Performance and Design Optimization of Two-Mirror Composite Concentrating PV Systems

Abstract

The reflectors of a linear solar concentrator investigated in this work consisted of two plane mirrors (2MCC), and they were designed in such a way that made all radiation within the acceptance angle (θ_a) arrive on flat-plate absorber, after less than two reflections. To investigate the performance of an east–west aligned 2MCC-based photovoltaic (PV) system (2MCPV), a mathematical procedure was suggested based on the three-dimensional radiation transfer and was validated by the ray-tracing analysis. Analysis indicated that the performance of 2MCPV was dependent on the geometry of 2MCC, the reflectivity of mirrors (ρ), and solar resources in a site, thus, given θ_a, an optimal geometry of 2MCC for maximizing the annual collectible radiation (ACR) and annual electricity generation (AEG) of 2MCPV in a site could be respectively found through iterative calculations. Calculation results showed that when the ρ was high, the optimal design of 2MCC for maximizing its geometric concentration (C_g) could be utilized for maximizing the ACR and AEG of 2MCPV. As compared to similar compound parabolic concentrator (CPC)-based PV systems, the 2MCPV with the tilt-angle of the aperture yearly fixed (1T-2MCPV), annually generated more electricity when the ρ was high; and the one with the tilt-angle adjusted yearly four times at three tilts (3T-2MCPV), performed better when θ_a < 25° and ρ > 0.7, even in sites with poor solar resources.

1. Introduction

In recent decades, the energy consumption over the world has increased continuously due to the increase in standard of living and comfort level of peoples, which led to the increase of emission of carbon dioxide and environmental issues. The use of renewable energy resources, such as solar, wind, biomass, and geothermal can minimize the utilization of fossils fuels and environmental issues. Among all renewable sources, solar energy has gained tremendous interest due to their abundant availability in most parts of the world and because they are free from pollution [1,2]. However, compared to conventional techniques for electricity generation, the cost of the electricity from a PV system is higher and this limits its application [3]. Apart from further finding cheap materials and advance techniques for the production of solar cells, sun-tracking, and radiation concentrating techniques are widely used to reduce the cost of electricity generated by PV systems. Sun-tracking increases electricity generation from PV panels by maximizing radiation collection [4,5,6], but sophisticated devices in sun-tracking and control are required. In recent years, compound parabolic concentrators (CPCs) and V-trough concentrators were widely tested for increasing the electricity from solar cells. Compared to non-concentrating PV systems, a low concentrating PV system can reduce the cost of electricity by up to 40% [7]. Tests by Mallick et al. showed that the use of an asymmetric CPC (2.01×) increased the maximum power point of solar cells by 62% [8,9]. The experimental study by Brogren et al. indicated that the use of a CPC (3×) increased the maximum power output of solar cells by 90% [10]. A comparative study by Yousef et al. showed that, compared to similar solar panels, the electricity from a CPC (2.4×)-based PV system (CPV) was 52% and 33% higher, with and without cooling of solar cells, respectively [11]. These research studies showed that the use of CPCs can increase electricity generation of solar cells but the increments were much less than the geometric concentration ratio (C_g) of CPCs, a result mainly caused by the optical loss of CPCs due to multi-reflections of solar rays on the way to solar cells [12,13]. To simplify the fabrication of CPCs and make solar flux on solar cells more uniform, Tang proposed a concentrator (CPC-A) as the alternative to CPC where the parabolic reflectors are replaced with multi-mirrors [14].

Compared to CPCs, V-trough concentrators are easier in fabrication, and solar flux on the base is more uniform. Measurements by Sangani and Solanki indicated that the use of a V-trough concentrator (2×) increased the electricity generation of solar cells by 44% [15,16]. Theoretical and experimental studies showed that a properly designed V-trough PV system (VPV) was particularly suitable for water pumping [17,18]. A study by Vilela et al. showed that, the single axis sun-tracking VPV (2.2×) increased the ACR by a factor of 1.74 but increased the pumped water by a factor of 2.49, as compared to similar fixed solar panels [19]. Recent work of the authors showed that, as compared to fixed PV panels, the increase of AEG from inclined north–south axis multi-position sun-tracking VPV is even larger than the C_g [20]. However, V-trough is not an ideal solar concentrator, the C_g is restricted, and the optical efficiency is low [20,21,22]. To reduce the optical loss due to reflections, the reflection number of solar rays within V-trough should be restricted, but the C_g of such V-tough is restricted [20]. To increase the C_g of V-trough concentrators, Mannan and Bannerot proposed a two-mirror composite concentrator (2MCC) [23], but its optical performance was not investigated in this work. As an alternative to CPC, composite multi-mirror concentrators (n-MCC) was designed by Mullick et al. [24] based on the edge-ray principle. It is found that the maximum C_g of n-MCC increases with an increase of mirror number (n) and approaches to that of CPC as n is close to infinite. However, with an increase of n, the optical efficiency decreases, and solar flux on the absorber becomes more uneven [14].

