossref]

J. S. Price, X. Sheng, B. M. Meulblok, J. A. Rogers, and N. C. Giebink, “Wide-angle planar microtracking for quasi-static microcell concentrating photovoltaics,” Nat. Commun. 6(1), 6223 (2015).
[Crossref]

2014 (1)

N. León, C. Ramírez, and H. García, “Rotating Prism Array for Solar Tracking,” Energy Procedia 57, 265–274 (2014).
[Crossref]

2013 (1)

2012 (2)

W. Lin, P. Benitez, and J. C. Miñano, “Beam-steering array optics designs with the SMS method,” Proc. SPIE 8485, 848505 (2012).
[Crossref]

J. Köster and S. Rahmann, “Snakemake—a scalable bioinformatics workflow engine,” Bioinformatics 28(19), 2520–2522 (2012).
[Crossref]

2011 (4)

P. Kotsidas, V. Modi, and J. M. Gordon, “Nominally stationary high-concentration solar optics by gradient-index lenses,” Opt. Express 19(3), 2325–2334 (2011).
[Crossref]

W. T. Xie, Y. J. Dai, R. Z. Wang, and K. Sumathy, “Concentrated solar energy applications using Fresnel lenses: A review,” Renewable Sustainable Energy Rev. 15(6), 2588–2606 (2011).
[Crossref]

F. Duerr, Y. Meuret, and H. Thienpont, “Tracking integration in concentrating photovoltaics using laterally moving optics,” Opt. Express 19(S3), A207–A218 (2011).
[Crossref]

S. Valyukh, I. Valyukh, and V. Chigrinov, “Liquid-Crystal Based Light Steering Optical Elements,” Photonics Lett. Pol. 3(2), 88–90 (2011).
[Crossref]

2009 (2)

H. Li and Q. Zhang, “Multiobjective Optimization Problems With Complicated Pareto Sets, MOEA/D and NSGA-II,” IEEE Trans. Evol. Computat. 13(2), 284–302 (2009).
[Crossref]

J. Xiang, N. Wu, J. Zhang, and L. Wu, “Design of driving and control system based on Voice Coil Actuation for linear motion of micro-lens array,” Proc. SPIE 7133, 713330 (2009).
[Crossref]

2008 (1)

J. Bourderionnet, M. Rungenhagen, D. Dolfi, and H. D. Tholl, “Continuous laser beam steering with micro-optical arrays: Experimental results,” Proc. SPIE 7113, 71130Z (2008).
[Crossref]

2007 (1)

2006 (1)

2004 (1)

2003 (1)

D. Buie, A. G. Monger, and C. J. Dey, “Sunshape distributions for terrestrial solar simulations,” Sol. Energy 74(2), 113–122 (2003).
[Crossref]

1998 (2)

I. Ono, S. Kobayashi, and K. Yoshida, “Global and multi-objective optimization for lens design by real-coded genetic algorithms,” Proc. SPIE 3482, 110–121 (1998).
[Crossref]

E. Watson, D. Miller, and K. Barnard, “Analysis of fill factor improvement using microlens arrays,” Proc. SPIE 3276, 123–134 (1998).
[Crossref]

1993 (1)

E. A. Watson, “Analysis of beam steering with decentered microlens arrays,” Proc. SPIE 32(11), 2665–2670 (1993).
[Crossref]

1990 (1)

K. M. Flood, B. Cassarly, C. Sigg, and J. Finlan, “Continuous wide angle beam steering using translation of binary microlens arrays and a liquid crystal phased array,” Proc. SPIE 1211, 296–304 (1990).
[Crossref]

. Contributors, S.

P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and S. . Contributors, “SciPy 1.0: Fundamental algorithms for scientific computing in Python,” Nat. Methods 17(3), 261–272 (2020).
[Crossref]

Akatay, A.

Aksnes, A.

H. J. D. Johnsen, A. Aksnes, and J. Torgersen, “Pushing the limits of beam-steering lens arrays,” Proc. SPIE 11120, 10 (2019).
[Crossref]

H. J. D. Johnsen, J. Torgersen, and A. Aksnes, “Solar tracking using beam-steering lens arrays,” Proc. SPIE 10758, 4 (2018).
[Crossref]

H. J. D. Johnsen, A. Aksnes, and J. Torgersen, “Zemax OpticStudio Models of Beam-Steering Lens Arrays, figshare,” (2020). https://doi.org/10.6084/m9.figshare.12505784

Apostoleris, H.

H. Apostoleris, M. Stefancich, and M. Chiesa, “Tracking-integrated systems for concentrating photovoltaics,” Nat. Energy 1(4), 16018 (2016).
[Crossref]

Archibald, A. M.

P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and S. . Contributors, “SciPy 1.0: Fundamental algorithms for scientific computing in Python,” Nat. Methods 17(3), 261–272 (2020).
[Crossref]

Ataman, C.

Barnard, K.

