# Optimal Design of a Secondary Optical Element for a Noncoplanar Two-Reflector Solar Concentrator

Authors: Yi-Cheng Chen; Chia-Chi You
Journal: International Journal of Photoenergy
 (2015)

---

## Abstract

This paper presents the results of a parametric design process used to achieve an optimal secondary optical element (SOE) in a noncoplanar solar concentrator composed of two reflectors. The noncoplanar solar concentrator comprises a primary parabolic mirror (M1) and a secondary hyperbolic mirror (M2). The optical performance (i.e., acceptance angle, optical efficiency, and irradiance distribution) of concentrators with various SOEs was compared using ray-tracing simulation. The parametric design process for the SOE was divided into two phases, and an optimal SOE was obtained. The sensitivity to assembly errors of the solar concentrator when using the optimal SOE was studied and the findings are discussed.


---

## Body

## 1. Introduction

Solar photovoltaic (PV) system is becoming a promising option for sustainable electricity generation as fossil fuels are decreasing on our planet. Concentrating photovoltaic (CPV) system is an important tool to improve the efficiency and to reduce the area of solar cell. The CPV systems can concentrate a large amount of sunlight into a small III-V solar cell area by using lenses or mirrors [1–5]. The primary attraction of CPV systems includes the reduced usage of expensive semiconductor solar cell and increasing the concentration ratio.The CPV systems usually use single or dual-axis tracking to enhance performance. For CPV system, the acceptance angleθ
90
% is defined as the incidence angle corresponding to 90% of the maximum optical efficiency η at normal incidence. In addition, the definition of acceptance angle θ
50
% is the incidence angle corresponding to 50% of the maximum optical efficiency η at normal incidence. A high optical efficiency η (>70%) and a wide acceptance angle (θ
90
%
>
1
°) are commonly required for a CPV system. To avoid localized hot spots on the solar cell, a secondary optical element (SOE) is typically used in front of a solar cell for a solar concentrator. An SOE also improves the acceptance angle of a solar concentrator as well as irradiance uniformity on the solar cell [1, 5–7].The optics used to collect sun energy in solar concentrators comprise three major types: refractive (using a refractive Fresnel lens), reflective (using a reflective parabolic mirror), and two-reflector (using a parabolic primary mirror and a hyperbolic secondary mirror) [6–15]. Gordon and Feuermann explored the design of two-reflector concentrator to provide radiative transfer at the thermodynamic limit [9, 12]. Chen et al. numerically investigated a solar concentrator combining a primary paraboloidal and a secondary hyperboloidal mirror by using self-developed ray-tracing simulation tool [6, 14]. Their study obtained a high concentration ratio and optimal mirror shapes of the solar concentrator.Figure1(a) shows a typical two-reflector solar concentrator that consists of a primary mirror (M1), a secondary mirror (M2), a solar cell, and an SOE. The initial design parameters of the two-reflector solar concentrator were as follows: the diameter of M1 = 120 mm, the diameter of the central hole of M1 = 20 mm, and the size of the square solar cell was 5.5 mm by 5.5 mm. The geometric concentration ratio (GCR) is defined as the ratio of the entrance aperture (determined by the primary mirror diameter) and the area defined by the solar cell. The GCR was 373x in this study. The other design parameters of the two mirrors were preliminarily determined based on the equations proposed in [9, 12, 13]. Figure 1 illustrates the design and dimensions of the noncoplanar two-reflector concentrator with a refractive pyramid-shaped SOE. Ray-tracing simulations were performed to evaluate the optical performance, the selected two-reflector concentrator was noncoplanar (Figure 1), and the numerical aperture (NA2) of M2 is 0.25. It is called “noncoplanar” because the rim of M1 and the backside of M2 are not in the same plane, as shown in Figure 1(b).Figure 1
Illustration of the noncoplanar two-reflector solar concentrator.
(a)
(b)Based on the design of the two reflectors, ray-tracing simulation was used to predict the optical performance of the concentrator using various types of SOE. For each type of SOE, a parametric design process was performed to determine the optimal design parameters of the SOE. The SOE design process consisted of two phases, Phase I and Phase II. In Phase I, the focal point of M2 was positioned at the surface of the solar cell [15]. In Phase II, the focal point of M2 was positioned at the top surface of the SOE to improve irradiance uniformity on the cell. The predicted performance of the solar concentrator using the optimal SOE design is presented and discussed. The sensitivity of the concentrator to various misalignments of optical elements (M1, M2, and SOE) is also discussed.