To investigate the optical performance of a concentrator, the ray-tracing analysis is commonly employed [25,26], but it is time-consuming, even impossible when one investigates effects of geometry of a static solar concentrator on the long-term performance [14]. Similar to CPCs and V-trough concentrator, the optical performance of a linear n-MCC is uniquely determined by the projected incident angle (PIA, termed as θ_p) of solar rays on the cross-section [12,27,28], thus, two-dimensional radiation transfer can reasonably predict the optical performance [12]. However, the 2-D model cannot reasonably predict photovoltaic performance as the photovoltaic efficiency of solar cells is sensitive to the incident angle (IA) of solar rays on solar cells [6,12]. In this work, a 3-D radiation transfer model was developed based on the vector algebra and imaging principle of mirrors. The objective of this work was to find the optimal design of 2MCC based PV system (2MCPV) for maximizing ACR and AEG and compare its performance with a similar CPC based PV system (CPV), which was identical in C_g and θ_a to 2MCC.

2. Geometric Characteristics of n-MCC

Reflectors of a solar concentrator are commonly designed based on the edge-ray principle, namely, edge rays reflecting from reflectors are required to hit at the ends of a plat absorber or to be tangent to a curved absorber [29]. As shown in Figure 1, each of the two reflectors of the linear concentrator investigated here consisted of two plane mirrors, and they were designed in such a way that it made the solar ray incident on the upper end of two mirrors at θ_p = θ_a arrive on the left end of the flat-plate absorber. Obviously, for such solar concentrator (2MCC), all radiation irradiating on the mirrors at θ_p ≤ θ_a would arrive on the absorber, after less than two reflections. According to the reflection law of light, the geometry of 2MCC should be subjected to (the width of the absorber was set to be 1 for simplifying the analysis):

C_g = 2(h₁ + h₂)tanα₂ − 1 = 2htanα₂ − 1

(1)

where h = h₁ + h₂ is the height of 2MCC, h₁ and h₂ are the vertical height of lower and upper mirrors, respectively, and they are given by:

$$h_1 = \frac{\cos {\alpha }_1\cos {\gamma }_1}{\sin ({\theta }_a+{\gamma }_1)}$$

(2)

$$h_2 = \frac{h_1(\tan {\alpha }_1-\tan {\alpha }_2)}{\tan {\alpha }_2-\tan {\gamma }_2}$$

(3)

where the γ₁ and γ₂ are, respectively, the tilt-angle of lower and upper mirrors relative to the normal of absorber (x-axis), and α₁ = θ_a + 2γ₁ and α₂ = θ_a + 2γ₂ are, respectively, the incident angle of solar rays on the absorber, after reflection from lower and upper mirrors, when θ_p = θ_a. Similarly, for the concentrator consisted of n-mirrors (n-MCC), the geometry was subjected to:

C_g = 2h_ntanα_n − 1

(4)

The vertical height (h₁) of the bottom mirror is given by Equation (2), and that of the jth mirror counting from the bottom is given by:

$$h_j = \frac{(\tan {\alpha }_{j-1}-\tan {\alpha }_j)}{\tan {\alpha }_j-\tan {\gamma }_j}{\displaystyle \sum _{i=1}^{j-1}{h}_i}·(j=2,3,\dots ,n)$$

(5)

α_j = θ_a + 2γ_j (j = 1, 2, …, n)

(6)

It is known from Equations (1)–(3) that, given θ_a, the geometry of 2MCC is dependent on γ₁ and γ₂, thus, a set of γ₁ and γ₂ for maximizing C_g could be found through the iterative calculations. Similarly, C_g of n-MCC was sensitive to the tilt-angle of n-mirrors and a set of γ_i (i = 1, 2, …, n) for maximizing C_g could be found though multi-loops iterative calculations.

As seen in Figure 2, given θ_a, the C_g of n-MCC increased with an increase of mirror numbers (n), as indicated by Mullick [24]. The optimal geometry of 2MCC and 3MCC for maximizing C_g is presented in Figure 3. It can be seen that the optimal tilt-angle of bottom mirror of n-MCC for maximizing C_g (termed as γ_1,g) was highly sensitive to θ_a and decreased with an increase of θ_a, but that of the top mirror (termed as γ_n,g) was weakly sensitive to θ_a, furthermore, γ_1,g was always larger than nγ_n,g. It was also seen that the maximum C_g of 2MCC and 3MCC decreased with an increase of θ_a. To ensure C_g > 2, the θ_a of 2MCC and 3MCC should be less than 17.7° and 21.3°, respectively. Just like CPCs, linear n-MCC was commonly oriented in the east–west direction. Therefore, to ensure C_g > 2 and the sun within θ_a of 2MCC and 3MCC for more than 7 h in all days of a year, the tilt-angle of 2MCCs’ aperture should be yearly adjusted four times at three tilts, and that of 3MCC’s aperture should be adjusted more than two times in a year [27,30].