E. Watson, D. Miller, and K. Barnard, “Analysis of fill factor improvement using microlens arrays,” Proc. SPIE 3276, 123–134 (1998).
[Crossref]

Benitez, P.

W. Lin, P. Benitez, and J. C. Miñano, “Beam-steering array optics designs with the SMS method,” Proc. SPIE 8485, 848505 (2012).
[Crossref]

Benitez, P. G.

R. Winston, J. C. Minano, P. G. Benitez, W. N. Shatz John, C. Bortz, and J. C. Bortz, Nonimaging Optics (Elsevier Science, 2005).

Blank, J.

J. Blank and K. Deb, “Pymoo: Multi-Objective Optimization in Python,” IEEE Access 8, 89497–89509 (2020).

Bortz, C.

R. Winston, J. C. Minano, P. G. Benitez, W. N. Shatz John, C. Bortz, and J. C. Bortz, Nonimaging Optics (Elsevier Science, 2005).

Bortz, J. C.

R. Winston, J. C. Minano, P. G. Benitez, W. N. Shatz John, C. Bortz, and J. C. Bortz, Nonimaging Optics (Elsevier Science, 2005).

Bourderionnet, J.

J. Bourderionnet, M. Rungenhagen, D. Dolfi, and H. D. Tholl, “Continuous laser beam steering with micro-optical arrays: Experimental results,” Proc. SPIE 7113, 71130Z (2008).
[Crossref]

Brett, M.

P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and S. . Contributors, “SciPy 1.0: Fundamental algorithms for scientific computing in Python,” Nat. Methods 17(3), 261–272 (2020).
[Crossref]

Bright, J.

P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and S. . Contributors, “SciPy 1.0: Fundamental algorithms for scientific computing in Python,” Nat. Methods 17(3), 261–272 (2020).
[Crossref]

Buie, D.

D. Buie, A. G. Monger, and C. J. Dey, “Sunshape distributions for terrestrial solar simulations,” Sol. Energy 74(2), 113–122 (2003).
[Crossref]

Burovski, E.

P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and S. . Contributors, “SciPy 1.0: Fundamental algorithms for scientific computing in Python,” Nat. Methods 17(3), 261–272 (2020).
[Crossref]

Campbell, R.

R. Campbell and M. Machado, “LOW cost CPV = Embedded CPV with internal tracker,” in 2010 35th IEEE Photovoltaic Specialists Conference, (IEEE, 2010), pp. 003003–003007.

Carey, C. J.

P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and S. . Contributors, “SciPy 1.0: Fundamental algorithms for scientific computing in Python,” Nat. Methods 17(3), 261–272 (2020).
[Crossref]

Cassarly, B.

K. M. Flood, B. Cassarly, C. Sigg, and J. Finlan, “Continuous wide angle beam steering using translation of binary microlens arrays and a liquid crystal phased array,” Proc. SPIE 1211, 296–304 (1990).
[Crossref]

Chiesa, M.

H. Apostoleris, M. Stefancich, and M. Chiesa, “Tracking-integrated systems for concentrating photovoltaics,” Nat. Energy 1(4), 16018 (2016).
[Crossref]

Chigrinov, V.

S. Valyukh, I. Valyukh, and V. Chigrinov, “Liquid-Crystal Based Light Steering Optical Elements,” Photonics Lett. Pol. 3(2), 88–90 (2011).
[Crossref]

Cimrman, R.

P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and S. . Contributors, “SciPy 1.0: Fundamental algorithms for scientific computing in Python,” Nat. Methods 17(3), 261–272 (2020).
[Crossref]

Cournapeau, D.

P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and S. . Contributors, “SciPy 1.0: Fundamental algorithms for scientific computing in Python,” Nat. Methods 17(3), 261–272 (2020).
[Crossref]

Dai, Y. J.

W. T. Xie, Y. J. Dai, R. Z. Wang, and K. Sumathy, “Concentrated solar energy applications using Fresnel lenses: A review,” Renewable Sustainable Energy Rev. 15(6), 2588–2606 (2011).
[Crossref]

Dannberg, P.

de Sousa, F. L.

Deb, K.

J. Blank and K. Deb, “Pymoo: Multi-Objective Optimization in Python,” IEEE Access 8, 89497–89509 (2020).

Dey, C. J.

D. Buie, A. G. Monger, and C. J. Dey, “Sunshape distributions for terrestrial solar simulations,” Sol. Energy 74(2), 113–122 (2003).
[Crossref]

DiDomenico, L. D.

Dolfi, D.

J. Bourderionnet, M. Rungenhagen, D. Dolfi, and H. D. Tholl, “Continuous laser beam steering with micro-optical arrays: Experimental results,” Proc. SPIE 7113, 71130Z (2008).
[Crossref]

Duerr, F.