## 2. Design Process of SOE

### 2.1. Phase I of SOE Design

In Phase I of the design process, the focal point of M2 was positioned on the surface of the receiver (solar cell). To increase the acceptance angle of the two-reflector solar concentrator, three types of SOEs were considered: (1) Type I: refractive pyramid-shaped SOE, (2) Type II: reflective pyramid-shaped SOE, and (3) Type III: refractive cone-shaped SOE, as depicted in Figures2(a)–2(c), respectively. The transmittance of refractive SOE was assumed to be 96%, and the reflectance of reflective SOE was assumed to be 98%. Figures 2(a)–2(c) also depict the two major SOE design parameters, parameter B and parameter H. Parameter B represents the edge length of the top area of the two pyramid-shaped SOEs (Figures 2(a) and 2(b)) and represents the diameter of the top area of the cone-shaped SOE (Figure 2(c)). Parameter H represents the height. The dimensions of the bottoms of the pyramid-shaped SOEs were the same as the solar cell (5.5 mm × 5.5 mm). The diameters of the bottoms of the cone-shaped SOEs were 5.5 mm.Figure 2
Three types of SOEs: (a) refractive pyramid, (b) reflective pyramid, and (c) refractive cone.
(a)
(b)
(c)The effects of the design parametersH and B on the improvement of the optical efficiency of the solar concentrator were studied using ray-tracing simulation. A parametric design process was conducted using each of the three SOE types to obtain preliminary optimal SOE design parameters. The SOE designs were narrowing down to the three most promising designs, and the optical performance of the three optimal SOE designs was simulated. The results are compared and discussed.


#### 2.1.1. SOE I: Refractive Pyramid

Figures3(a) and 3(b) depict the effects of SOE I design parameters B (top edge length) and H (height) on the optical efficiency of the solar concentrator at incidence angles of 0° (normal incidence) and 1°, respectively. As shown in Figure 3(a), when H was fixed, optical efficiency at normal incidence decreased as B increased. Based on the simulation results shown in Figures 3(a) and 3(b), the feasible parameter sets H and B, (10, 12), (13, 12), (15, 12), and (18, 14), were selected for further investigation. The optical efficiency of the solar concentrator with SOE I of parameter set (18, 14) was good (74.9%) at incidence angle of 1°; however, the optical efficiency was only 80.8% at incidence angle of 0° (normal incidence). Therefore the parameter set (18, 14) was not selected as a feasible parameter set for SOE I.Figure 3
Effects of refractive pyramid SOE parameters on optical efficiency: (a) incidence angle = 0° (normal incidence) and (b) incidence angle = 1°.
(a)
(b)


#### 2.1.2. SOE II: Reflective Pyramid

Figures4(a) and 4(b) show the effects of SOE II design parameters B and H on the optical efficiency of the solar concentrator at incidence angles of 0° (normal incidence) and 1°, respectively. Figure 4(a) indicates that optical efficiency at normal incidence decreased with an increase in H or B, except when H
=
13 and B
=
16. However, Figure 4(b) shows that the optical efficiency of SOE II was lower than 70% at an incidence angle of 1°. This phenomenon limits the applicability of SOE II.Figure 4
Effects of reflective pyramid SOE parameters on the optical efficiency: (a) incidence angle = 0° (normal incidence) and (b) incidence angle = 1°.
(a)
(b)


#### 2.1.3. Discussion of Pyramid-Shaped SOE

Based on a comparison of Figures3(b) and 4(b), the refractive pyramid SOE generally exhibited higher optical efficiency than a reflective pyramid SOE at an incidence angle of 1°. This implies that using SOE I improved the acceptance angle of the concentrator more than SOE II. Consequently, the feasible parameter sets of SOE I were used to simulate the optical efficiency of the concentrator at various incidence angles; the results are depicted in Figure 5. Among the four feasible SOE I designs, two designs with the parameter sets H and B, (13, 12) and (15, 12), were preliminarily selected as optimal designs because their optical efficiencies were stable and remained higher than 60% even at an incidence angle of 1.2°.Figure 5
Optical efficiency with a refractive pyramid SOE under various incidence angles.