3. Mathematical Procedure to Predict the Performance of 2MCPV

The 2MCC orientation was assumed to be in the east–west direction and its aperture to be tilted at β from the horizon. To make the analysis easier, a coordinate system with the x-axis normal to the aperture, the y-axis parallel to the horizon and pointing to the east, and the z-axis pointing to the northern sky was employed (see Figure 4). In this coordinate system, the unit vector of the incident solar rays was expressed by [6,31]:

$$n_s = (n_x, n_y,n_z)$$

(7)

$$\left\{\begin{array}{l}{n}_x=\cos \delta \cos \omega \cos (\lambda -\beta )+\sin \delta \sin (\lambda -\beta )\hfill \\ n_y=-\cos \delta \sin \omega \hfill \\ n_z=-\cos \delta \cos \omega \sin (\lambda -\beta )+\sin \delta \cos (\lambda -\beta )\hfill \end{array}\right.$$

(8)

where λ is the site latitude, ω is the hour angle, and δ is the declination of the sun, which was determined by the day of the year [29]. The PIA of solar rays on the cross-section of the linear 2MCC was calculated by:

$$\tan {\theta }_p = \left|\frac{n_z}{n_x}\right|$$

(9)

For the 2MCC symmetric about the normal to aperture, the performance for solar rays n_s = (n_x, n_y, ± n_z) was identical. Therefore, to simplify the analysis, it was assumed that solar rays were always incidental onto the right mirrors (see Figure 4), namely, n_swas always set to be $n_s = (n_x,n_y, -\left|n_z\right|).$

3.1. Optical and Photovoltaic Efficiency of 2MCC-Based PV System

The radiation on the absorber of 2MCC include five components—radiation directly incident on the absorber (I₁) and those irradiating first on four mirrors and then arriving on the absorber after reflections (I₂, I₃, I₄, I₅). Therefore, the optical efficiency of 2MCC was expressed by:

f = (I₁ + I₂ + I₃ + I₄ + I₅)/I_ap = f₁ + f₂ + f₃ + f₄ + f₅

(10)

where I_ap is the radiation incident on the aperture; f_i (i = 1, 2, 3, 4, 5) is the energy fraction of the radiation on the absorber contributed by I_i. Similarly, the PV conversion efficiency of the 2MCC-based PV system (2MCPV) was expressed by:

η = (P₁ + P₂ + P₃ + P₄ + P₅)/I_ap = η₁ + η₂ + η₃ + η₄ + η₅

(11)

where P_i (i = 1, 2, 3, 4, 5) is the electricity generated by I_i, and the η_i (i = 1, 2, 3, 4, 5) is the PV conversion efficiency of 2MCPV contributed by P_i.

3.1.1. Radiation Directly Irradiating on the Absorber

As seen from Figure 4, the absorber was fully irradiated when θ_p ≤ γ₀ and partially irradiated as γ₀ < θ_p < α₂, thus, f₁ = I₁/I_ap = Δ_z1/C_g was calculated by:

$$f_1 = \left\{\begin{array}{cc}1/C_g\hfill & ({\theta }_p\le {\gamma }_0)\\ h(\tan {\alpha }_2-\tan {\theta }_p)/C_g\hfill & ({\gamma }_0<{\theta }_p<{\alpha }_2)\hfill \\ 0& ({\theta }_p\ge {\alpha }_2)\end{array}\right.$$

(12)

As shown in Figure 4, the γ₀ could be calculated by:

tanγ₀ = 0.5(C_g − 1)/h

(13)

The electricity from 2MCPV generated by I₁ was given by P₁ = I₁η_pv(θ_in,1), thus, the one had:

η₁ = f₁η_pv(θ_in,1)

