Optical characterization of nonimaging dish concentrator for the application of dense-array concentrator photovoltaic system Ming-Hui Tan,1,* Kok-Keong Chong,1,2 and Chee-Woon Wong1,3 1
Faculty of Engineering and Science, Universiti Tunku Abdul Rahman, Off Jalan Genting Kelang, Setapak, 53300 Kuala Lumpur, Malaysia 2
e-mail: [email protected]
3
e-mail: [email protected]
*Corresponding author: [email protected] Received 10 October 2013; revised 9 December 2013; accepted 11 December 2013; posted 16 December 2013 (Doc. ID 199239); published 17 January 2014
Optimization of the design of a nonimaging dish concentrator (NIDC) for a dense-array concentrator photovoltaic system is presented. A new algorithm has been developed to determine configuration of facet mirrors in a NIDC. Analytical formulas were derived to analyze the optical performance of a NIDC and then compared with a simulated result obtained from a numerical method. Comprehensive analysis of optical performance via analytical method has been carried out based on facet dimension and focal distance of the concentrator with a total reflective area of 120 m2 . The result shows that a facet dimension of 49.8 cm, focal distance of 8 m, and solar concentration ratio of 411.8 suns is the most optimized design for the lowest cost-per-output power, which is US$1.93 per watt. © 2014 Optical Society of America OCIS codes: (230.4040) Mirrors; (220.1770) Concentrators; (220.4298) Nonimaging optics. http://dx.doi.org/10.1364/AO.53.000475
1. Introduction
The depletion of fossil fuel and global warming issues has pressed mankind to explore alternative sources of energy that are safe, clean, and renewable. The invention of multijunction solar cells capable of achieving 40% conversion efficiency has increased the potential of a concentrator photovoltaic (CPV) system to replace the current source of energy [1]. Due to the high cost of solar cell materials, a CPV system can allow a large area of solar cells to be substituted by reflectors or lenses in the concentrator. For the application of this system, a solar concentrator plays a vital role by applying geometrical optics in 1559-128X/14/030475-12$15.00/0 © 2014 Optical Society of America
the design of the reflectors or lens to focus high solar flux on the CPV module at the receiver [2–4]. Over the last few decades, a parabolic dish concentrator has been widely used for solar energy applications, especially for applications involving high solar concentration and high-temperature collection. For a large dish concentrator system, using a single parabolic mirror as a reflector is costly due to the special technology needed to fabricate special mirrors with a thickness from 0.7 to 1.0 mm and to obtain the parabolic shape [5]. For a large collective area of parabolic dish concentrator, multifaceted spherical or concave mirrors are arranged on the approximate parabolic shape structure as a reflector instead of using a single piece of parabolic reflector [6,7]. Despite alignment of mirrors being required in a faceted parabolic concentrator, its fabrication is cheaper and simpler 20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS
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as compared with a single precise parabolic reflector. The implementation of a parabolic concentrator in a dense-array CPV system encounters the difficulty to produce good uniformity of solar illumination on the receiver, but instead the tendency is more toward Gaussian distribution [6,7]. On the other hand, a dense-array CPV module that receives nonuniform illumination will face a severe drop in efficiency due to the current mismatched problem as opposed to the module under uniform illumination [8–12]. Beside the aforementioned parabolic concentrators, researchers have proposed various kinds of solar concentrators for the sake of producing uniform flux distribution on the target. Mills and Morrison advocate the use of an advanced form of linear Fresnel reflector that produces better uniformity of solar irradiation at a solar concentration ratio lower than 100 suns [13]. For a moderate solar concentration ratio, modular Fresnel lenses are applied in a CPV system to generate better uniformity of solar irradiance. Conversely, the transmission efficiency of the modular Fresnel lenses are less than 80% due to the absorption by the lens material and the reflection at the surface of the lens [14–16]. In addition, two-stage solar concentrators that comprise a parabolic dish and secondary optics or homogenizer have been introduced for the application of a CPV system [17–19]. The introduction of an additional optical device will inflict additional losses, which can reach 10% or more for typical designs as well as additional cost and complexity [20]. Alternatively, Chong et al. proposed a nonimaging planar concentrator (NIPC) consisting of numerous flat facet mirrors, which are arranged in a square array and spaced evenly at the same height, to superimpose the sunlight at the receiver, which can produce a reasonably high solar concentration ratio and uniform solar irradiance [1,21]. However, as the ratio of focal distance to width of the NIPC decreases, the sunlight blocking effect among adjacent facet mirrors will increase. For those facet mirrors especially located further from the center of the NIPC, higher tilting angles are required to reflect sunlight toward the common target, but it will cause more sunlight directed from an adjacent facet mirror to be blocked. Inspired by adopting the merit ideas from both NIPC (to produce reasonably uniform illumination with the use of flat facet mirrors) and parabolic dish concentrator (to avoid the sunlight blocking and shadowing effects with the geometry gradually increase in height), the nonimaging dish concentrator (NIDC) with single-stage focusing has been proposed by extracting the best designs from both solar concentrators [22,23]. In this paper, a newly developed computational algorithm is introduced to determine the facet mirror configuration for the NIDC, which eliminate blocking and shadowing effects from the adjacent facet mirrors. Besides, analytical formulas have been derived to simplify and expedite the process of characterization study of NIDC. In addition, the calculated results using analytical formulas 476
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are also verified with the simulated results obtained from a numerical method using a ray-tracing technique. Comprehensive analysis of optical performance via analytical method has been carried out for various facet dimensions and focal distances with a total reflective area of 120 m2 in order to optimize the optical design of NIDC in terms of cost-peroutput power of the dense-array CPV system. 2. Optical Principle of Nonimaging Dish Concentrator
NIDC employs multiple identical square or rectangular facet mirrors acting as optical apertures to gather the solar irradiance from the sun and to superimpose all the images at the common receiver. In our study, NIDC consists of a 2m × 2n array of identical facet mirrors arranged at different heights to prevent sunlight blocking and shadowing effects among adjacent facet mirrors. For a geometry representation, a Cartesian coordinate system is used to identify the position of all the facet mirrors within the concentrator where the origin is defined as a central point of the NIDC. The central line of the NIDC starting from the origin lies along the row direction is defined as the X axis; the central line of the NIDC from the origin lies along the column direction is defined as the Y axis; the Z axis is defined from the origin pointing to the target direction, which is also perpendicular to both the X axis and Y axis. Referring to Fig. 1, the origin of the global coordinate system, O0; 0; 0, is located at the center of the concentrator, and the locality of each facet mirror in the NIDC can be indexed as i; j, where the i and j represent the position of the facet mirror at the ith column and jth row of the NIDC, respectively. From Fig. 2, the NIDC can be divided into four quadrants, which are the top right, top left, bottom right, and bottom left. The global coordinate of the central point of any facet mirror is written as 2
3 Mx Mi;j 4 M y 5 : M z i;j
(1)
Fig. 1. Cartesian coordinate system representing the coordinate, incident angle θi;j , and the two tilted angles, σ i;j and γ i;j , of i, j mirror where F is the focal distance of the NIDC and O is the origin, which is the center of the concentrator frame.
" σ i;j sin−1
# Mx q ; 2 cos θi;j M 2x M 2y F − M z 2
(6)
# My q : (7) F − M z M 2x M 2y F − M z 2
" γ i;j tan
−1
3. Methodology of Designing NIDC Geometry
Fig. 2. Initial facet mirror’s configuration of the NIDC. The concentrator can be divided into four quadrants, which are the top right, top left, bottom right, and bottom left.
The coordinate of the receiver is defined as 2 3 2 3 Tx 0 T 4 T y 5 4 0 5; Tz F
(2)
According to the previous design of a NIPC, all the facet mirrors are arranged in an array where all the adjacent facets are spaced evenly and located in the same height. From Fig. 3, as the distance of the facet mirror from the center of the concentrator increases, the sunlight blocking effect among the adjacent mirrors becomes worse due to large tilted angles. Furthermore, the horizontal gap (Gi ) between the adjacent facet mirrors is wider owing to large tilted angles as shown in Fig. 4. To optimize optical efficiency of the new solar concentrator, the sunlight blocking and shadowing effects among the adjacent facet mirrors must be eliminated. To do so, the facet mirrors must be gradually lifted upward along the z direction for the facet mirrors located from the
where F is the shortest distance between the center of the receiver and the center of the concentrator frame (the origin of coordinate system), which is also known as the focal distance of NIDC. As the NIDC tracks the sun during its operational period, the unit vector of the incident sunray can be written as 2 3 0 I⃗ 4 0 5: (3) 1 The unit vector of the normal of each facet mirror can be derived as 3 2 Nx ⃗ ⃗ ⃗ i;j 4 N y 5 I Ri;j N 2 cos θi;j N z i;j 2 3 −M x p Nx 2 cos θi;j M 2x M 2y F−M z 2 6 7 −M y 6N 7 p 6 7 y 6 (4) 2 cos θi;j M 2x M 2y F−M z 2 7 ; p 4 5 F−M z M 2x M 2y F−M z 2 p Nz 2 2 2 2 cos θi;j
M x M y F−M z
Fig. 3. Cross-sectional view of the planar concentrator. Blocking effect starts to occur at the fifth row of the facet mirror counting from the center of the concentrator.