#### 2.1.4. SOE III: Refractive Cone

Figures6(a) and 6(b) illustrate the effects of SOE III design parameters B (diameter of top area) and H (height) on optical efficiency at incidence angles of 0° (normal incidence) and 1°, respectively. Figure 6(a) shows that optical efficiency at normal incidence decreased as H increased. However, at an incidence angle of 1°, an increase in H resulted in increased optical efficiency (Figure 6(b)). Based on the parametric analyses of SOE III (Figure 6), feasible parameter sets H and B, (10, 12), (13, 12), (15, 12), and (18, 12), were selected for further investigation.Figure 6
Effects of refractive cone SOE parameters on the optical efficiency: (a) incidence angle = 0° (normal incidence) and (b) incidence angle = 1°.
(a)
(b)Figure7 illustrates the optical efficiency of the concentrator using the four feasible designs of SOE III at various incidence angles. Among the four feasible designs of SOE III, the design with the parameter set (H
=
18, B
=
14) was chosen as a preliminary optimal design because of its higher optical efficiency at large incidence angles. Based on a comparison of Figures 5 and 7, the optical efficiency of concentrators using SOE III generally dropped more rapidly than that of concentrators using SOE I as the deviation of the incidence angle increased.Figure 7
Optical efficiency with a refractive cone SOE under various incidence angles.


#### 2.1.5. Discussion of Phase I SOE Design

The performance simulations of the two optimal designs of SOE I and one optimal design of SOE III are compared and summarized in Figure8 and Table 1. Figure 8 shows the optical efficiency of the concentrator using the three SOEs at various incidence angles. Table 1 summarizes the design parameters of the three optimal SOE designs and the performance of the concentrator using each of the designs, including optical efficiencies at incidence angles of 0° and 1° and acceptance angles θ
90
% and θ
50
%. As shown in Table 1 and Figure 8, at normal incidence, the refractive pyramid SOE design Q1 with the parameter sets H and B, (13, 12), enabled the concentrator to achieve the highest optical efficiency, 84.24%, compared with two other optimal SOE designs, Q2 (82.83%) and Q3 (82%). SOE Q3 also enabled the concentrator to achieve the highest corresponding acceptance angle θ
90
%, 1.06°, whereas the acceptance angles θ
90
% for Q2 and Q1 were 1.03° and 1.00°, respectively.Table 1
Optical performance of the two-reflector solar concentrator with preliminary optimal SOEs from Phase I and Phase II.
Phase
SOE
H (mm)
B (mm)
η
opt (at  0°)
η
opt (at  1°)
θ
90
%
θ
50
%
I
Q1
Refractive pyramid
13
12
84.24%
75.64%
1.00
1.39
Q2
Refractive pyramid
15
12
82.83%
75.00%
1.03
1.54
Q3
Refractive cone
18
14
82.00%
74.82%
1.06
1.51
II
D1*
Refractive pyramid
15
10
83.52%
80.88%
1.11
1.51
D2
Refractive cone
18
12
83.26%
78.08%
0.97
1.63
*The optimal design founded in this study.Figure 8
Optical efficiency with 3 optimal SOEs of Phase I under various incidence angles.The SOE designs produced in Phase I enabled the concentrator to achieve adequate acceptance angles and optical efficiencies. However, the uniformity of the irradiance distribution on the receiver was observed to be insufficient. In Phase II, the focal point of M2 was positioned at the top surface of the SOE to improve irradiance uniformity on the solar cell.


### 2.2. Phase II of SOE Design

In Phase II, the focal point of M2 was positioned on the top surface of the SOE to improve irradiance uniformity. Based on the results of Phase I, the preliminary optimal SOE types were observed to be the refractive pyramid and refractive cone. Consequently, Phase II considered only these two types of SOE to determine an SOE design enabling the concentrator to achieve both superior acceptance angles and irradiance uniformity. The two SOEs selected were (1) Type I: refractive pyramid-shaped and (2) Type II: refractive cone-shaped.


#### 2.2.1. SOE I: Refractive Pyramid

Figures9(a) and 9(b) illustrate the effects of SOE I design parameters B (edge length of top area) and H (height) on the optical efficiency of the concentrator at incidence angles 0° (normal incidence) and 1°, respectively. As shown in Figure 9(a), when H is fixed, optical efficiency at normal incidence decreased as B increased. Based on the simulation results shown in Figures 9(a) and 9(b), the feasible parameter sets H and B, (13, 10) and (15, 10), respectively, were selected. These feasible parameter sets exhibited the highest optical efficiency at normal incidence and incidence angle deviations.Figure 9
Effects of refractive pyramid SOE parameters (Phase II) on the optical efficiency: (a) incidence angle = 0° (normal incidence) and (b) incidence angle = 1°.
(a)
(b)


#### 2.2.2. SOE II: Refractive Cone

Figures10(a) and 10(b) show the effects of the SOE III design parameters B (diameter of top area) and H (height) on optical efficiency at incidence angles 0° and 1°, respectively. Figure 10(a) shows that, at normal incidence, optical efficiency decreased when H increased. At an incidence angle of 1°, however, an increase in H resulted in increased optical efficiency, as shown in Figure 10(b). Based on the simulation results of Figure 10, the feasible parameter set H and B, (18, 12), was selected.Figure 10
Effects of refractive cone SOE parameters (Phase II) on the optical efficiency: (a) incidence angle = 0° (normal incidence) and (b) incidence angle = 1°.
(a)
(b)