(14)

where η_pv(θ_in,1) is the PV efficiency of solar cells as a function of θ_in,1. The IA of the solar rays directly irradiating on the solar cells is given by:

$$\cos {\theta }_{in,1} = n_s·n_{abs} = n_s·(1,0,0) = n_x$$

(15)

as the unit vector of the normal to solar cells of 2MCPV is n_abs = (1,0,0) in the suggested coordinate system. The electricity from 2MCPV is generally affected by many factors such as cell temperature and electricity load [32]. To simplify the analysis and facilitate investigating the effects of the geometry of 2MCC on the PV performance of 2MCPV, it was assumed that, except the IA (θ_in), effects of the other factors on the PV efficiency of solar cells were not considered, and the PV efficiency of solar cells in percentage was subjected to the correlation, as follows [33]:

$$n_{pv} = \left\{\begin{array}{cc}15.5494+0.002325{\theta }_{in}-0.00301{\theta }_{in}^2+9.4685\times {10}^{-5}{\theta }_{in}^3-1.134\times {10}^{-6}{\theta }_{in}^4\hfill & ({\theta }_{in}\le {65}^{°})\hfill \\ 41.52-0.4784{\theta }_{in}\hfill & ({65}^{°}<{\theta }_{in}<{90}^{°})\hfill \end{array}\right.$$

(16)

3.1.2. Radiation Irradiating on Right Lower Mirror and Arriving on the Absorber

According to the geometric characteristics of 2MCC and the imaging principle of mirrors, one knows that all radiation irradiating on the right lower mirror arrives on the absorber after one reflection when θ_p ≤ θ_a, whereas for the case of θ_p > θ_a, a fraction of radiation would arrive after more than one reflection, as seen in Figure 5 [22]. For the case of θ_p ≤ θ_a, the radiation incident on right lower mirror and arriving on the absorber after reflection was I₂ = Δz₂ρ, hence, one had:

f₂ = Δz₂ρ/C_g = h₁(tanθ_p + tanγ₁)ρ/C_g (θ_p ≤ θ_a)

(17)

η₂ = Δz₂ρη_pv(θ_in,21)/C_g = f₂η_pv(θ_in,21) (θ_p ≤ θ_a)

(18)

where η_pv(θ_in,21) is the PV efficiency of solar cells as the function of θ_in,21 and is determined by Equation (16). The IA of the solar ray on solar cells in this case (see left of Figure 5) was given by:

cosθ_in,21 = n_s · (cos2γ₁,0,sin2γ₁)= n_x cos2γ₁+n_z sin2γ₁

(19)

as the unit vector of the normal to first right-image of the absorber formed by lower mirrors is (cos2γ₁,0,sin2γ₁), due to the first right-image making an angle of 2γ₁ from the absorber [12,20,22].

When solar rays are incident on the aperture at θ_p > θ_a, a fraction of radiation incident on the right lower mirror might arrive on the absorber after more than one reflection, as shown in the right side of Figure 5. The PIA of solar rays on the absorber after multi-reflections within the V-trough formed by the two lower mirrors was equal to (2γ₁k + θ_p) and was required to be less than 90°, thus the maximum reflection number of solar rays within the V-trough was given by:

k_max = Int [(90 − θ_p)/2γ₁] ≤ Int [(90 − θ_a)/2γ₁]

(20)

as multi-reflections take place only when θ_p > θ_a. Calculations showed that γ₁ of the 2MCC optimized for maximizing the C_g was larger than 17° for θ_a < 40°, thus k_max was 2. This meant that more than two reflections would not take place for θ_a < 40°. In practical applications, θ_a should be less than 40° to ensure C_g > 1.3. Therefore, f₂ and η₂ for the case of θ_p > θ_a could be calculated by:

f₂ = (Δz₂₁ ρ + Δz₂₂ ρ ²)/C_g

(21)

η₂ = [Δz₂₁ ρ η_pv(θ_in,21) + Δz₂₂ ρ²η_pv(θ_in,22)]/C_g

(22)

where Δz₂₁ and Δz₂₂ are the radiation that irradiates on the right lower mirror and arrives on the absorber after one and two reflections, respectively (see Figure 5). They were calculated by [20,22]:

Δz₂₁ = F_R,1[1 − tan(2γ₁)tanθ_p]

(23)

Δz₂₂ = F_R,2[1 − tan(4γ₁)tanθ_p]

(24)

where F_R1 and F_R2 are the z-coordinate differences between two ends of the right first and second images irradiated by the incident radiation, respectively, and they were given by:

$$F_{R,k} = \left\{\begin{array}{ll}0\hfill & (z_{e,k}-z_{s,k}\le 0)\hfill \\ z_{e,k}-z_{s,k}\hfill & (0<z_{e,k}-z_{s,k}\le \cos 2k{\gamma }_1)\hfill \\ \cos 2k{\gamma }_1\hfill & (z_{e,k}-z_{s,k}>\cos 2k{\gamma }_1)\hfill \end{array}\right.·(k = 1,2)$$

(25)

where

$$\left\{\begin{array}{l}{z}_{s,1}=Max[0.5,(0.5C_{g}c\tan {\theta }_p+0.5\tan 2{\gamma }_1-h)/(\tan 2{\gamma }_1-c\tan {\theta }_p)]\hfill \\ z_{e,1}=0.5+\cos 2{\gamma }_1\hfill \end{array}\right.$$