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Supplementary Material (1)

NameDescription
» Code 1       Zemax OpticStudio models of the two selected beam-steering lens array designs reported in the paper High-performance stationary solar tracking through multi-objective optimization of beam-steering lens arrays. Details of how these models relate to th

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Figures (13)

Fig. 1.
Fig. 1. (a) Conceptual illustration of how a beam-steering system can be combined with conventional concentrator optics. (b) Example of a beam-steering lens array: An afocal stack of lens arrays, which redirects sunlight utilizing relative movement between these lens arrays.
Fig. 2.
Fig. 2. Paraxial working principle of a simple beam-steering lens array.
Fig. 3.
Fig. 3. (a) Assumed orientation of beam-steering lens array during optimization. (b) Simulated angular distribution of normalized yearly direct irradiation on a lens array with the fixed orientation from a, installed at a latitude of 40°.
Fig. 4.
Fig. 4. The global MOEA/D-DE optimization algorithm is combined with the SLSQP local optimization algorithm to form a hybrid optimization algorithm. The choice between local and global search is based on the number of objective function evaluations in the local search, $n_{fev,local}$ compared to the total number of objective function evaluations, $n_{fev,total}$. We used $k=0.5$, which means that $50\%$ of the search effort is dedicated to the local searches.
Fig. 5.
Fig. 5. Set of the best performing optimized configurations, mapping out the trade-off between efficiency and divergence half-angle. The value in brackets below the divergence half angle represents the ideal geometric concentration at this divergence half-angle, according to Eq. (3).
Fig. 6.
Fig. 6. Beam-steering lens array with ◀▶↓◀▶ configuration optimized for $2^{\circ }$ divergence half-angle (oe-28-14-20503-i001 in Fig. 5), drawn at $0^{\circ }$ and $40^{\circ }$ angles of incidence respectively. The black arrows indicate tracking motion. The drawing shows a 2 -dimensional slice of the optimized system, which consists of hexagonally packed three-dimensional lens arrays.
Fig. 7.
Fig. 7. Simplified beam-steering lens array with ◀▶↓▶ configuration optimized for $3.5^{\circ }$ divergence half-angle (oe-28-14-20503-i002 in Fig. 5), drawn at $0^{\circ }$ and $40^{\circ }$ angles of incidence respectively. The black arrows indicate tracking motion. The drawing shows a 2-dimensional slice of the optimized system, which consists of hexagonally packed three-dimensional lens arrays.
Fig. 8.
Fig. 8. (a) The two selected beam-steering lens arrays have >80% efficiency for $\pm 40^{\circ }$angle of incidence, and a gradual drop-off in efficiency at larger angles of incidence. The dashed curves show nominal performance, while the continuous curves show expected performance with the chosen set of error distributions. (b) When the systems are placed in a fixed orientation as described in Section 3.2, this efficiency distribution corresponds to average yearly efficiency of 73.5% for the high concentration design and 75.6% for the simplified design, respectively.
Fig. 9.
Fig. 9. Ray-traced 3D-model of the two selected beam-steering lens arrays, both shown at a $40^{\circ }$ angle of incidence. (a) ◀▶↓▶ configuration optimized for a permitted divergence of $3.5^{\circ }$, (b) ◀▶↓◀▶ configuration optimized for a permitted divergence of $2.0^{\circ }$.
Fig. 10.
Fig. 10. (a) Optimized Pareto fronts for the ◀▶↓◀▶ configuration, assuming different scaled versions of the error distributions in Table 3. 100% corresponds to the values reported in the table. The achievable performance is strongly influenced by the scale of manufacturing errors, as can be seen by the large changes to the Pareto fronts. (b) The selected designs from a are simulated at different error distributions to see how sensitive they are to errors. The system optimized for nominal performance has the highest performance at zero manufacturing errors, but is also the most sensitive to such errors.
Fig. 11.
Fig. 11. Full set of trade-off curves for different configurations of beam-steering lens arrays optimized as described in Section 3.5.
Fig. 12.
Fig. 12. Full set of trade-off curves for different configurations of beam-steering lens arrays optimized as described in Section 3.5, but assuming nominal performance instead of as-built performance.
Fig. 13.
Fig. 13. Comparing internal Python model and Zemax OpticStdio model of the selected beam-steering lens array designs. The Zemax OpticStudio model assumes nominal performance and constant azimuth angle, and the simulated efficiencies agree well with the internal Python model when these assumptions are replicated.

Tables (3)

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Table 1. Proposed symbols for classifying different beam-steering lens array configurations.

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Table 2. List of free variables during optimization of the beam-steering lens arrays

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Table 3. Manufacturing errors and manufacturing constraints assumed during optimization.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

Δ x = f 1 tan ϕ .
M = f 1 f 2 = 1 + 2 f 1 d tan ϕ m a x ,
C m a x = 1 ( sin θ ) 2
min f ( x , θ m a x ) = ( η ¯ ( x , θ m a x ) , θ m a x )
such that g j ( x ) 0 ,
η ¯ ( x , θ m a x ) = E o u t ( x , θ m a x ) E i n ,
η ¯ ( x , θ m a x ) = 0 π e ( ϕ ) × η ( x , θ m a x , ϕ ) d ϕ ,

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