#### 2.2.3. Discussion of Phase II SOE Design

Figure11 depicts optical efficiency at various incidence angles of the two optimal SOEs selected in Phase II. Table 1 summarizes the optical performance of the solar concentrator when using these two SOEs, including optical efficiency at incidence angles 0° and 1° and acceptance angles θ
90
% and θ
50
%. As shown in Table 1 and Figure 11, D1 enabled the solar concentrator to achieve a higher optical efficiency (83.52%) than D2 (83.25%) at normal incidence. D1 also enabled the concentrator to achieve a higher acceptance angle θ
90
% (1.11°) than D2 (0.97°).Figure 11
Optical efficiency with 2 optimal SOEs of Phase II under various incidence angles.After comparing the performance results of the SOEs, it was observed that D1 yielded the optimal optical performance. The irradiance distribution of D1 on the receiver under normal incidence is shown in Figure12; the irradiance uniformity is 67.14%.Figure 12
Irradiance distribution of the two-reflector solar concentrator with optimal SOE D1 of Phase II (normal incidence).


## 2.1. Phase I of SOE Design

In Phase I of the design process, the focal point of M2 was positioned on the surface of the receiver (solar cell). To increase the acceptance angle of the two-reflector solar concentrator, three types of SOEs were considered: (1) Type I: refractive pyramid-shaped SOE, (2) Type II: reflective pyramid-shaped SOE, and (3) Type III: refractive cone-shaped SOE, as depicted in Figures2(a)–2(c), respectively. The transmittance of refractive SOE was assumed to be 96%, and the reflectance of reflective SOE was assumed to be 98%. Figures 2(a)–2(c) also depict the two major SOE design parameters, parameter B and parameter H. Parameter B represents the edge length of the top area of the two pyramid-shaped SOEs (Figures 2(a) and 2(b)) and represents the diameter of the top area of the cone-shaped SOE (Figure 2(c)). Parameter H represents the height. The dimensions of the bottoms of the pyramid-shaped SOEs were the same as the solar cell (5.5 mm × 5.5 mm). The diameters of the bottoms of the cone-shaped SOEs were 5.5 mm.Figure 2
Three types of SOEs: (a) refractive pyramid, (b) reflective pyramid, and (c) refractive cone.
(a)
(b)
(c)The effects of the design parametersH and B on the improvement of the optical efficiency of the solar concentrator were studied using ray-tracing simulation. A parametric design process was conducted using each of the three SOE types to obtain preliminary optimal SOE design parameters. The SOE designs were narrowing down to the three most promising designs, and the optical performance of the three optimal SOE designs was simulated. The results are compared and discussed.


### 2.1.1. SOE I: Refractive Pyramid

Figures3(a) and 3(b) depict the effects of SOE I design parameters B (top edge length) and H (height) on the optical efficiency of the solar concentrator at incidence angles of 0° (normal incidence) and 1°, respectively. As shown in Figure 3(a), when H was fixed, optical efficiency at normal incidence decreased as B increased. Based on the simulation results shown in Figures 3(a) and 3(b), the feasible parameter sets H and B, (10, 12), (13, 12), (15, 12), and (18, 14), were selected for further investigation. The optical efficiency of the solar concentrator with SOE I of parameter set (18, 14) was good (74.9%) at incidence angle of 1°; however, the optical efficiency was only 80.8% at incidence angle of 0° (normal incidence). Therefore the parameter set (18, 14) was not selected as a feasible parameter set for SOE I.Figure 3
Effects of refractive pyramid SOE parameters on optical efficiency: (a) incidence angle = 0° (normal incidence) and (b) incidence angle = 1°.
(a)
(b)


### 2.1.2. SOE II: Reflective Pyramid

Figures4(a) and 4(b) show the effects of SOE II design parameters B and H on the optical efficiency of the solar concentrator at incidence angles of 0° (normal incidence) and 1°, respectively. Figure 4(a) indicates that optical efficiency at normal incidence decreased with an increase in H or B, except when H
=
13 and B
=
16. However, Figure 4(b) shows that the optical efficiency of SOE II was lower than 70% at an incidence angle of 1°. This phenomenon limits the applicability of SOE II.Figure 4
Effects of reflective pyramid SOE parameters on the optical efficiency: (a) incidence angle = 0° (normal incidence) and (b) incidence angle = 1°.
(a)
(b)