i;j
where Ri;j is the unit vector of the reflected solar ray for the i, j facet mirror, and θi;j is the incident angle of the sunray, which can be expressed as 2 M x M 2y 1∕2 1 −1 tan : (5) θi;j F − Mz 2 i;j Finally, there are two tilted angles of the i, j facet mirror to superimpose its solar image on the common receiver: σ i;j is a tilted angle about Y axis, and γ i;j is a tilted angle about the X axis. The two tilted angles can be derived as
Fig. 4. Conceptual drawing that defines the initial positions of facet mirrors with the same height and then obtains final positions of facet mirrors with gradually increased height to eliminate sunlight blocking and shadowing by adjacent mirrors. Gi is initial gap between the facet mirror, and G is the gap between facet mirrors after reposition. (a) Along Y and Z axes. (b) Along X and Z axes. 20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS
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central to peripheral position of the new concentrator as described in Fig. 4. The lifting of the facet mirror is not done physically, but it is virtually performed during the computational process. Moreover, gaps between the adjacent facet mirrors should also be minimized from Gi to G in order to optimize the total effective area of NIDC. The positions of the facet mirrors are computed in a way as to maintain the gap spacing among the adjacent facet mirrors at a fixed distance via the coordinate transformation method in both X and Y axes. In practice, small gaps among the facet mirrors are still required and unavoidable to absorb the manufacturing defect by providing tolerance for the installation of facets. In the computational algorithm of designing the NIDC geometry, we start with defining all the facet mirrors arranged in a 2D array at the same height. Then we obtain final positions of facet mirrors one by one from the central region toward the peripheral region with gradually increased height as to eliminate sunlight blocking and shadowing among adjacent mirrors as well as to minimize the gap between adjacent mirrors. The final positions of the facets that form a dish will be different from that of the initially defined position, and the process of rearrangement of mirror positions will cause the variation in the tilting angles of the facet mirror as to maintain the solar image aiming at the receiver. Therefore, in the process of computing and designing the geometry of NIDC, an iterative method must be used to calculate the final position of each facet mirror in which the details will be discussed in the following section. From Fig. 2, all facet mirrors in the NIDC can be subdivided into four major quadrants and the origin, O, is located at the center of the NIDC. The configuration of facet mirrors in the four quadrants is symmetric to each other, and therefore we only need to consider any of the four quadrants for the simulation to save the computational time. From Fig. 2, the topright quadrant of the concentrator is chosen as a reference for performing simulation, and the facet mirror closest to the origin is defined as M1;1 ; the facet mirror located furthest from the origin is defined as Mm;n , provided that m is the number of column and n is the number of row in the NIDC. A.
Coordinate Transformation of Facet Mirror
As the facet mirrors are used to reflect the sunray to the receiver or target, each of them must be tilted at particular tilting angles so that the solar image can be accurately superimposed on the common receiver. The new orientation of the facet mirrors in 3D spaces can be modeled by applying coordinate transformation. For the convenience of representing 3D coordinate transformation in the mathematical formulas, one more dimensionality of space is inserted to express all the transformation functions in a linear relationship. The initial coordinate of the individual facet mirror is first defined in a local coordinate system as shown in Fig. 5 wherein the origin of the coordinate system, 478
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Fig. 5. Initial coordinate of the facet mirror where the origin is at the middle of the facet mirror surface, and Pi is the center point of the four edges of the facet mirror.
M0; 0; 0, is attached to the center of the facet mirror surface instead of the center of the concentrator. Then it will perform the rotational transformations for the tilted angles σ i;j and γ i;j before it is transformed to the global coordinate system whereby the origin is defined at the center of concentrator. In this modeling, the facet mirror is represented by the central points at the four boundaries of the facet mirror (see Fig. 4). The four points representing the i, j facet mirror can be treated as a vector in the coordinate space, and it is written as 3 2 Pkx 6 Pky 7 7 (8) Pk i;j 6 4 Pkz 5 ; 1 i;j where k 1 (central point at upper boundary of the facet mirror), 2 (central point at right boundary of the facet mirror), 3 (central point at bottom boundary of the facet mirror), 4 (central point at left boundary of the facet mirror). After the transformation, the final coordinate of the four points representing the i, j facet mirror will be treated as a vector in matrix form written as: 2 3 Hkx 6 Hky 7 7 (9) Hk i;j 6 4 Hkz 5 : 1 i;j In the coordinate transformation, there are two tilted angles, σ i;j and γ i;j , rotating about two orthogonal axes with respect to the facet mirror itself. Last, a translational coordinate transformation is required to transform the position of facet mirror from the local coordinate system (attached to the local facet mirror) to the global coordinate system (attached to the concentrator). The rotational transformation matrix for the tilted angle σ i;j can be written as 2 3 cos σ i;j 0 − sin σ i;j 0 60 1 0 07 7 σ i;j 6 (10) 4 sin σ i;j 0 cos σ i;j 0 5: 0 0 0 1 The rotational transformation matrix for the tilted angle γ i;j can be written as
2
1 0 6 0 cos γ i;j γ i;j 6 4 0 sin γ i;j 0 0
0 − sin γ i;j cos γ i;j 0
3 0 07 7: 05 1
(11)
Hk MT Pk ;
The translational transformation matrix based on the coordinate of the facet mirror M x ; M y ; M z i;;j in the global coordinate system can be written as 2
1 60 T 1 i;j 6 40 0
0 1 0 0
0 0 1 0
3 Mx My 7 7: Mz 5 1
The final orientation of the facet mirror can be known by applying coordinate transformations on the initial coordinate in the following formula: (13)
where the coordinate transformation matrices for the facet mirror is MT T 1 i;j γ i;j σ i;j :
(14)
B. Computational Algorithm
(12)
Figure 6 shows a flow chart to illustrate the special computational algorithm to compute the geometry of NIDC by determining the coordinate and orientation
Fig. 6. Flow chart to illustrate the special computational algorithm to compute the geometry of the NIDC, which defines the initial orientations of all facet mirrors at the same height and then obtains final orientations of facet mirrors with gradually increased height, so that all the facet images remain superimposition at the common receiver without sunlight blocking and shadowing among adjacent mirrors. 20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS
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of all facet mirrors. During the initial state of computational algorithm, all the facet mirrors are defined in an array in which the central point of facets are spaced evenly with M z; 0, as shown in Fig. 2. Then each of the facet mirrors is virtually lifted up one by one from the central region toward the peripheral region along the x, y, and z directions in order to eliminate the sunlight blocking and shadowing effects while keeping fixed gaps among the facet mirrors. The final positions of the facet mirrors that form a dish will be different from that of the initially defined position in the horizontal plane, and the process of computing the final facet mirrors positions will cause the variation in the tilted angles of each facet mirror as to keep the solar image aiming at the receiver. Therefore, in the process of designing the geometry of the NIDC, an iterative method is used to calculate the final position as well as the two tilted angles, σ i;j and γ i;j , of each facet mirror. After obtaining the tilted angles of the facet mirror, coordinate transformation is applied to determine the orientation of the facet mirrors. The process of designing the position of facet mirrors is twofold: the first involves virtually lifting of the facet mirrors arranged in row sequence ( j 1; 2; 3; …:n.) along the same column in the Y and Z axes; the second involves lifting of facet mirrors arranged in column sequence (i 1; 2; 3; …:m.) along the same row in the Y and Z axes. Figure 7(a) shows how to determine the new position of facet mirrors in the (j 1)th row along the Y and Z axes, and it is considered free from sunlight blocking and shadowing effects when the following formulas are fulfilled: H 3z i;j1 H 1z i;j ;
(15a)
H 3y i;j1 H 1y i;j G:
(15b)
Figure 7(b) shows how to determine the new position of facet mirrors in the (i 1)th along the Y and Z axes, and it is considered free from sunlight blocking and shadowing effects when the following formulas are fulfilled: H 4z i1;j H 2z i;j ;
(16a)
H 4x i1;j H 2x i;j G:
(16b)
In the simulation, important parameters must be inserted, such as focal distance of the NIDC (F), width of the facet mirror (w), and gap between the adjacent facet mirrors (G). As the squared facet mirror is considered in our case study, the length is equal to the width of the facet mirror. The gap between the adjacent facet mirrors is set to be 5% of the width or length of a single facet mirror. The coordinate of the facet mirror, which is located closest to the origin, M1;1 , is ½w ½G; ½w ½G; 0. Considering the concentrator consisted of 2m columns and 2n rows, the computational algorithm will start to compute the final position of the facet mirrors located at the first column, i 1, with the row sequence starting from j 1 to j n. The same procedure continues for the following column from i 2 to i m. The detail of the computational algorithm to simulate the configuration of the facet mirrors in the NIDC is summarized in the flow chart as shown in Fig. 6. Figure 8 shows a 3D drawing of the final NIDC configuration, which consisted of 196 facet mirrors and is generated based on the computational algorithm. The computational algorithm is also performed for a different number of arrays and different facet dimensions with the detailed specifications as listed in Table 2. 4. Optical Performance Analysis of NIDC Using Analytical Method
For optical characterization of NIDC, two major optical parameters are needed to be taken into account, which are the average solar concentration ratio at the uniform illumination and the area of uniform illumination at the receiver. These parameters affect the power generation of the CPV module, which is attached to the receiver. On top of that, the total number of CPV cells required in the CPV module is also an important measure in the cost calculation for the whole CPV system.
Fig. 7. Conceptual drawing to illustrate how the facet mirrors are virtually lifted up for eliminating sunlight blocking and shadowing in the computational algorithm. (a) Along Y and Z axes. (b) Along X and Z axes. 480
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Fig. 8. Example of NIDC configuration completed by using the computational algorithm, which consists of 196 facet mirrors.
