Tailored free-form optics with movement to integrate tracking in concentrating photovoltaics

Fabian Duerr; Youri Meuret; Hugo Thienpont

doi:10.1364/OE.21.00A401

Optics Express
Vol. 21,
Issue S3,
pp. A401-A411
(2013)
•https://doi.org/10.1364/OE.21.00A401

Tailored free-form optics with movement to integrate tracking in concentrating photovoltaics

Fabian Duerr, Youri Meuret, and Hugo Thienpont

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Fabian Duerr, Youri Meuret, and Hugo Thienpont, "Tailored free-form optics with movement to integrate tracking in concentrating photovoltaics," Opt. Express 21, A401-A411 (2013)

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Abstract

The economic use of high-efficiency solar cells in photovoltaics requires high concentration of sunlight and therefore precise dual-axis tracking of the sun. Due to their size and bulkiness, these trackers are less adequate for small- to mid-scale installations like flat rooftops. Our approach to combine concentrating and tracking of sunlight utilizes two laterally moving lens arrays. The presented analytic optics design method allows direct calculation of the free-form lens surfaces while incorporating the lateral movement. The obtained concentration performance exceeds a factor of 500. This demonstrates that one can benefit from high-efficiency solar cells and more compact and flexible single-axis trackers at the same time.

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Fig. 1 Today’s utility-scale concentrating photovoltaic systems are less suitable for small-to mid-scale installations due to their size and bulkiness (Photo courtesies of (a) Jeff Aubin, (b) Soitec Concentrix and (c) SolFocus)

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Fig. 2 Examples of different tracking integration concepts: using (a) rotary or (b) rectilinear motion; (c) a motion-free example based on liquid crystal (LC) light steering.

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Fig. 3 Shown are different solar trackers used in both PV and CPV; (a) dual-axis tracker, (b) polar aligned single-axis tracker, and (c) horizontal single-axis tracker.

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Fig. 4 (a) Schematic assembly of a conventional CPV module for pedestal-mounted dual-axis trackers. (b) Our tracking integrated CPV module for polar aligned single-axis trackers. (c) The laterally moving lens design in this paper is based on two plan-convex lenses with rectangular apertures, suitable for lens array structures.

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Fig. 5 (a) The 2D analytic design of the single thick lens is based on the convergence points construction in (b). (c) This analytic lens design can be extended for tracking integration by including separated plano-convex lenses and (d) a shifted second lens for off-axis rays.

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Fig. 6 Introduction of all necessary initial values in (a) and (b), and functions in (c) and (d) to derive the conditional equations from Fermat’s principle in three dimensions.

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Fig. 7 Ray tracing results for explicitly calculated solutions in two and three dimensions demonstrate perfect coupling of on- and off-axis ray sets, as intended by the analytic design.

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Fig. 8 Comparison of the concentration performance for three different designs methods: the analytic free-form solution of this paper offers superior performance over the full field of view when compared with the extended SMS3D design and the rotational symmetric SMS2D design; all designs share rectangular lens and rectangular receiver apertures.

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Equations (11)

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d_{1} = {\vec{v}}_{0} \cdot ({\vec{p}}_{1} - {\vec{w}}_{0}), d_{2} = n_{2} | {\vec{p}}_{2} - {\vec{p}}_{1} |, d_{3} = | {\vec{p}}_{3} - {\vec{p}}_{2} |, d_{4} = n_{3} | {\vec{p}}_{4} - {\vec{p}}_{3} |, d_{5} = | {\vec{p}}_{5} - {\vec{p}}_{4} |

{\hat{d}}_{1} = {\vec{v}}_{1} \cdot ({\vec{q}}_{1} - {\vec{w}}_{0}), {\hat{d}}_{2} = n_{2} | {\vec{q}}_{2} - {\vec{q}}_{1} |, {\hat{d}}_{3} = | {\vec{q}}_{3} - {\vec{q}}_{2} |, {\hat{d}}_{4} = n_{3} | {\vec{q}}_{4} - {\vec{q}}_{3} |, {\hat{d}}_{5} = | {\vec{q}}_{5} - {\vec{q}}_{4} |