### 2.1.3. Discussion of Pyramid-Shaped SOE

Based on a comparison of Figures3(b) and 4(b), the refractive pyramid SOE generally exhibited higher optical efficiency than a reflective pyramid SOE at an incidence angle of 1°. This implies that using SOE I improved the acceptance angle of the concentrator more than SOE II. Consequently, the feasible parameter sets of SOE I were used to simulate the optical efficiency of the concentrator at various incidence angles; the results are depicted in Figure 5. Among the four feasible SOE I designs, two designs with the parameter sets H and B, (13, 12) and (15, 12), were preliminarily selected as optimal designs because their optical efficiencies were stable and remained higher than 60% even at an incidence angle of 1.2°.Figure 5
Optical efficiency with a refractive pyramid SOE under various incidence angles.


### 2.1.4. SOE III: Refractive Cone

Figures6(a) and 6(b) illustrate the effects of SOE III design parameters B (diameter of top area) and H (height) on optical efficiency at incidence angles of 0° (normal incidence) and 1°, respectively. Figure 6(a) shows that optical efficiency at normal incidence decreased as H increased. However, at an incidence angle of 1°, an increase in H resulted in increased optical efficiency (Figure 6(b)). Based on the parametric analyses of SOE III (Figure 6), feasible parameter sets H and B, (10, 12), (13, 12), (15, 12), and (18, 12), were selected for further investigation.Figure 6
Effects of refractive cone SOE parameters on the optical efficiency: (a) incidence angle = 0° (normal incidence) and (b) incidence angle = 1°.
(a)
(b)Figure7 illustrates the optical efficiency of the concentrator using the four feasible designs of SOE III at various incidence angles. Among the four feasible designs of SOE III, the design with the parameter set (H
=
18, B
=
14) was chosen as a preliminary optimal design because of its higher optical efficiency at large incidence angles. Based on a comparison of Figures 5 and 7, the optical efficiency of concentrators using SOE III generally dropped more rapidly than that of concentrators using SOE I as the deviation of the incidence angle increased.Figure 7
Optical efficiency with a refractive cone SOE under various incidence angles.


### 2.1.5. Discussion of Phase I SOE Design

The performance simulations of the two optimal designs of SOE I and one optimal design of SOE III are compared and summarized in Figure8 and Table 1. Figure 8 shows the optical efficiency of the concentrator using the three SOEs at various incidence angles. Table 1 summarizes the design parameters of the three optimal SOE designs and the performance of the concentrator using each of the designs, including optical efficiencies at incidence angles of 0° and 1° and acceptance angles θ
90
% and θ
50
%. As shown in Table 1 and Figure 8, at normal incidence, the refractive pyramid SOE design Q1 with the parameter sets H and B, (13, 12), enabled the concentrator to achieve the highest optical efficiency, 84.24%, compared with two other optimal SOE designs, Q2 (82.83%) and Q3 (82%). SOE Q3 also enabled the concentrator to achieve the highest corresponding acceptance angle θ
90
%, 1.06°, whereas the acceptance angles θ
90
% for Q2 and Q1 were 1.03° and 1.00°, respectively.Table 1
Optical performance of the two-reflector solar concentrator with preliminary optimal SOEs from Phase I and Phase II.
Phase
SOE
H (mm)
B (mm)
η
opt (at  0°)
η
opt (at  1°)
θ
90
%
θ
50
%
I
Q1
Refractive pyramid
13
12
84.24%
75.64%
1.00
1.39
Q2
Refractive pyramid
15
12
82.83%
75.00%
1.03
1.54
Q3
Refractive cone
18
14
82.00%
74.82%
1.06
1.51
II
D1*
Refractive pyramid
15
10
83.52%
80.88%
1.11
1.51
D2
Refractive cone
18
12
83.26%
78.08%
0.97
1.63
*The optimal design founded in this study.Figure 8
Optical efficiency with 3 optimal SOEs of Phase I under various incidence angles.The SOE designs produced in Phase I enabled the concentrator to achieve adequate acceptance angles and optical efficiencies. However, the uniformity of the irradiance distribution on the receiver was observed to be insufficient. In Phase II, the focal point of M2 was positioned at the top surface of the SOE to improve irradiance uniformity on the solar cell.