Numerical simulation using a ray-tracing technique has been widely deployed in the analysis of the optical performance of a nonimaging solar concentrator [1,21]. The main hindrance of the numerical simulation is a long computational time if reasonably high accuracy is required. An analytical model proposed by Wong et al. to analyze the optical characteristics of a NIPC has reduced the computational time dramatically with good accuracy [24]. Since the nonimaging geometries between planar and dish concentrators are quite different, the analytical formulas of NIDC are described in detail. Furthermore, the simulated results of optical performance of NIDC using a numerical method are compared with the calculated result using analytical methods. A pillbox sun shape is considered in the optical performance analysis for solar disk effect by ignoring the effect of the sun’s atmosphere, which gives the disc a limb darkening or decreasing the brightness toward the edge of the disc. The approach of simulating the sun shape in this context has been fully digital and simplified by considering the sunlight that strikes on the mirror surface as digitized cone rays. The reflectivity of the facet mirror is treated as unity. A cross-sectional view of the NIDC to show how the individual square facet mirror with a dimension of w reflects the incident sunlight to the receiver is illustrated in Fig. 9. It is also shown that the total solar energy falling on both the facet mirror is fully reflected to the receiver provided that no energy is lost during the process. Conversely, the solar concentration ratio received by the receiver is decreased because the projected solar image of the facet mirror is spread across a longer distance on the receiver, La , compared to the effective distance, Lb , of the facet mirror exposed to solar flux. Due to the cosine effect, the intensity of solar flux, which falls on the receiver, is lower than that of the incident ray from the sun. Assuming that the incident sunlight is designated with solar concentration ratio of one sun, the solar concentration ratio at the receiver surface will be reduced by the cosine of angle between the incident ray
Fig. 9. Cross-section view of the NIDC, which shows how the individual facet mirror with dimension w × w reflects and superimposes the sunray to the receiver.
and reflected ray, 2θi;j . Therefore the average solar concentration ratio at the uniform illumination area, C, after superimposing all the solar images of facet mirrors on the receiver can be expressed as C
2m X 2n X
cos 2θi;j :
(17)
i1 j1
Based on trigonometry theorem with the solar disk half angle (ε) of 4.65 mrad, the length of the solar image of the i, j facet, which is cast on the receiver surface, can be calculated by L l1 l2 l 3 W sin θi;j tan2θi;j ε W cos θi;j F − M z − 2 W (18) − F − M z sin θi;j tan2θi;j − ε; 2 where
W l1 F − M z − sin θi;j tan2θi;j ε 2 − tan2θi;j − ε
(19)
W sin θi;j tan2θi;j − ε l2 W cos θi;j F − M z − 2 W sin θi;j tan2θi;j ε; (20) − F − Mz 2 W sin θi;j tan2θi;j ε l3 F − M z 2
− tan2θi;j − ε:
(21)
Due to the cosine effect, the solar image for the facet mirror at a larger incident angle will be larger than that of the facet mirror at a smaller incident angle. The facet mirror with the smallest incident angle will produce the smallest solar image on the receiver. Due to the solar disk effect, the uniform illumination area is equal to l22 . On the other hand, the facet mirror with the largest incident angle will produce the largest solar image on the receiver, which will determine the final image size of the whole solar concentrator. The total image area, Aove , and the uniform illumination area, Auni , on the receiver can be calculated by using the following formulas: Aove L2 8 92 Wcos θmax > > > > > > > > < F − M − W sin θ = ε tan2θ z max max 2 ; > > > > > > W > : − F − M z 2 sin θmax tan2θmax − ε > ; (22) 20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS
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Auni l22 8 92 > > W cos θ > > min > > > > > > < F − M − W sin θ = − ε tan2θ z min min 2 ; > > > > > > W > > > : − F − M z 2 sin θmin tan2θmin ε > ; (23) where the maximum incident angle of the facet mirror is θmax , which is also equal to θm;n , and the minimum incident angle of the facet mirror is θmin , which is also equal to θ1;1 . Since dense-array CPV cells operate the best at uniform illumination for achieving the optimal power output, we consider the CPV cells to be installed only in the uniform illumination region of the receiver in the study. The total number of CPV cells at the uniform illumination region of the NIDC can be computed as l g 2 N CPV int 2 ; bg
(24)
where b is the width of single CPV cell, g is the gap between the two adjacent CPV cells, and l2 can be calculated using Eq. (20) by substituting θi;j θmin . The output power of the CPV system can be expressed as follows: Pout ηCPV × I × C × N CPV × ACPV ;
(25)
where ηcpv is the conversion efficiency of the CPV cell, I is the incident solar irradiance (Watt∕m2 ), and Acpv is the area of single CPV cell (m2 ). The total cost of the system per power output of the CPV system (US $/Watt) can be determined by using the following formula: Sc∕w
N CPV × CCPV Cfacets Cmech ; Pout
(26)
where Ccpv is the cost per single CPV cell (assuming the cost is US$6.5/cell), Cfacets is the total cost of all the facet mirrors, and Cmech is the total cost of the solar concentrator structure, including mechanical structure, pedestal, and dual-axis tracking mechanism, etc. 5. Results and Discussion
To analyze the optical performance of NIDC, a numerical method is usually used to simulate the solar flux distribution with reasonable good accuracy. Figures 10(a) and 10(b) show the numerical simulation result of the solar flux distribution in 3D and 2D plots, accordingly, for the case of NIDC with an array of 22 × 22 facets, and each facet of 49.8 cm × 49.8 cm at a focal distance of 10 m. It is well know that the numerical simulation based on ray-tracing 482
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Fig. 10. Numerical simulation results of solar flux distribution for the case of 22 × 22 array of 49.8 cm facet dimension with 10 m focal distance. (a) 3D plot. (b) 2D plot.
technique that involves handling a huge number of databases is time consuming. Alternatively, an analytical model is proposed to expedite the work. To validate the analytical model, comparisons between the numerical and analytical methods have been carried out. Figures 11(a) and 11(b) reveal a comparison study between two methods in terms of average solar concentration ratio at uniform illumination and uniform illumination area, respectively. For the average solar concentration ratio at uniform illumination, the relative standard deviation of the numerical simulation is less than 0.09%, and the variation result between the two methods is less than 0.005%. The very small variation in average solar concentration shows excellent agreement between both methods. On the other hand, the accuracy in calculating the uniform illumination area depends on the resolution defined for the receiver in terms of pixel per cm for the numerical simulation. The error of uniform illumination area caused by resolution of the receiver for a numerical method can be calculated by using the following formula:
Fig. 11. In the case of 22 × 22 array of 49.8 cm facet dimension with focal distance from 6 to 10 m. (a) Graph shows the average solar concentration ratio at uniform illumination area for both numerical and analytical methods versus focal distances. (b) Graph shows the area of uniform illumination for numerical and analytical methods versus focal distances.
Aerror %
l2nm − lnm − 2∕Res2 × 100%; l2nm
(27)
where lnm is the simulated length of uniform illumination region using a numerical method, and Res is resolution of the receiver (pixel per cm) used in the numerical simulation. The errors of the uniform illumination area due to the resolution of the receiver for focal distances from 6, 7, 8, 9, to 10 m are 4.4%, 4.5%, 4.6%, 4.7%, and 4.8%, respectively. The difference between analytical and numerical methods for the area of uniform illumination at focal distances from 6, 7, 8, 9, to 10 m are 3.5%, 3.1%, 2.7%, 2.4%, and 2.1%, correspondingly. This difference is mainly caused by the receiver’s resolution (pixel per cm) used in the numerical method simulation. If the resolution of the receiver applied to the numerical simulation is increased, the simulated result will be more accurate and closer to the analytical value. Figure 11(b) shows that the variations in results for two computational methods fall within the range of errors for the numerical simulation.
Since the two major optical parameters needed to be considered in the optical design of NIDC can also be calculated using the analytical model, the following study of system performance is carried out based on the analytical method. Despite the analytical model being unable to plot the solar flux distribution on the receiver, the average solar concentration ratio at the uniform illumination and also the uniform illumination area are sufficient for optical engineers to analyze the performance of NIDC as well as a densearray CPV system. To understand the overall performance of the whole system, the synthesized parameters including the output power, total number of CPV cells, and the cost per output power can be derived from the two major optical parameters. As a result, the performance of the NIDC can be optimized by changing the designed parameters such as focal distance of the concentrator (F) and the dimension of the facet mirror (w) via the aforementioned methodology. The detailed specifications of the NIDC used in the study are presented in Table 1. From Table 2, 15 sets of different arrays with their corresponding total number of facet mirrors and dimensions have been considered in our performance analysis in such a way that the total reflective area of the solar concentrator remains almost the same for all the cases, which is about 120 m2 . Figure 12 illustrates the average solar concentration ratio at the uniform illumination versus facet dimension at different focal distances. It shows that different facet dimensions can accommodate for Table 1.
Specification and Design Parameters of the NIDC
Total reflective area Area of single CPV cell, Acpv Gap between CPV cell, g Incident solar radiation, I Cost of single CPV cell, CCPV Cost of concentrator structure, Cmech Cost of 120 m2 facet mirror, Cfacets
Table 2.