(26)

$$\left\{\begin{array}{l}{z}_{s,2}=Max\{0.5+\cos 2{\gamma }_1,[0.5C_{g}c\tan {\theta }_p+(0.5+\cos 2{\gamma }_1)\tan 4{\gamma }_1-h-\sin 2{\gamma }_1]/(\tan 4{\gamma }_1-c\tan {\theta }_p)\}\hfill \\ z_{e,2}=Min\{0.5+\cos 2{\gamma }_1+\cos 4{\gamma }_1,[\sin 2{\gamma }_1+(0.5+\cos 2{\gamma }_1)\tan 4{\gamma }_1-0.5C_{1}c\tan {\theta }_p-h_1]/(\tan 4{\gamma }_1-c\tan {\theta }_p)\}\hfill \end{array}\right.$$

(27)

The C₁ = 2h₁tanα₁ − 1 was the geometric concentration of 1MCC (V-trough formed by two lower mirrors of 2MCC). The θ_in,22 in Equation (22) was the IA of the solar rays after two reflections and was determined by:

cosθ_in,22 = n_s· (cos4γ₁,0,sin4γ₁) = n_x cos4γ₁ + n_z sin4γ₁

(28)

as the unit vector of the normal to second right-image was expressed by (cos4γ₁, 0, sin4γ₁). It was noted that f₂ and η₂ were set to be zero when θ_p ≥ 90 − 4γ₁ or θ_p ≥ θ_max. The θ_max was the angle of the edge-ray that irradiated on lower mirrors and arrived on the absorber, after more than one reflection, and was determined by:

θ_max = Max(θ₁,θ₂)

(29)

where θ₁ and θ₂ are, respectively, the angle of rays passing the left/right-end of the aperture and the right/left-end of first and second right/left images, and was calculated by:

tanθ₁ = (0.5 + 0.5C_g + cos2γ₁)/(h + sin2γ₁)

(30)

tanθ₂ = (0.5 + 0.5C_g + cos2γ₁ + cos4γ₁)/(h + sin2γ₁ + sin4γ₁)

(31)

3.1.3. Radiation Irradiating on the Left Lower Mirror and Arriving on the Absorber

First, it should be addressed that all radiation irradiating on the left mirrors would arrive on the absorber after less than two reflections, because θ_p was negative, thus, it was less than θ_a, as the radiation was always assumed to be incident onto the right mirrors, in this work. As seen in Figure 6, the left lower mirror was fully irradiated when θ_p ≤ γ₂, and partially irradiated for γ₂ < θ_p < γ₀. Thus one had:

$$f_3 = \Delta z3/C_g = \left\{\begin{array}{ll}{h}_1(\tan {\gamma }_1-\tan {\theta }_p)\rho /C_g\hfill & ({\theta }_p\le {\gamma }_2)\hfill \\ (0.5C_g-0.5-h\tan {\theta }_p)\rho /C_g\hfill & ({\gamma }_2<{\theta }_p<{\gamma }_0)\hfill \\ 0\hfill & ({\theta }_p\ge {\gamma }_0)\hfill \end{array}\right.$$

(32)

The IA of the solar rays after reflection was given by:

cosθ_in,3 = n_s· (cos2γ₁,0, −sin2γ₁) = n_x cos2γ₁ − n_z sin2γ₁

(33)

as the unit vector of the normal to the image formed by left lower mirror was (cos2γ₁,0, −sin2γ₁) [12]. Therefore, the η₃ was given by:

η₃ = f₃η_pv(θ_in,3)

(34)

3.1.4. Radiation Irradiating on Left Upper Mirror and Arriving on Solar Cells

As shown in Figure 7, the CD’ was the image of the lower mirror (CD) formed by the upper mirror (AC). According to the imaging principle of mirrors, one knows that when θ_p ≥ ${\gamma }_{A{D}^{\prime }}$, all radiation incident on the upper mirror would redirect onto the lower mirror first and then redirect onto solar cells; whereas for θ_p < ${\gamma }_{A{D}^{\prime }}$, radiation incident on the upper part (AE) of the upper mirror directly redirected onto solar cells and that on the lower part (EC) redirected onto the lower mirror (CD) first, then redirected onto the solar cells. Therefore, f₄ could be calculated by:

$$f_4 = \left\{\begin{array}{ll}(\tan {\gamma }_2-\tan {\theta }_p)[(h_2-h_{EC})\rho +h_{EC}{\rho }^2]/C_g\hfill & ({\theta }_p<{\gamma }_{A{D}^{\prime }})\hfill \\ h_2(\tan {\gamma }_2-\tan {\theta }_p){\rho }^2/C_g\hfill & ({\gamma }_{A{D}^{\prime }}\le {\theta }_p<{\gamma }_2)\hfill \\ 0\hfill & ({\theta }_p\ge {\gamma }_2)\hfill \end{array}\right.$$