## 2.1.1. SOE I: Refractive Pyramid

Figures3(a) and 3(b) depict the effects of SOE I design parameters B (top edge length) and H (height) on the optical efficiency of the solar concentrator at incidence angles of 0° (normal incidence) and 1°, respectively. As shown in Figure 3(a), when H was fixed, optical efficiency at normal incidence decreased as B increased. Based on the simulation results shown in Figures 3(a) and 3(b), the feasible parameter sets H and B, (10, 12), (13, 12), (15, 12), and (18, 14), were selected for further investigation. The optical efficiency of the solar concentrator with SOE I of parameter set (18, 14) was good (74.9%) at incidence angle of 1°; however, the optical efficiency was only 80.8% at incidence angle of 0° (normal incidence). Therefore the parameter set (18, 14) was not selected as a feasible parameter set for SOE I.Figure 3
Effects of refractive pyramid SOE parameters on optical efficiency: (a) incidence angle = 0° (normal incidence) and (b) incidence angle = 1°.
(a)
(b)


## 2.1.2. SOE II: Reflective Pyramid

Figures4(a) and 4(b) show the effects of SOE II design parameters B and H on the optical efficiency of the solar concentrator at incidence angles of 0° (normal incidence) and 1°, respectively. Figure 4(a) indicates that optical efficiency at normal incidence decreased with an increase in H or B, except when H
=
13 and B
=
16. However, Figure 4(b) shows that the optical efficiency of SOE II was lower than 70% at an incidence angle of 1°. This phenomenon limits the applicability of SOE II.Figure 4
Effects of reflective pyramid SOE parameters on the optical efficiency: (a) incidence angle = 0° (normal incidence) and (b) incidence angle = 1°.
(a)
(b)


## 2.1.3. Discussion of Pyramid-Shaped SOE

Based on a comparison of Figures3(b) and 4(b), the refractive pyramid SOE generally exhibited higher optical efficiency than a reflective pyramid SOE at an incidence angle of 1°. This implies that using SOE I improved the acceptance angle of the concentrator more than SOE II. Consequently, the feasible parameter sets of SOE I were used to simulate the optical efficiency of the concentrator at various incidence angles; the results are depicted in Figure 5. Among the four feasible SOE I designs, two designs with the parameter sets H and B, (13, 12) and (15, 12), were preliminarily selected as optimal designs because their optical efficiencies were stable and remained higher than 60% even at an incidence angle of 1.2°.Figure 5
Optical efficiency with a refractive pyramid SOE under various incidence angles.


## 2.1.4. SOE III: Refractive Cone

Figures6(a) and 6(b) illustrate the effects of SOE III design parameters B (diameter of top area) and H (height) on optical efficiency at incidence angles of 0° (normal incidence) and 1°, respectively. Figure 6(a) shows that optical efficiency at normal incidence decreased as H increased. However, at an incidence angle of 1°, an increase in H resulted in increased optical efficiency (Figure 6(b)). Based on the parametric analyses of SOE III (Figure 6), feasible parameter sets H and B, (10, 12), (13, 12), (15, 12), and (18, 12), were selected for further investigation.Figure 6
Effects of refractive cone SOE parameters on the optical efficiency: (a) incidence angle = 0° (normal incidence) and (b) incidence angle = 1°.
(a)
(b)Figure7 illustrates the optical efficiency of the concentrator using the four feasible designs of SOE III at various incidence angles. Among the four feasible designs of SOE III, the design with the parameter set (H
=
18, B
=
14) was chosen as a preliminary optimal design because of its higher optical efficiency at large incidence angles. Based on a comparison of Figures 5 and 7, the optical efficiency of concentrators using SOE III generally dropped more rapidly than that of concentrators using SOE I as the deviation of the incidence angle increased.Figure 7
Optical efficiency with a refractive cone SOE under various incidence angles.