120 m2 1 × 1 cm2 1 mm 1000 Watt∕m2 US$6.5 US$32,000 US$3,000
Specifications Used in the Simulation for Different Arrays of Facet Mirrors
Array of Facet Mirrors
Dimension of Facet Mirror w × w
12 × 12 14 × 14 16 × 16 18 × 18 20 × 20 22 × 22 24 × 24 26 × 26 28 × 28 30 × 30 32 × 32 34 × 34 36 × 36 38 × 38 40 × 40
91.3 × 78.2 × 68.5 × 60.9 × 54.8 × 49.8 × 45.6 × 42.1 × 39.1 × 36.5 × 34.2 × 32.2 × 30.4 × 28.8 × 27.4 ×
cm × 91.3 cm cm × 78.2 cm cm × 68.5 cm cm × 60.9 cm cm × 54.8 cm cm × 49.8 cm cm × 45.6 cm cm × 42.1 cm cm × 39.1 cm cm × 36.5 cm cm × 34.2 cm cm × 32.2 cm cm × 30.4 cm cm × 28.8 cm cm × 27.4 cm
Total Number of Facet Mirrors 144 196 256 324 400 484 576 676 784 900 1024 1156 1296 1444 1600
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Fig. 12. Bar chart to show the comparisons of average solar concentration ratio at uniform illumination area versus facet dimensions at different focal distances.
different average solar concentration ratios from 109.46 suns to 1439.2 suns for the focal distances ranging from 6 to 10 m. Moreover, it also shows that the increment of focal distance will increase the average solar concentration ratio while the increment of the facet dimension will reduce the average solar concentration ratio. Bar charts for the uniform illumination area and the total number of CPV solar cells attached on the receiver versus facet dimension at different focal distances are plotted in Figs. 13 and 14, respectively. The uniform illumination area and the total number of CPV cells are directly proportional to the facet dimension. In contrast, the uniform illumination area is inversely proportional to the focal distance of the concentrator, and similarly it also happens to the charts for the number of CPV cells versus focal distance of the concentrator in most of the cases. The exceptional cases are facet dimension 91.3 cm with focal distances 8 and 9 m and facet dimension 78.2 cm with focal distances 7 and 8 m etc., in which the number of CPV solar cells remains the same, even though their focal distances are different. These phenomena happen because the increment of the
Fig. 13. Bar chart to show the comparisons of uniform illumination areas versus facet dimensions at different focal distances. 484
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Fig. 14. Bar chart to show the comparisons of a number of CPV cells used on the receiver versus facet dimensions at different focal distances.
uniform illumination area is not large enough to insert a new row or column of CPV solar cells in the receiver. We assume the CPV cell employed in the study is an Emcore CTJ photovoltaic cell, 10 mm × 10 mm, and the efficiency of the cell is based on specification provided by Emcore, as shown in Fig. 15 [25]. Figure 16 shows that different facet dimensions can cater for different system output power from 12.99 to 26.83 kW at different focal distances ranging from 6 to 10 m. It also shows that the system output power is higher at a lower solar concentration ratio compared to that of higher solar concentration ratio. Unfortunately, the usage of CPV cells in the NIDC system will be more for lower solar concentration ratio compared to that of higher solar concentration ratio. It will increase the total cost of the whole system. To address practical concerns, the following study is done to justify the facet dimensions and focal distances of the concentrator to provide the optimal performance. The total cost per output power of the system is assessed for different facet dimensions and focal distances, provided that the total reflective area is fixed.
Fig. 15. Graph to show the cell conversion efficiencies versus solar concentration ratio for Emcore CTJ photovoltaic cell, 10 mm × 10 mm [25].
To reduce the cost of the system, innovation must be implemented in the design of the concentrator. As the cost of CPV cell and concentrator varies with territories and quantity, engineers can apply the above-mentioned methodology to evaluate the cost per output power of the system in their respective countries. 6. Conclusions
Fig. 16. Bar chart to show the comparisons of system output power versus facet dimensions at different focal distances.
This assessment is significant for designing an optimized dense-array CPV system by considering the facet dimension and focal distance of NIDC as well as the required number of CPV cells on the receiver. The estimated cost of a single Emcore CTJ photovoltaic cell and the corresponding direct bond copper substrate is US$6.5. The costs of the CPV cells vary, depending on the purchase quantity; these costs have high potential to be reduced in the future. The estimated cost of the concentrator is US $35,000 for a collective area of 120 m2 in Malaysia, and the value may vary from country to country. Figure 17 has been plotted to show the total cost of the system per output power versus facet dimension at different focal distances. It shows that the solar concentrator with a facet dimension of 49.8 cm and focal distance of 8 m operating at a solar concentration ratio of 411.8 suns provides the lowest cost per output power, which is US$1.93 per watt compared to other designs. Based on these parameters, we further break down the cost per output power of the system to three different parts, which are the CPV cell, facet mirror, and the concentrator structure, and the cost per output power of the breakdown are $0.409/W, $0.065/W, and $1.459/W respectively. This shows that the concentrator structure accounts for the majority of the cost in the system.
Fig. 17. Bar chart to show the comparisons of total system cost-per-output power versus facet dimensions at different focal distances.
The optical principle of NIDC and a computational algorithm to design the facet mirror configuration have been developed. An analytical model for NIDC has been formulated. The computational results between the numerical and analytical methods to determine the maximum solar concentration ratio and the area of uniform illumination are compared. The variation between the two methods for the average solar concentration ratio at uniform illumination is less than 0.005% while the variation between the two methods for the area of uniform illumination at focal distances from 6, 7, 8, 9, to 10 m are 3.5%, 3.1%, 2.7%, 2.4%, and 2.1%, correspondingly. It shows that the analytical model is proficient to provide reasonable precise results without consuming lengthy computational time. In the study, the optical performance of NIDC with a reflective area of 120 m2 for various facet dimensions and the focal distances ranging from 6 to 10 m is considered. Depending on the facet dimensions and the focal distance, the average solar concentration ratio at the uniform illumination region can vary, ranging from 109 suns to 1439 suns, and the system output power ranges from 12.99 to 26.83 kW. NIDC with a facet dimension of 49.8 cm and a focal distance of 8 m operating at solar concentration ratio of 411.8 suns offers the lowest cost per output power, which is US$1.93 per watt compared to other configurations. The results show that NIDC with single-stage focusing, which combines the advantages from NIPC and a parabolic dish concentrator, is capable of producing a uniform illumination area with a high solar concentration ratio at the receiver. The optical characteristics and simplicity in design have recommended that NIDC is suitable for the application of a dense-array CPV system. In fact, all the concentrator contains exactly the same basic driving mechanisms such as motor, gearbox, optical encoder, and control system. The necessitated manufacturing, assembly, and alignment procedures are needed for all concentrators. Comparing the construction cost of different concentrators is not the main aim of this paper because it depends on many factors such as the labor cost (which is dependent on country), the quantity of concentrator to be manufactured, type or grade of materials used, etc. Despite alignment of mirrors being required for multifaceted mirrors in NIDC, its fabrication is cheaper and simpler as compared with a precise large reflective area of a parabolic dish. For a large parabolic dish concentrator with a total reflective area of more than 100 m2 , a single parabolic reflector 20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS
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is not possible to be fabricated, and therefore multifaceted parabolic mirrors are still required. If it is compared with NIDC, a multifaceted parabolic dish is more expensive because curved facet mirrors are used, and the alignment process is more complicated with the involvement of curved mirrors. On the other hand, a continuous parabolic concentrator produces a solar flux distribution that closely resembles the Gaussian distribution. It may cause the dense-array CPV module encounters a severe drop in efficiency due to the current mismatched problem. From the analysis in the paper, the solar concentrator structure of the CPV system contributes the major cost in the CPV system. Therefore the system cost can be very much dependent on the innovative design of the solar concentrator structure and will remain a great challenge for solar-energy engineering in the future. The authors would like to express their gratitude to the Ministry of Energy, Green Technology and Water (AAIBE Trust Fund) and the Ministry of Science, Technology and Innovation (e-Science Fund with project number 03-02-11-SF0143) for their financial support. The authors would also like to thank the project advisor Ir. Dr. Philip Tan Chee Lin for his many insightful suggestions.
9. 10.
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13. 14.
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16.