(35)

where ${\gamma }_{A{D}^{\prime }}$ is the angle of line $A{D}^{\prime }$ relative to the x-axis and determined by:

$$\tan {\gamma }_{A{D}^{\prime }} = \frac{h_2\tan {\gamma }_2-C{D}^{\prime }\sin ({\gamma }_1-2{\gamma }_2)}{h_2+C{D}^{\prime }\cos ({\gamma }_1-2{\gamma }_2)}$$

(36)

as the image $C{D}^{\prime }$ of lower mirror made an angle of γ₁ − 2γ₂ relative to x-axis. The $C{D}^{\prime }$ in Equation (36) was the width of image $C{D}^{\prime }$ and was given by $C{D}^{\prime }$ = h₁/cosγ₁. The h_EC in Equation (34) was the vertical height of the mirror EC and was calculated by:

$$h_{EC} = \frac{C{D}^{\prime }[\sin ({\gamma }_1-2{\gamma }_2)+\cos ({\gamma }_1-2{\gamma }_2)\tan {\theta }_p]}{\tan {\gamma }_2-\tan {\theta }_p}$$

(37)

The IA of the solar rays directly redirecting onto the solar cells was calculated by:

cosθ_in,41 = n_s· (cos2γ₂,0, −sin2γ₂) = n_x cos2γ₂ – n_z sin2γ₂

(38)

as the vector of the normal to the image of solar cells formed by left upper mirror was (cos2γ₂,0, −sin2γ₂). The IA of the solar rays redirecting onto the lower mirror (CD) first, then onto the solar cells was calculated by [12]:

cosθ_in,42 = n_x cos (2γ₁ − 2γ₂) + n_z sin (2γ₁ − 2γ₂)

(39)

On knowing θ_in,41 and θ_in,42, one could calculate η₄ as follows:

$${\eta }_4 = \left\{\begin{array}{ll}(\tan {\gamma }_2-\tan {\theta }_p)[(h_2-h_{EC})\rho {\eta }_{pv}({\theta }_{in,41})+h_{EC}{\rho }^2{\eta }_{pv}({\theta }_{in,42})]/C_g\hfill & ({\theta }_p<{\gamma }_{A{D}^{\prime }})\hfill \\ h_2(\tan {\gamma }_2-\tan {\theta }_p){\rho }^2{\eta }_{pv}({\theta }_{in,42})/C_g\hfill & ({\gamma }_{A{D}^{\prime }}\le {\theta }_p<{\gamma }_2)\hfill \\ 0\hfill & ({\theta }_p\ge {\gamma }_2)\hfill \end{array}\right.$$

(40)

3.1.5. Radiation Irradiating on the Right Upper Mirror and Arriving on the Absorber

As seen in Figure 8, the methods through which the solar rays irradiate on the right upper mirror and arrive on the absorber, differed for different θ_p and were divided into four cases, as follows.

• Case A: θ_p < γ₁ − 2γ₂

As shown in Figure 8a, the image $B{D}^{\prime }$ of the right lower mirror (BD) formed by the right upper mirror made an angle of γ₁ − 2γ₂ from the x-axis, therefore, when the solar rays were incident on the upper mirror at θ_p < γ₁ − 2γ₂, the radiation irradiating on the upper part (AE) of the mirror directly redirected onto the solar cells, and that on the lower part (EB) redirected onto the lower mirror (BD) first, and then onto the solar cells. The f₅ and η₅ in this case were calculated by

$$f_5 = \frac{\Delta z_{AE}\rho +\Delta z_{EB}{\rho }^2}{C_g} = \frac{(\tan {\gamma }_2+\tan {\theta }_p)(h_2-h_{EB}+h_{EB}\rho )\rho }{C_g}$$

(41)

$${\eta }_5 = \frac{(\tan {\gamma }_2+\tan {\theta }_p)[(h_2-h_{EB}){\eta }_{pv}({\theta }_{in,51})+h_{EB}\rho {\eta }_{pv}({\theta }_{in,52})]\rho }{C_g}$$

(42)

The vertical height of EB was given by:

$$h_{EB} = \frac{[\sin ({\gamma }_1-2{\gamma }_2)-\cos ({\gamma }_1-2{\gamma }_2)\tan {\theta }_p]h_1}{(\tan {\gamma }_2+\tan {\theta }_p)\cos {\gamma }_1}$$

(43)