## 2.1.5. Discussion of Phase I SOE Design

The performance simulations of the two optimal designs of SOE I and one optimal design of SOE III are compared and summarized in Figure8 and Table 1. Figure 8 shows the optical efficiency of the concentrator using the three SOEs at various incidence angles. Table 1 summarizes the design parameters of the three optimal SOE designs and the performance of the concentrator using each of the designs, including optical efficiencies at incidence angles of 0° and 1° and acceptance angles θ
90
% and θ
50
%. As shown in Table 1 and Figure 8, at normal incidence, the refractive pyramid SOE design Q1 with the parameter sets H and B, (13, 12), enabled the concentrator to achieve the highest optical efficiency, 84.24%, compared with two other optimal SOE designs, Q2 (82.83%) and Q3 (82%). SOE Q3 also enabled the concentrator to achieve the highest corresponding acceptance angle θ
90
%, 1.06°, whereas the acceptance angles θ
90
% for Q2 and Q1 were 1.03° and 1.00°, respectively.Table 1
Optical performance of the two-reflector solar concentrator with preliminary optimal SOEs from Phase I and Phase II.
Phase
SOE
H (mm)
B (mm)
η
opt (at  0°)
η
opt (at  1°)
θ
90
%
θ
50
%
I
Q1
Refractive pyramid
13
12
84.24%
75.64%
1.00
1.39
Q2
Refractive pyramid
15
12
82.83%
75.00%
1.03
1.54
Q3
Refractive cone
18
14
82.00%
74.82%
1.06
1.51
II
D1*
Refractive pyramid
15
10
83.52%
80.88%
1.11
1.51
D2
Refractive cone
18
12
83.26%
78.08%
0.97
1.63
*The optimal design founded in this study.Figure 8
Optical efficiency with 3 optimal SOEs of Phase I under various incidence angles.The SOE designs produced in Phase I enabled the concentrator to achieve adequate acceptance angles and optical efficiencies. However, the uniformity of the irradiance distribution on the receiver was observed to be insufficient. In Phase II, the focal point of M2 was positioned at the top surface of the SOE to improve irradiance uniformity on the solar cell.


## 2.2. Phase II of SOE Design

In Phase II, the focal point of M2 was positioned on the top surface of the SOE to improve irradiance uniformity. Based on the results of Phase I, the preliminary optimal SOE types were observed to be the refractive pyramid and refractive cone. Consequently, Phase II considered only these two types of SOE to determine an SOE design enabling the concentrator to achieve both superior acceptance angles and irradiance uniformity. The two SOEs selected were (1) Type I: refractive pyramid-shaped and (2) Type II: refractive cone-shaped.


### 2.2.1. SOE I: Refractive Pyramid

Figures9(a) and 9(b) illustrate the effects of SOE I design parameters B (edge length of top area) and H (height) on the optical efficiency of the concentrator at incidence angles 0° (normal incidence) and 1°, respectively. As shown in Figure 9(a), when H is fixed, optical efficiency at normal incidence decreased as B increased. Based on the simulation results shown in Figures 9(a) and 9(b), the feasible parameter sets H and B, (13, 10) and (15, 10), respectively, were selected. These feasible parameter sets exhibited the highest optical efficiency at normal incidence and incidence angle deviations.Figure 9
Effects of refractive pyramid SOE parameters (Phase II) on the optical efficiency: (a) incidence angle = 0° (normal incidence) and (b) incidence angle = 1°.
(a)
(b)


### 2.2.2. SOE II: Refractive Cone

Figures10(a) and 10(b) show the effects of the SOE III design parameters B (diameter of top area) and H (height) on optical efficiency at incidence angles 0° and 1°, respectively. Figure 10(a) shows that, at normal incidence, optical efficiency decreased when H increased. At an incidence angle of 1°, however, an increase in H resulted in increased optical efficiency, as shown in Figure 10(b). Based on the simulation results of Figure 10, the feasible parameter set H and B, (18, 12), was selected.Figure 10
Effects of refractive cone SOE parameters (Phase II) on the optical efficiency: (a) incidence angle = 0° (normal incidence) and (b) incidence angle = 1°.
(a)
(b)


### 2.2.3. Discussion of Phase II SOE Design

Figure11 depicts optical efficiency at various incidence angles of the two optimal SOEs selected in Phase II. Table 1 summarizes the optical performance of the solar concentrator when using these two SOEs, including optical efficiency at incidence angles 0° and 1° and acceptance angles θ
90
% and θ
50
%. As shown in Table 1 and Figure 11, D1 enabled the solar concentrator to achieve a higher optical efficiency (83.52%) than D2 (83.25%) at normal incidence. D1 also enabled the concentrator to achieve a higher acceptance angle θ
90
% (1.11°) than D2 (0.97°).Figure 11
Optical efficiency with 2 optimal SOEs of Phase II under various incidence angles.After comparing the performance results of the SOEs, it was observed that D1 yielded the optimal optical performance. The irradiance distribution of D1 on the receiver under normal incidence is shown in Figure12; the irradiance uniformity is 67.14%.Figure 12
Irradiance distribution of the two-reflector solar concentrator with optimal SOE D1 of Phase II (normal incidence).