References 1. K. K. Chong, C. W. Wong, F. L. Siaw, and T. K. Yew, “Optical characterization of nonimaging planar concentrator for the application in concentrator photovoltaic system,” J. Sol. Energy Eng. 132, 011011 (2010). 2. A. W. Bett, B. Burger, F. Dimroth, G. Siefer, and H. Lerchenmuller, “High-concentration PV using III-V solar cells,” in Conference Record of the IEEE Fourth World Conference on Photovoltaic Energy Conversion (2006), pp. 615–620. 3. G. S. Kinsey, R. A. Sherif, H. L. Cotal, P. Pien, R. R. King, R. J. Brandt, W. G. Wise, E. L. Labios, K. F. Wan, M. Haddad, J. M. Lacey, C. M. Fetzer, P. Verlinden, K. Lasich, and N. H. Karam, “Multijunction solar cells for dense-array concentrators,” in Conference Record of the IEEE Fourth World Conference on Photovoltaic Energy Conversion (2006), pp. 625–627. 4. C. T. Kuo, H. Y. Shin, H. F. Hong, C. H. Wu, C. D. Lee, I. T. Lung, and Y. T. Hsu, “Development of the high concentration III-V photovoltaic system at INER, Taiwan,” Renew. Energy 34, 1931–1933 (2009). 5. E. Kussul, T. Baidyk, F. Lara-Rosano, J. M. Saniger, N. Bruce, and C. Estrada, “Micro-facet solar concentrator,” Int. J. Sustain. Energy 27, 61–71 (2008). 6. G. Johnston, K. Lovegrove, and A. Luzzi, “Optical performance of spherical reflecting elements for use with paraboloidal dish concentrator,” Sol. Energy 74, 133–140 (2003). 7. S. Ulmer, P. Heller, and W. Reinalter, “Slope measurements of parabolic dish concentrators using color-coded targets,” J. Sol. Energy Eng. 130, 011015 (2008). 8. V. Andreev, V. Grilikhes, V. Rumyantsev, N. Timoshina, and M. Shvarts, “Effect of non-uniform light intensity distribution on temperature coefficients of concentrators solar cells,” in Third
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World Conference on Photovoltaic Energy Conversion, (2003), pp. 881–884. J. S. Coventry, “Performance of a concentrating photovoltaic/thermal solar collector,” Sol. Energy 78, 211–222 (2005). E. Franklin and J. Coventry, “Effects of highly non-uniform illumination distribution on electrical performance of solar cells,” in ANZSES Solar Conference, New Castle, Australia (2002). K. Nishioka, T. Takamoto, T. Agui, M. Kaneiwa, Y. Uraoka, and T. Fuyuki, “Annual output estimation of concentrator photovoltaic systems using high-efficiency InGaP/InGaAs/Ge triple-junction solar cells based on experimental solar cell’s characteristics and field-test meteorological data,” Sol. Energy Mater. Sol. Cells 90, 57–67 (2006). D. Faiman, S. Biryukov, and K. K. Pearlmutter, “PETAL: a research pathway to fossil-competitive solar electricity,” in Conference Record of the 29th IEEE Photovoltaic Specialists Conference (2002), pp. 1384–1387. D. R. Mills and G. L. Morrison, “Compact linear Fresnel reflector solar thermal power plants,” Sol. Energy 68, 263–283 (2000). K. Araki, M. Kondo, H. Uozumi, N. J. Ekins-Daukes, T. Egami, M. Hiramatsu, Y. Miyazaki, and M. Yamaguchi, “Packaging III-V tandem solar cells for practical terrestrial applications achievable to 27% of module efficiency by conventional machine assemble technology,” Sol. Energy Mater. Sol. Cells 90, 3320–3326 (2006). Y. Kemmoku, T. Sakakibara, M. Hiramatsu, Y. Miyazaki, and T. Egami, “Field test of a concentrator photovoltaic system with flat Fresnel lens,” in Third World Conference on Photovoltaic Energy Conversion (2003), pp. 2379–2382. K. Ryu, J. G. Rhee, K. M. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Energy 80, 1580–1587 (2006). H. Ries, J. M. Gordon, and M. Lasken, “High-flux photovoltaic solar concentrators with kaleidoscope-based optical designs,” Sol. Energy 60, 11–16 (1997). G. Segev and A. Kribus, “Performance of CPV modules based on vertical multi-juctNion cells under non-uniform illuminations,” Sol. Energy 88, 120–128 (2013). Z. Wang, H. Zhang, D. Wen, W. Zhao, and Z. Zhou, “Characterization of the InDaP/InGaAs/Ge triple-junction solar cell with a two-stage dish-style concentration system,” Energy Convers. Manage. 76, 177–184 (2013). K. Kreske, “Optical design of a solar flux homogenizer for concentrator photovoltaics,” Appl. Opt. 41, 2053–2058 (2002). K. K. Chong, F. L. Siaw, C. W. Wong, and G. S. Wong, “Design and construction of non-imaging planar concentrator for concentrator photovoltaic system,” Renew. Energy 34, 1364–1370 (2009). K.-K. Chong, C.-W. Wong, T.-K. Yew, and M.-H. Tan, “Solar concentrator assembly,” U.S. Patent Application 13/901,519 (filed on May 23, 2013, pending). K.-K. Chong, C.-W. Wong, T.-K. Yew, and M.-H. Tan, “Solar concentrator assembly,” Malaysian Patent No. PI 2012002439 (filed on May 31, 2012, pending). C.-W. Wong, K.-K. Chong, and T.-K. Yew, “Analytical model of non-imaging planar concentrator for the application in densearray concentrator photovoltaic system,” in 1st International Symposium on Innovative Technologies in Engineering and Science (2013), pp. 679–686. Emcore, 2012. “CTJ photovoltaic cell – 10 mm × 10 mm, triple junction solar cell for terrestrial applications,” http://www .emcore.com.
Faculty of Engineering and Science, Universiti Tunku Abdul Rahman, Off Jalan Genting Kelang, Setapak, 53300 Kuala Lumpur, Malaysia 2
e-mail: [email protected]
3
e-mail: [email protected]
*Corresponding author: [email protected] Received 10 October 2013; revised 9 December 2013; accepted 11 December 2013; posted 16 December 2013 (Doc. ID 199239); published 17 January 2014
Optimization of the design of a nonimaging dish concentrator (NIDC) for a dense-array concentrator photovoltaic system is presented. A new algorithm has been developed to determine configuration of facet mirrors in a NIDC. Analytical formulas were derived to analyze the optical performance of a NIDC and then compared with a simulated result obtained from a numerical method. Comprehensive analysis of optical performance via analytical method has been carried out based on facet dimension and focal distance of the concentrator with a total reflective area of 120 m2 . The result shows that a facet dimension of 49.8 cm, focal distance of 8 m, and solar concentration ratio of 411.8 suns is the most optimized design for the lowest cost-per-output power, which is US$1.93 per watt. © 2014 Optical Society of America OCIS codes: (230.4040) Mirrors; (220.1770) Concentrators; (220.4298) Nonimaging optics. http://dx.doi.org/10.1364/AO.53.000475
1. Introduction
The depletion of fossil fuel and global warming issues has pressed mankind to explore alternative sources of energy that are safe, clean, and renewable. The invention of multijunction solar cells capable of achieving 40% conversion efficiency has increased the potential of a concentrator photovoltaic (CPV) system to replace the current source of energy [1]. Due to the high cost of solar cell materials, a CPV system can allow a large area of solar cells to be substituted by reflectors or lenses in the concentrator. For the application of this system, a solar concentrator plays a vital role by applying geometrical optics in 1559-128X/14/030475-12$15.00/0 © 2014 Optical Society of America
the design of the reflectors or lens to focus high solar flux on the CPV module at the receiver [2–4]. Over the last few decades, a parabolic dish concentrator has been widely used for solar energy applications, especially for applications involving high solar concentration and high-temperature collection. For a large dish concentrator system, using a single parabolic mirror as a reflector is costly due to the special technology needed to fabricate special mirrors with a thickness from 0.7 to 1.0 mm and to obtain the parabolic shape [5]. For a large collective area of parabolic dish concentrator, multifaceted spherical or concave mirrors are arranged on the approximate parabolic shape structure as a reflector instead of using a single piece of parabolic reflector [6,7]. Despite alignment of mirrors being required in a faceted parabolic concentrator, its fabrication is cheaper and simpler 20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS
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as compared with a single precise parabolic reflector. The implementation of a parabolic concentrator in a dense-array CPV system encounters the difficulty to produce good uniformity of solar illumination on the receiver, but instead the tendency is more toward Gaussian distribution [6,7]. On the other hand, a dense-array CPV module that receives nonuniform illumination will face a severe drop in efficiency due to the current mismatched problem as opposed to the module under uniform illumination [8–12]. Beside the aforementioned parabolic concentrators, researchers have proposed various kinds of solar concentrators for the sake of producing uniform flux distribution on the target. Mills and Morrison advocate the use of an advanced form of linear Fresnel reflector that produces better uniformity of solar irradiation at a solar concentration ratio lower than 100 suns [13]. For a moderate solar concentration ratio, modular Fresnel lenses are applied in a CPV system to generate better uniformity of solar irradiance. Conversely, the transmission efficiency of the modular Fresnel lenses are less than 80% due to the absorption by the lens material and the reflection at the surface of the lens [14–16]. In addition, two-stage solar concentrators that comprise a parabolic dish and secondary optics or homogenizer have been introduced for the application of a CPV system [17–19]. The introduction of an additional optical device will inflict additional losses, which can reach 10% or more for typical designs as well as additional cost and complexity [20]. Alternatively, Chong et al. proposed a nonimaging planar concentrator (NIPC) consisting of numerous flat facet mirrors, which are arranged in a square array and spaced evenly at the same height, to superimpose the sunlight at the receiver, which can produce a reasonably high solar concentration ratio and uniform solar irradiance [1,21]. However, as the ratio of focal distance to width of the NIPC decreases, the sunlight blocking effect among adjacent facet mirrors will increase. For those facet mirrors especially located further from the center of the NIPC, higher tilting angles are required to reflect sunlight toward the common target, but it will cause more sunlight directed from an adjacent facet mirror to be blocked. Inspired by adopting the merit ideas from both NIPC (to produce reasonably uniform illumination with the use of flat facet mirrors) and parabolic dish concentrator (to avoid the sunlight blocking and shadowing effects with the geometry gradually increase in height), the nonimaging dish concentrator (NIDC) with single-stage focusing has been proposed by extracting the best designs from both solar concentrators [22,23]. In this paper, a newly developed computational algorithm is introduced to determine the facet mirror configuration for the NIDC, which eliminate blocking and shadowing effects from the adjacent facet mirrors. Besides, analytical formulas have been derived to simplify and expedite the process of characterization study of NIDC. In addition, the calculated results using analytical formulas 476
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are also verified with the simulated results obtained from a numerical method using a ray-tracing technique. Comprehensive analysis of optical performance via analytical method has been carried out for various facet dimensions and focal distances with a total reflective area of 120 m2 in order to optimize the optical design of NIDC in terms of cost-peroutput power of the dense-array CPV system. 2. Optical Principle of Nonimaging Dish Concentrator
NIDC employs multiple identical square or rectangular facet mirrors acting as optical apertures to gather the solar irradiance from the sun and to superimpose all the images at the common receiver. In our study, NIDC consists of a 2m × 2n array of identical facet mirrors arranged at different heights to prevent sunlight blocking and shadowing effects among adjacent facet mirrors. For a geometry representation, a Cartesian coordinate system is used to identify the position of all the facet mirrors within the concentrator where the origin is defined as a central point of the NIDC. The central line of the NIDC starting from the origin lies along the row direction is defined as the X axis; the central line of the NIDC from the origin lies along the column direction is defined as the Y axis; the Z axis is defined from the origin pointing to the target direction, which is also perpendicular to both the X axis and Y axis. Referring to Fig. 1, the origin of the global coordinate system, O0; 0; 0, is located at the center of the concentrator, and the locality of each facet mirror in the NIDC can be indexed as i; j, where the i and j represent the position of the facet mirror at the ith column and jth row of the NIDC, respectively. From Fig. 2, the NIDC can be divided into four quadrants, which are the top right, top left, bottom right, and bottom left. The global coordinate of the central point of any facet mirror is written as 2
3 Mx Mi;j 4 M y 5 : M z i;j
(1)
Fig. 1. Cartesian coordinate system representing the coordinate, incident angle θi;j , and the two tilted angles, σ i;j and γ i;j , of i, j mirror where F is the focal distance of the NIDC and O is the origin, which is the center of the concentrator frame.