\begin{array}{l} D_{1} = \frac{\partial}{\partial x} (d_{1} + d_{2}) = 0, & D_{2} = \frac{\partial}{\partial y} (d_{1} + d_{2}) = 0, & D_{3} = \frac{\partial}{\partial p_{2 x}} (d_{2} + d_{3}) = 0, \\ D_{4} = \frac{\partial}{\partial p_{2 y}} (d_{2} + d_{3}) = 0, & D_{5} = \frac{\partial}{\partial p_{3 x}} (d_{3} + d_{4}) = 0, & D_{6} = \frac{\partial}{\partial p_{3 y}} (d_{3} + d_{4}) = 0, \\ D_{7} = \frac{\partial}{\partial s} (d_{4} + d_{5}) = 0, & D_{8} = \frac{\partial}{\partial t} (d_{4} + d_{5}) = 0 \end{array}

\begin{array}{l} D_{9} = \frac{\partial}{\partial x} ({\hat{d}}_{1} + {\hat{d}}_{2}) = 0, & D_{10} = \frac{\partial}{\partial y} ({\hat{d}}_{1} + {\hat{d}}_{2}) = 0, & D_{11} = \frac{\partial}{\partial q_{2 x}} ({\hat{d}}_{2} + {\hat{d}}_{3}) = 0, \\ D_{12} = \frac{\partial}{\partial q_{2 y}} ({\hat{d}}_{2} + {\hat{d}}_{3}) = 0, & D_{13} = \frac{\partial}{\partial q_{3 x}} ({\hat{d}}_{3} + {\hat{d}}_{4}) = 0, & D_{14} = \frac{\partial}{\partial q_{3 y}} ({\hat{d}}_{3} + {\hat{d}}_{4}) = 0, \\ D_{15} = \frac{\partial}{\partial u} ({\hat{d}}_{4} + {\hat{d}}_{5}) = 0, & D_{16} = \frac{\partial}{\partial v} ({\hat{d}}_{4} + {\hat{d}}_{5}) = 0 \end{array}

f (x, y) = \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} f_{i, j} {(x - x_{0})}^{i} y^{2 j}, g (x, y) = \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} g_{i, j} {(x - x_{1})}^{i} y^{2 j}

s (x, y) = \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} s_{i, j} {(x - x_{0})}^{i} y^{2 j}, t (x, y) = \sum_{i = 0}^{\infty} \sum_{j = 1}^{\infty} t_{i, j} {(x - x_{0})}^{i} y^{(2 j - 1)}

with s (x, y) = (p_{2 x}, p_{3 x}, q_{2 x}, q_{3 x}, s, y) (x, y) and t (x, y) = (p_{2 y}, p_{3 y}, q_{2 y}, q_{3 y}, t, v) (x, y)

lim_{x \to x_{0}} lim_{y \to 0} \frac{\partial^{n}}{\partial x^{n}} \frac{\partial^{m}}{\partial y^{m}} D_{i} = 0 (i = 1, 3, \dots, 15) m {n \in ℕ_{1}, m = 0}

lim_{x \to x_{0}} lim_{y \to 0} \frac{\partial^{n}}{\partial x^{n}} \frac{\partial^{m}}{\partial y^{m}} D_{i} = 0 (i = 2, 4, \dots 16), {n = 0, m \in ℕ | odd number m}

\begin{array}{l} lim_{x \to x_{0}} lim_{y \to 0} \frac{\partial^{n}}{\partial x^{n}} \frac{\partial^{m}}{\partial y^{m}} D_{i} = 0 (i = 2, 4, \dots, 16), {n \in ℕ_{1}, m \in ℕ | odd number m} \\ lim_{x \to x_{0}} lim_{y \to 0} \frac{\partial^{n - 1}}{\partial x^{n - 1}} \frac{\partial^{m + 1}}{\partial y^{m + 1}} D_{i} = 0 (i = 3, 5, 7, 11, 13, 15), {n \in ℕ_{1}, m \in ℕ | odd number m} \end{array}

M_{x} (\begin{matrix} f_{n + 1, 0} \\ g_{n + 1, 0} \\ s_{n, 0} \end{matrix}) = {\vec{b}}^{(n, 0)}, M_{y} (\begin{matrix} f_{0, m + 1} \\ g_{0, m + 1} \\ t_{0, m} \end{matrix}) = {\vec{b}}^{(0, m)}, M_{x y} (\begin{matrix} f_{n, m + 1} \\ g_{n, m + 1} \\ s_{n - 1, m + 1} \\ t_{n, m} \end{matrix}) = {\vec{b}}^{(n, m)}

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Equations (11)

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