## 2.2.1. SOE I: Refractive Pyramid

Figures9(a) and 9(b) illustrate the effects of SOE I design parameters B (edge length of top area) and H (height) on the optical efficiency of the concentrator at incidence angles 0° (normal incidence) and 1°, respectively. As shown in Figure 9(a), when H is fixed, optical efficiency at normal incidence decreased as B increased. Based on the simulation results shown in Figures 9(a) and 9(b), the feasible parameter sets H and B, (13, 10) and (15, 10), respectively, were selected. These feasible parameter sets exhibited the highest optical efficiency at normal incidence and incidence angle deviations.Figure 9
Effects of refractive pyramid SOE parameters (Phase II) on the optical efficiency: (a) incidence angle = 0° (normal incidence) and (b) incidence angle = 1°.
(a)
(b)


## 2.2.2. SOE II: Refractive Cone

Figures10(a) and 10(b) show the effects of the SOE III design parameters B (diameter of top area) and H (height) on optical efficiency at incidence angles 0° and 1°, respectively. Figure 10(a) shows that, at normal incidence, optical efficiency decreased when H increased. At an incidence angle of 1°, however, an increase in H resulted in increased optical efficiency, as shown in Figure 10(b). Based on the simulation results of Figure 10, the feasible parameter set H and B, (18, 12), was selected.Figure 10
Effects of refractive cone SOE parameters (Phase II) on the optical efficiency: (a) incidence angle = 0° (normal incidence) and (b) incidence angle = 1°.
(a)
(b)


## 2.2.3. Discussion of Phase II SOE Design

Figure11 depicts optical efficiency at various incidence angles of the two optimal SOEs selected in Phase II. Table 1 summarizes the optical performance of the solar concentrator when using these two SOEs, including optical efficiency at incidence angles 0° and 1° and acceptance angles θ
90
% and θ
50
%. As shown in Table 1 and Figure 11, D1 enabled the solar concentrator to achieve a higher optical efficiency (83.52%) than D2 (83.25%) at normal incidence. D1 also enabled the concentrator to achieve a higher acceptance angle θ
90
% (1.11°) than D2 (0.97°).Figure 11
Optical efficiency with 2 optimal SOEs of Phase II under various incidence angles.After comparing the performance results of the SOEs, it was observed that D1 yielded the optimal optical performance. The irradiance distribution of D1 on the receiver under normal incidence is shown in Figure12; the irradiance uniformity is 67.14%.Figure 12
Irradiance distribution of the two-reflector solar concentrator with optimal SOE D1 of Phase II (normal incidence).


## 3. Sensitivity Analysis of Misalignments of Optical Elements

The effects of concentrator assembly errors on performance when using the optimal SOE, D1, were studied. The assembly errors considered included (1) tilt of M1, (2) axial displacement of M2 away from cell, (3) axial displacement of M2 toward cell, (4) radial displacement of M2, (5) tilt of M2, (6) axial displacement of cell away from M2, and (7) axial displacement of cell toward M2. Misalignments of element displacements were measured in mm, and misalignments of element tilt were measured in degrees.Figure13(a) shows how optical efficiency is affected by various element displacements, and Figure 13(b) shows how optical efficiency is affected by various element tilts. The displacement of M2 was the most crucial factor affecting optical efficiency, as shown in Figure 13(a). Figure 13(b) shows that optical efficiency drops dramatically if the tilt of M1 is larger than 0.8°.Figure 13
Optical efficiency under different assembly errors: (a) element displacement and (b) element tilt.
(a)
(b)Figure14 summarizes the sensitivity of optical efficiency to the seven assembly errors. Changes in optical efficiency resulting from a 1 mm element displacement or 1° of element tilt are plotted in Figure 14. As Figure 14 shows, optical efficiency is most sensitive to M1 tilt (Error 1), displacement of M2 toward the cell (Error 3), and displacement of M2 away from the cell (Error 2) accordingly.Figure 14
Change of optical efficiency under 7 assembly errors.


## 4. Conclusions

This study used a parametric design process to determine the optimal SOE for a noncoplanar, two-reflector solar concentrator. The optical performance of the solar concentrator when using several proposed SOE designs was investigated using ray-tracing simulation. Two design phases were conducted and five preliminary optimal SOE designs were selected: Q1, Q2, and Q3 from Phase I and D1 and D2 from Phase II. After comparing the optical performance of the SOE designs, D1 (refractive pyramid SOE) was selected as the optimal SOE for the noncoplanar solar concentrator. The optimal SOE, D1, enabled the concentrator to achieve a large acceptance angle (θ
90
%
=
1.11
°) and high optical efficiency (η
=
83.52
%) at normal incidence. The sensitivity to assembly errors of the solar concentrator when using the optimal SOE was also studied, and the results indicated that the tilt of M1 was the most critical assembly error. The parametric design process presented in this study can be further applied to achieve an optimal SOE for solar concentrators with various values of GCR, NA and dimensions.



---

*Source: 861353-2015-03-05.xml*