The IA of the solar rays on the solar cells reflecting from AE (θ_in,51) and those reflecting from EB (θ_in,52) were calculated by [12]:

cos θ_in,51 = n_s ·(cos2γ₂,0, sin2γ₂) = n_x cos2γ₂ + n_z sin2γ₂

(44)

cosθ_in,52 = n_x cos(2γ₁ − 2γ₂) – n_z sin(2γ₁ − 2γ₂)

(45)

• Case B: γ₁ − 2γ₂≤θ_p ≤ θ_a

In this case, all radiation incident on the right upper mirror directly redirected onto the solar cells, as shown in Figure 8b, therefore, one had:

f₅ = Δz_ABρ/C_g = h₂(tanγ₂ + tanθ_p)ρ/C_g

(46)

η₅ = f₅η_pv(θ_in,51)

(47)

• Case C: θ_a < θ_p < θ_max − 2γ₂

As shown in Figure 8c,d, the angle of line AC relative to the x-axis was equal to α₂ and the angle of rays reflecting from right upper mirrors was θ_p + 2γ₂. Therefore, when α₂ < θ_p + 2γ₂ < θ_max, i.e., θ_a < θ_p < θ_max − 2γ₂, the radiation incident on the lower part (EB) of the right upper mirror would arrive on the solar cells after one reflection and that on the upper part (AE) would redirect onto the left lower mirror first then onto the absorber, after more than one reflection. Thus, the radiation incident on right upper mirror at θ_a < θ_p < θ_max − 2γ₂ and arriving on the solar cells included two components—directly redirecting on solar cells (I_5,1) and redirecting to the left lower mirror first, then onto the solar cells (I_5,2). The I_5,1 could be calculated by:

$$I_{\mathrm{5,1}}/I_{ap} = {\Delta }_{EB}\rho /C_g = \left\{\begin{array}{ll}(h_2-h_{AE})(\tan {\gamma }_2+\tan {\theta }_p)\rho /C_g\hfill & ({\theta }_a<{\theta }_p\le {\theta }_a+2{\gamma }_1-2{\gamma }_2)\hfill \\ 0\hfill & ({\theta }_p>{\theta }_a+2{\gamma }_1-2{\gamma }_2)\hfill \end{array}\right.$$

(48)

as the IA of the ray reflecting from the lower end (B) of the right upper mirror should be less than α₁, namely θ_p + 2γ₂ ≤ θ_a + 2γ₁ (θ_p ≤ θ_a + 2γ₁ − 2γ₂), thus, no radiation arrived on the solar cells after reflection from the right upper mirror, when θ_p > θ_a + 2γ₁ − 2γ₂. The electricity generated by I_5,1 was given by:

P_5,1/I_ap = I_5,1 η_pv(θ_in,51)

(49)

As shown in Figure 8d, when θ_a < θ_p < θ_max − 2γ₂, the rays reflecting from the upper part (AE) of the right upper mirror redirected onto the left lower mirror first and then onto the solar cells, after more than one reflection. The I_5,2 and P_5,2 were calculated by:

I_5,2/I_ap = (Δz_L1ρ² + Δz_L2ρ³)/C_g

(50)

P_5,1/I_ap = [Δz_L1ρ²η_pv(θ_in,L1) + Δz_L2ρ³ η_pv(θ_in,L2)]/C_g

(51)

where Δz_L1 and Δz_L2 were the radiation reflecting from the right upper mirror and arriving on the solar cells after one- and two-reflection, respectively, and were calculated as follows:

Δz_L1 = F_L1cos(θ_p + 2γ₂) [1 − tan(2γ₁)tan(θ_p + 2γ₂)]/cosθ_p

(52)

Δz_L2 = F_L2cos(θ_p + 2γ₂) [1 − tan(4γ₁)tan(θ_p + 2γ₂)]/cosθ_p

(53)

where F_L,1 and F_L,2 are the z-coordinate differences between the two ends of the first and second left-image of the absorber irradiated by the radiation from AB, respectively, and they were given by:

$$F_{L,k} = \left\{\begin{array}{ll}0\hfill & ({\zeta }_{s,k}-{\zeta }_{e,k}\le 0)\hfill \\ {\zeta }_{s,k}-{\zeta }_{e,k}\hfill & (0<{\zeta }_{s,k}-{\zeta }_{e,k}\le \cos 2k{\gamma }_1) (k = \mathrm{1,2})\hfill \\ \cos 2k{\gamma }_1\hfill & ({\zeta }_{s,k}-{\zeta }_{e,k}>\cos 2k{\gamma }_1)\hfill \end{array}\right.$$