" σ i;j sin−1
# Mx q ; 2 cos θi;j M 2x M 2y F − M z 2
(6)
# My q : (7) F − M z M 2x M 2y F − M z 2
" γ i;j tan
−1
3. Methodology of Designing NIDC Geometry
Fig. 2. Initial facet mirror’s configuration of the NIDC. The concentrator can be divided into four quadrants, which are the top right, top left, bottom right, and bottom left.
The coordinate of the receiver is defined as 2 3 2 3 Tx 0 T 4 T y 5 4 0 5; Tz F
(2)
According to the previous design of a NIPC, all the facet mirrors are arranged in an array where all the adjacent facets are spaced evenly and located in the same height. From Fig. 3, as the distance of the facet mirror from the center of the concentrator increases, the sunlight blocking effect among the adjacent mirrors becomes worse due to large tilted angles. Furthermore, the horizontal gap (Gi ) between the adjacent facet mirrors is wider owing to large tilted angles as shown in Fig. 4. To optimize optical efficiency of the new solar concentrator, the sunlight blocking and shadowing effects among the adjacent facet mirrors must be eliminated. To do so, the facet mirrors must be gradually lifted upward along the z direction for the facet mirrors located from the
where F is the shortest distance between the center of the receiver and the center of the concentrator frame (the origin of coordinate system), which is also known as the focal distance of NIDC. As the NIDC tracks the sun during its operational period, the unit vector of the incident sunray can be written as 2 3 0 I⃗ 4 0 5: (3) 1 The unit vector of the normal of each facet mirror can be derived as 3 2 Nx ⃗ ⃗ ⃗ i;j 4 N y 5 I Ri;j N 2 cos θi;j N z i;j 2 3 −M x p Nx 2 cos θi;j M 2x M 2y F−M z 2 6 7 −M y 6N 7 p 6 7 y 6 (4) 2 cos θi;j M 2x M 2y F−M z 2 7 ; p 4 5 F−M z M 2x M 2y F−M z 2 p Nz 2 2 2 2 cos θi;j
M x M y F−M z
Fig. 3. Cross-sectional view of the planar concentrator. Blocking effect starts to occur at the fifth row of the facet mirror counting from the center of the concentrator.
i;j
where Ri;j is the unit vector of the reflected solar ray for the i, j facet mirror, and θi;j is the incident angle of the sunray, which can be expressed as 2 M x M 2y 1∕2 1 −1 tan : (5) θi;j F − Mz 2 i;j Finally, there are two tilted angles of the i, j facet mirror to superimpose its solar image on the common receiver: σ i;j is a tilted angle about Y axis, and γ i;j is a tilted angle about the X axis. The two tilted angles can be derived as
Fig. 4. Conceptual drawing that defines the initial positions of facet mirrors with the same height and then obtains final positions of facet mirrors with gradually increased height to eliminate sunlight blocking and shadowing by adjacent mirrors. Gi is initial gap between the facet mirror, and G is the gap between facet mirrors after reposition. (a) Along Y and Z axes. (b) Along X and Z axes. 20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS
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central to peripheral position of the new concentrator as described in Fig. 4. The lifting of the facet mirror is not done physically, but it is virtually performed during the computational process. Moreover, gaps between the adjacent facet mirrors should also be minimized from Gi to G in order to optimize the total effective area of NIDC. The positions of the facet mirrors are computed in a way as to maintain the gap spacing among the adjacent facet mirrors at a fixed distance via the coordinate transformation method in both X and Y axes. In practice, small gaps among the facet mirrors are still required and unavoidable to absorb the manufacturing defect by providing tolerance for the installation of facets. In the computational algorithm of designing the NIDC geometry, we start with defining all the facet mirrors arranged in a 2D array at the same height. Then we obtain final positions of facet mirrors one by one from the central region toward the peripheral region with gradually increased height as to eliminate sunlight blocking and shadowing among adjacent mirrors as well as to minimize the gap between adjacent mirrors. The final positions of the facets that form a dish will be different from that of the initially defined position, and the process of rearrangement of mirror positions will cause the variation in the tilting angles of the facet mirror as to maintain the solar image aiming at the receiver. Therefore, in the process of computing and designing the geometry of NIDC, an iterative method must be used to calculate the final position of each facet mirror in which the details will be discussed in the following section. From Fig. 2, all facet mirrors in the NIDC can be subdivided into four major quadrants and the origin, O, is located at the center of the NIDC. The configuration of facet mirrors in the four quadrants is symmetric to each other, and therefore we only need to consider any of the four quadrants for the simulation to save the computational time. From Fig. 2, the topright quadrant of the concentrator is chosen as a reference for performing simulation, and the facet mirror closest to the origin is defined as M1;1 ; the facet mirror located furthest from the origin is defined as Mm;n , provided that m is the number of column and n is the number of row in the NIDC. A.
Coordinate Transformation of Facet Mirror
As the facet mirrors are used to reflect the sunray to the receiver or target, each of them must be tilted at particular tilting angles so that the solar image can be accurately superimposed on the common receiver. The new orientation of the facet mirrors in 3D spaces can be modeled by applying coordinate transformation. For the convenience of representing 3D coordinate transformation in the mathematical formulas, one more dimensionality of space is inserted to express all the transformation functions in a linear relationship. The initial coordinate of the individual facet mirror is first defined in a local coordinate system as shown in Fig. 5 wherein the origin of the coordinate system, 478
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Fig. 5. Initial coordinate of the facet mirror where the origin is at the middle of the facet mirror surface, and Pi is the center point of the four edges of the facet mirror.
M0; 0; 0, is attached to the center of the facet mirror surface instead of the center of the concentrator. Then it will perform the rotational transformations for the tilted angles σ i;j and γ i;j before it is transformed to the global coordinate system whereby the origin is defined at the center of concentrator. In this modeling, the facet mirror is represented by the central points at the four boundaries of the facet mirror (see Fig. 4). The four points representing the i, j facet mirror can be treated as a vector in the coordinate space, and it is written as 3 2 Pkx 6 Pky 7 7 (8) Pk i;j 6 4 Pkz 5 ; 1 i;j where k 1 (central point at upper boundary of the facet mirror), 2 (central point at right boundary of the facet mirror), 3 (central point at bottom boundary of the facet mirror), 4 (central point at left boundary of the facet mirror). After the transformation, the final coordinate of the four points representing the i, j facet mirror will be treated as a vector in matrix form written as: 2 3 Hkx 6 Hky 7 7 (9) Hk i;j 6 4 Hkz 5 : 1 i;j In the coordinate transformation, there are two tilted angles, σ i;j and γ i;j , rotating about two orthogonal axes with respect to the facet mirror itself. Last, a translational coordinate transformation is required to transform the position of facet mirror from the local coordinate system (attached to the local facet mirror) to the global coordinate system (attached to the concentrator). The rotational transformation matrix for the tilted angle σ i;j can be written as 2 3 cos σ i;j 0 − sin σ i;j 0 60 1 0 07 7 σ i;j 6 (10) 4 sin σ i;j 0 cos σ i;j 0 5: 0 0 0 1 The rotational transformation matrix for the tilted angle γ i;j can be written as
2
1 0 6 0 cos γ i;j γ i;j 6 4 0 sin γ i;j 0 0
0 − sin γ i;j cos γ i;j 0
3 0 07 7: 05 1
(11)
Hk MT Pk ;
The translational transformation matrix based on the coordinate of the facet mirror M x ; M y ; M z i;;j in the global coordinate system can be written as 2
1 60 T 1 i;j 6 40 0
0 1 0 0
0 0 1 0
3 Mx My 7 7: Mz 5 1
The final orientation of the facet mirror can be known by applying coordinate transformations on the initial coordinate in the following formula: (13)
where the coordinate transformation matrices for the facet mirror is MT T 1 i;j γ i;j σ i;j :
(14)
B. Computational Algorithm
(12)
Figure 6 shows a flow chart to illustrate the special computational algorithm to compute the geometry of NIDC by determining the coordinate and orientation
Fig. 6. Flow chart to illustrate the special computational algorithm to compute the geometry of the NIDC, which defines the initial orientations of all facet mirrors at the same height and then obtains final orientations of facet mirrors with gradually increased height, so that all the facet images remain superimposition at the common receiver without sunlight blocking and shadowing among adjacent mirrors. 20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS
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of all facet mirrors. During the initial state of computational algorithm, all the facet mirrors are defined in an array in which the central point of facets are spaced evenly with M z; 0, as shown in Fig. 2. Then each of the facet mirrors is virtually lifted up one by one from the central region toward the peripheral region along the x, y, and z directions in order to eliminate the sunlight blocking and shadowing effects while keeping fixed gaps among the facet mirrors. The final positions of the facet mirrors that form a dish will be different from that of the initially defined position in the horizontal plane, and the process of computing the final facet mirrors positions will cause the variation in the tilted angles of each facet mirror as to keep the solar image aiming at the receiver. Therefore, in the process of designing the geometry of the NIDC, an iterative method is used to calculate the final position as well as the two tilted angles, σ i;j and γ i;j , of each facet mirror. After obtaining the tilted angles of the facet mirror, coordinate transformation is applied to determine the orientation of the facet mirrors. The process of designing the position of facet mirrors is twofold: the first involves virtually lifting of the facet mirrors arranged in row sequence ( j 1; 2; 3; …:n.) along the same column in the Y and Z axes; the second involves lifting of facet mirrors arranged in column sequence (i 1; 2; 3; …:m.) along the same row in the Y and Z axes. Figure 7(a) shows how to determine the new position of facet mirrors in the (j 1)th row along the Y and Z axes, and it is considered free from sunlight blocking and shadowing effects when the following formulas are fulfilled: H 3z i;j1 H 1z i;j ;
(15a)
H 3y i;j1 H 1y i;j G:
(15b)
Figure 7(b) shows how to determine the new position of facet mirrors in the (i 1)th along the Y and Z axes, and it is considered free from sunlight blocking and shadowing effects when the following formulas are fulfilled: H 4z i1;j H 2z i;j ;
(16a)
H 4x i1;j H 2x i;j G:
(16b)
In the simulation, important parameters must be inserted, such as focal distance of the NIDC (F), width of the facet mirror (w), and gap between the adjacent facet mirrors (G). As the squared facet mirror is considered in our case study, the length is equal to the width of the facet mirror. The gap between the adjacent facet mirrors is set to be 5% of the width or length of a single facet mirror. The coordinate of the facet mirror, which is located closest to the origin, M1;1 , is ½w ½G; ½w ½G; 0. Considering the concentrator consisted of 2m columns and 2n rows, the computational algorithm will start to compute the final position of the facet mirrors located at the first column, i 1, with the row sequence starting from j 1 to j n. The same procedure continues for the following column from i 2 to i m. The detail of the computational algorithm to simulate the configuration of the facet mirrors in the NIDC is summarized in the flow chart as shown in Fig. 6. Figure 8 shows a 3D drawing of the final NIDC configuration, which consisted of 196 facet mirrors and is generated based on the computational algorithm. The computational algorithm is also performed for a different number of arrays and different facet dimensions with the detailed specifications as listed in Table 2. 4. Optical Performance Analysis of NIDC Using Analytical Method
For optical characterization of NIDC, two major optical parameters are needed to be taken into account, which are the average solar concentration ratio at the uniform illumination and the area of uniform illumination at the receiver. These parameters affect the power generation of the CPV module, which is attached to the receiver. On top of that, the total number of CPV cells required in the CPV module is also an important measure in the cost calculation for the whole CPV system.
Fig. 7. Conceptual drawing to illustrate how the facet mirrors are virtually lifted up for eliminating sunlight blocking and shadowing in the computational algorithm. (a) Along Y and Z axes. (b) Along X and Z axes. 480
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Fig. 8. Example of NIDC configuration completed by using the computational algorithm, which consists of 196 facet mirrors.
Numerical simulation using a ray-tracing technique has been widely deployed in the analysis of the optical performance of a nonimaging solar concentrator [1,21]. The main hindrance of the numerical simulation is a long computational time if reasonably high accuracy is required. An analytical model proposed by Wong et al. to analyze the optical characteristics of a NIPC has reduced the computational time dramatically with good accuracy [24]. Since the nonimaging geometries between planar and dish concentrators are quite different, the analytical formulas of NIDC are described in detail. Furthermore, the simulated results of optical performance of NIDC using a numerical method are compared with the calculated result using analytical methods. A pillbox sun shape is considered in the optical performance analysis for solar disk effect by ignoring the effect of the sun’s atmosphere, which gives the disc a limb darkening or decreasing the brightness toward the edge of the disc. The approach of simulating the sun shape in this context has been fully digital and simplified by considering the sunlight that strikes on the mirror surface as digitized cone rays. The reflectivity of the facet mirror is treated as unity. A cross-sectional view of the NIDC to show how the individual square facet mirror with a dimension of w reflects the incident sunlight to the receiver is illustrated in Fig. 9. It is also shown that the total solar energy falling on both the facet mirror is fully reflected to the receiver provided that no energy is lost during the process. Conversely, the solar concentration ratio received by the receiver is decreased because the projected solar image of the facet mirror is spread across a longer distance on the receiver, La , compared to the effective distance, Lb , of the facet mirror exposed to solar flux. Due to the cosine effect, the intensity of solar flux, which falls on the receiver, is lower than that of the incident ray from the sun. Assuming that the incident sunlight is designated with solar concentration ratio of one sun, the solar concentration ratio at the receiver surface will be reduced by the cosine of angle between the incident ray
Fig. 9. Cross-section view of the NIDC, which shows how the individual facet mirror with dimension w × w reflects and superimposes the sunray to the receiver.
and reflected ray, 2θi;j . Therefore the average solar concentration ratio at the uniform illumination area, C, after superimposing all the solar images of facet mirrors on the receiver can be expressed as C
2m X 2n X
cos 2θi;j :
(17)
i1 j1
Based on trigonometry theorem with the solar disk half angle (ε) of 4.65 mrad, the length of the solar image of the i, j facet, which is cast on the receiver surface, can be calculated by L l1 l2 l 3 W sin θi;j tan2θi;j ε W cos θi;j F − M z − 2 W (18) − F − M z sin θi;j tan2θi;j − ε; 2 where
W l1 F − M z − sin θi;j tan2θi;j ε 2 − tan2θi;j − ε
(19)
W sin θi;j tan2θi;j − ε l2 W cos θi;j F − M z − 2 W sin θi;j tan2θi;j ε; (20) − F − Mz 2 W sin θi;j tan2θi;j ε l3 F − M z 2
− tan2θi;j − ε:
(21)
Due to the cosine effect, the solar image for the facet mirror at a larger incident angle will be larger than that of the facet mirror at a smaller incident angle. The facet mirror with the smallest incident angle will produce the smallest solar image on the receiver. Due to the solar disk effect, the uniform illumination area is equal to l22 . On the other hand, the facet mirror with the largest incident angle will produce the largest solar image on the receiver, which will determine the final image size of the whole solar concentrator. The total image area, Aove , and the uniform illumination area, Auni , on the receiver can be calculated by using the following formulas: Aove L2 8 92 Wcos θmax > > > > > > > > < F − M − W sin θ = ε tan2θ z max max 2 ; > > > > > > W > : − F − M z 2 sin θmax tan2θmax − ε > ; (22) 20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS
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Auni l22 8 92 > > W cos θ > > min > > > > > > < F − M − W sin θ = − ε tan2θ z min min 2 ; > > > > > > W > > > : − F − M z 2 sin θmin tan2θmin ε > ; (23) where the maximum incident angle of the facet mirror is θmax , which is also equal to θm;n , and the minimum incident angle of the facet mirror is θmin , which is also equal to θ1;1 . Since dense-array CPV cells operate the best at uniform illumination for achieving the optimal power output, we consider the CPV cells to be installed only in the uniform illumination region of the receiver in the study. The total number of CPV cells at the uniform illumination region of the NIDC can be computed as l g 2 N CPV int 2 ; bg
(24)
where b is the width of single CPV cell, g is the gap between the two adjacent CPV cells, and l2 can be calculated using Eq. (20) by substituting θi;j θmin . The output power of the CPV system can be expressed as follows: Pout ηCPV × I × C × N CPV × ACPV ;
(25)
where ηcpv is the conversion efficiency of the CPV cell, I is the incident solar irradiance (Watt∕m2 ), and Acpv is the area of single CPV cell (m2 ). The total cost of the system per power output of the CPV system (US $/Watt) can be determined by using the following formula: Sc∕w
N CPV × CCPV Cfacets Cmech ; Pout
(26)
where Ccpv is the cost per single CPV cell (assuming the cost is US$6.5/cell), Cfacets is the total cost of all the facet mirrors, and Cmech is the total cost of the solar concentrator structure, including mechanical structure, pedestal, and dual-axis tracking mechanism, etc. 5. Results and Discussion
To analyze the optical performance of NIDC, a numerical method is usually used to simulate the solar flux distribution with reasonable good accuracy. Figures 10(a) and 10(b) show the numerical simulation result of the solar flux distribution in 3D and 2D plots, accordingly, for the case of NIDC with an array of 22 × 22 facets, and each facet of 49.8 cm × 49.8 cm at a focal distance of 10 m. It is well know that the numerical simulation based on ray-tracing 482
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Fig. 10. Numerical simulation results of solar flux distribution for the case of 22 × 22 array of 49.8 cm facet dimension with 10 m focal distance. (a) 3D plot. (b) 2D plot.
technique that involves handling a huge number of databases is time consuming. Alternatively, an analytical model is proposed to expedite the work. To validate the analytical model, comparisons between the numerical and analytical methods have been carried out. Figures 11(a) and 11(b) reveal a comparison study between two methods in terms of average solar concentration ratio at uniform illumination and uniform illumination area, respectively. For the average solar concentration ratio at uniform illumination, the relative standard deviation of the numerical simulation is less than 0.09%, and the variation result between the two methods is less than 0.005%. The very small variation in average solar concentration shows excellent agreement between both methods. On the other hand, the accuracy in calculating the uniform illumination area depends on the resolution defined for the receiver in terms of pixel per cm for the numerical simulation. The error of uniform illumination area caused by resolution of the receiver for a numerical method can be calculated by using the following formula:
Fig. 11. In the case of 22 × 22 array of 49.8 cm facet dimension with focal distance from 6 to 10 m. (a) Graph shows the average solar concentration ratio at uniform illumination area for both numerical and analytical methods versus focal distances. (b) Graph shows the area of uniform illumination for numerical and analytical methods versus focal distances.
Aerror %
l2nm − lnm − 2∕Res2 × 100%; l2nm
(27)
where lnm is the simulated length of uniform illumination region using a numerical method, and Res is resolution of the receiver (pixel per cm) used in the numerical simulation. The errors of the uniform illumination area due to the resolution of the receiver for focal distances from 6, 7, 8, 9, to 10 m are 4.4%, 4.5%, 4.6%, 4.7%, and 4.8%, respectively. The difference between analytical and numerical methods for the area of uniform illumination at focal distances from 6, 7, 8, 9, to 10 m are 3.5%, 3.1%, 2.7%, 2.4%, and 2.1%, correspondingly. This difference is mainly caused by the receiver’s resolution (pixel per cm) used in the numerical method simulation. If the resolution of the receiver applied to the numerical simulation is increased, the simulated result will be more accurate and closer to the analytical value. Figure 11(b) shows that the variations in results for two computational methods fall within the range of errors for the numerical simulation.
Since the two major optical parameters needed to be considered in the optical design of NIDC can also be calculated using the analytical model, the following study of system performance is carried out based on the analytical method. Despite the analytical model being unable to plot the solar flux distribution on the receiver, the average solar concentration ratio at the uniform illumination and also the uniform illumination area are sufficient for optical engineers to analyze the performance of NIDC as well as a densearray CPV system. To understand the overall performance of the whole system, the synthesized parameters including the output power, total number of CPV cells, and the cost per output power can be derived from the two major optical parameters. As a result, the performance of the NIDC can be optimized by changing the designed parameters such as focal distance of the concentrator (F) and the dimension of the facet mirror (w) via the aforementioned methodology. The detailed specifications of the NIDC used in the study are presented in Table 1. From Table 2, 15 sets of different arrays with their corresponding total number of facet mirrors and dimensions have been considered in our performance analysis in such a way that the total reflective area of the solar concentrator remains almost the same for all the cases, which is about 120 m2 . Figure 12 illustrates the average solar concentration ratio at the uniform illumination versus facet dimension at different focal distances. It shows that different facet dimensions can accommodate for Table 1.
Specification and Design Parameters of the NIDC
Total reflective area Area of single CPV cell, Acpv Gap between CPV cell, g Incident solar radiation, I Cost of single CPV cell, CCPV Cost of concentrator structure, Cmech Cost of 120 m2 facet mirror, Cfacets
Table 2.
120 m2 1 × 1 cm2 1 mm 1000 Watt∕m2 US$6.5 US$32,000 US$3,000
Specifications Used in the Simulation for Different Arrays of Facet Mirrors
Array of Facet Mirrors
Dimension of Facet Mirror w × w
12 × 12 14 × 14 16 × 16 18 × 18 20 × 20 22 × 22 24 × 24 26 × 26 28 × 28 30 × 30 32 × 32 34 × 34 36 × 36 38 × 38 40 × 40
91.3 × 78.2 × 68.5 × 60.9 × 54.8 × 49.8 × 45.6 × 42.1 × 39.1 × 36.5 × 34.2 × 32.2 × 30.4 × 28.8 × 27.4 ×
cm × 91.3 cm cm × 78.2 cm cm × 68.5 cm cm × 60.9 cm cm × 54.8 cm cm × 49.8 cm cm × 45.6 cm cm × 42.1 cm cm × 39.1 cm cm × 36.5 cm cm × 34.2 cm cm × 32.2 cm cm × 30.4 cm cm × 28.8 cm cm × 27.4 cm
Total Number of Facet Mirrors 144 196 256 324 400 484 576 676 784 900 1024 1156 1296 1444 1600
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Fig. 12. Bar chart to show the comparisons of average solar concentration ratio at uniform illumination area versus facet dimensions at different focal distances.
different average solar concentration ratios from 109.46 suns to 1439.2 suns for the focal distances ranging from 6 to 10 m. Moreover, it also shows that the increment of focal distance will increase the average solar concentration ratio while the increment of the facet dimension will reduce the average solar concentration ratio. Bar charts for the uniform illumination area and the total number of CPV solar cells attached on the receiver versus facet dimension at different focal distances are plotted in Figs. 13 and 14, respectively. The uniform illumination area and the total number of CPV cells are directly proportional to the facet dimension. In contrast, the uniform illumination area is inversely proportional to the focal distance of the concentrator, and similarly it also happens to the charts for the number of CPV cells versus focal distance of the concentrator in most of the cases. The exceptional cases are facet dimension 91.3 cm with focal distances 8 and 9 m and facet dimension 78.2 cm with focal distances 7 and 8 m etc., in which the number of CPV solar cells remains the same, even though their focal distances are different. These phenomena happen because the increment of the
Fig. 13. Bar chart to show the comparisons of uniform illumination areas versus facet dimensions at different focal distances. 484
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Fig. 14. Bar chart to show the comparisons of a number of CPV cells used on the receiver versus facet dimensions at different focal distances.
uniform illumination area is not large enough to insert a new row or column of CPV solar cells in the receiver. We assume the CPV cell employed in the study is an Emcore CTJ photovoltaic cell, 10 mm × 10 mm, and the efficiency of the cell is based on specification provided by Emcore, as shown in Fig. 15 [25]. Figure 16 shows that different facet dimensions can cater for different system output power from 12.99 to 26.83 kW at different focal distances ranging from 6 to 10 m. It also shows that the system output power is higher at a lower solar concentration ratio compared to that of higher solar concentration ratio. Unfortunately, the usage of CPV cells in the NIDC system will be more for lower solar concentration ratio compared to that of higher solar concentration ratio. It will increase the total cost of the whole system. To address practical concerns, the following study is done to justify the facet dimensions and focal distances of the concentrator to provide the optimal performance. The total cost per output power of the system is assessed for different facet dimensions and focal distances, provided that the total reflective area is fixed.
Fig. 15. Graph to show the cell conversion efficiencies versus solar concentration ratio for Emcore CTJ photovoltaic cell, 10 mm × 10 mm [25].
To reduce the cost of the system, innovation must be implemented in the design of the concentrator. As the cost of CPV cell and concentrator varies with territories and quantity, engineers can apply the above-mentioned methodology to evaluate the cost per output power of the system in their respective countries. 6. Conclusions
Fig. 16. Bar chart to show the comparisons of system output power versus facet dimensions at different focal distances.
This assessment is significant for designing an optimized dense-array CPV system by considering the facet dimension and focal distance of NIDC as well as the required number of CPV cells on the receiver. The estimated cost of a single Emcore CTJ photovoltaic cell and the corresponding direct bond copper substrate is US$6.5. The costs of the CPV cells vary, depending on the purchase quantity; these costs have high potential to be reduced in the future. The estimated cost of the concentrator is US $35,000 for a collective area of 120 m2 in Malaysia, and the value may vary from country to country. Figure 17 has been plotted to show the total cost of the system per output power versus facet dimension at different focal distances. It shows that the solar concentrator with a facet dimension of 49.8 cm and focal distance of 8 m operating at a solar concentration ratio of 411.8 suns provides the lowest cost per output power, which is US$1.93 per watt compared to other designs. Based on these parameters, we further break down the cost per output power of the system to three different parts, which are the CPV cell, facet mirror, and the concentrator structure, and the cost per output power of the breakdown are $0.409/W, $0.065/W, and $1.459/W respectively. This shows that the concentrator structure accounts for the majority of the cost in the system.
Fig. 17. Bar chart to show the comparisons of total system cost-per-output power versus facet dimensions at different focal distances.
The optical principle of NIDC and a computational algorithm to design the facet mirror configuration have been developed. An analytical model for NIDC has been formulated. The computational results between the numerical and analytical methods to determine the maximum solar concentration ratio and the area of uniform illumination are compared. The variation between the two methods for the average solar concentration ratio at uniform illumination is less than 0.005% while the variation between the two methods for the area of uniform illumination at focal distances from 6, 7, 8, 9, to 10 m are 3.5%, 3.1%, 2.7%, 2.4%, and 2.1%, correspondingly. It shows that the analytical model is proficient to provide reasonable precise results without consuming lengthy computational time. In the study, the optical performance of NIDC with a reflective area of 120 m2 for various facet dimensions and the focal distances ranging from 6 to 10 m is considered. Depending on the facet dimensions and the focal distance, the average solar concentration ratio at the uniform illumination region can vary, ranging from 109 suns to 1439 suns, and the system output power ranges from 12.99 to 26.83 kW. NIDC with a facet dimension of 49.8 cm and a focal distance of 8 m operating at solar concentration ratio of 411.8 suns offers the lowest cost per output power, which is US$1.93 per watt compared to other configurations. The results show that NIDC with single-stage focusing, which combines the advantages from NIPC and a parabolic dish concentrator, is capable of producing a uniform illumination area with a high solar concentration ratio at the receiver. The optical characteristics and simplicity in design have recommended that NIDC is suitable for the application of a dense-array CPV system. In fact, all the concentrator contains exactly the same basic driving mechanisms such as motor, gearbox, optical encoder, and control system. The necessitated manufacturing, assembly, and alignment procedures are needed for all concentrators. Comparing the construction cost of different concentrators is not the main aim of this paper because it depends on many factors such as the labor cost (which is dependent on country), the quantity of concentrator to be manufactured, type or grade of materials used, etc. Despite alignment of mirrors being required for multifaceted mirrors in NIDC, its fabrication is cheaper and simpler as compared with a precise large reflective area of a parabolic dish. For a large parabolic dish concentrator with a total reflective area of more than 100 m2 , a single parabolic reflector 20 January 2014 / Vol. 53, No. 3 / APPLIED OPTICS
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is not possible to be fabricated, and therefore multifaceted parabolic mirrors are still required. If it is compared with NIDC, a multifaceted parabolic dish is more expensive because curved facet mirrors are used, and the alignment process is more complicated with the involvement of curved mirrors. On the other hand, a continuous parabolic concentrator produces a solar flux distribution that closely resembles the Gaussian distribution. It may cause the dense-array CPV module encounters a severe drop in efficiency due to the current mismatched problem. From the analysis in the paper, the solar concentrator structure of the CPV system contributes the major cost in the CPV system. Therefore the system cost can be very much dependent on the innovative design of the solar concentrator structure and will remain a great challenge for solar-energy engineering in the future. The authors would like to express their gratitude to the Ministry of Energy, Green Technology and Water (AAIBE Trust Fund) and the Ministry of Science, Technology and Innovation (e-Science Fund with project number 03-02-11-SF0143) for their financial support. The authors would also like to thank the project advisor Ir. Dr. Philip Tan Chee Lin for his many insightful suggestions.
9. 10.
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