(54)

where

$$\left\{\begin{array}{l}{\zeta }_{s,1}=Min[-0.5,\frac{h_1-0.5\tan 2{\gamma }_1-0.5C_{1}c\tan ({\theta }_p+2{\gamma }_2)}{\tan 2{\gamma }_1-c\tan ({\theta }_p+2{\gamma }_2)}]\hfill \\ {\zeta }_{e,1}=Max[-0.5-\cos 2{\gamma }_1,\frac{h-0.5\tan 2{\gamma }_1-0.5C_{g}c\tan ({\theta }_p+2{\gamma }_2)}{\tan 2{\gamma }_1-c\tan ({\theta }_p+2{\gamma }_2)}]\hfill \end{array}\right.$$

(55)

$$\left\{\begin{array}{l}{\zeta }_{s,2}=Min[-0.5-\cos 2{\gamma }_1,\frac{h_1+\sin 2{\gamma }_1-(0.5+\cos 2{\gamma }_1)\tan 4{\gamma }_1-0.5C_{1}c\tan ({\theta }_p+2{\gamma }_2)}{\tan 4{\gamma }_1-c\tan ({\theta }_p+2{\gamma }_2)}]\hfill \\ {\zeta }_{e,2}=Max[-0.5-\cos 2{\gamma }_1-\cos 4{\gamma }_1,\frac{h+\sin 2{\gamma }_1-(0.5+\cos 2{\gamma }_1)\tan 4{\gamma }_1-0.5C_{g}c\tan ({\theta }_p+2{\gamma }_2)}{\tan 4{\gamma }_1-c\tan ({\theta }_p+2{\gamma }_2)}]\hfill \end{array}\right.$$

(56)

The IA of the solar rays from AE and arriving on the solar cells after one- and two-reflection was equal to that of rays from AE on the first and second left-image of solar cells, formed by the lower mirrors, respectively, and they were calculated by [12]:

cosθ_in,L1 = r_AE ·(cos2γ₁,0, −sin2γ₁) = n_x cos(2γ₁ + 2γ₂) + n_z sin(2γ₁ + 2γ₂)

(57)

cosθ_in,L2 = r_AE ·(cos4γ₁,0, −sin4γ₁) = n_x cos(4γ₁ + 2γ₂) + n_z sin(4γ₁ + 2γ₂)

(58)

as the unit vector of the ray reflecting from AE was given by r_AE = (cos2γ₂, n_y, n_z + 2cosθ_in,AEcosγ₂) and the IA of the solar rays on the AE was given by cosθ_in,AE = n_s ·(sinγ₂,0, −cosγ₂) = n_x sinγ₂ − n_z cosγ₂. It was noted that Δz_L1 = 0 when θ_p > 90° − 2γ₁ − 2γ₂ and Δz_L2 = 0 as θ_p > 90° − 4γ₁ − 2γ₂, because the PIA of the solar rays from the AE on the first and second left-images was equal to θ_p + 2γ₂ + 2γ₁ and θ_p + 2γ₂ + 4γ₁, respectively, and they were required to be less than 90°. On obtaining I_5,1, P_5,1, I_5,2 and P_5,2, one had:

f₅ = (I_5,1 + I_5,2)/I_ap

(59)

η₅ = (P_5,1 + P_5,2)/I_ap

(60)

The analysis in above showed that the optical efficiency of 2MCPV only depended on θ_p, but the PV efficiency depended on θ_p and n_x.

3.2. Annual Optical and Photovoltaic Performance of 2MCPVs

It was assumed that the diffuse radiation from the sky was isotopic and that reflection from the ground was not considered. Therefore, the radiation on the unit area of the solar cells of 2MCPV at any moment of a day could be calculated by:

$$I_{abs} = C_g{I}_{b}g({\theta }_{ap})f\cos {\theta }_{ap}+I_{abs,d} = C_g{I}_{b}g({\theta }_{ap})n_{x}f + I_{abs,d}$$

(61)

as the IA of the solar rays on the aperture was given by cosθ_ap = n_s ·(1,0,0) = n_x. The electricity generated by the unit area of the solar cells of 2MCPVs at any time of a day was calculated by:

$$P = C_g{I}_{b}g({\theta }_{ap})n_x\eta +P_d$$

(62)

where I_b is the intensity of beam radiation, the $g({\theta }_{ap})$, a control function, is 1 for cosθ_ap > 0 otherwise zero. The I_abs,d in Equation (61) is the sky diffuse radiation collected by the unit area of the solar cells and could be calculated on the basis of the two-dimensional sky diffuse radiation, as follows:

$$I_{abs,d} = 0.5I_d{C}_g{\displaystyle {\int }_{-(0.5\pi -\beta )}^{0.5\pi }f({\theta }_p)\cos {\theta }_{p}d{\theta }_p} = C_d{I}_d$$

(63)

as the optical efficiency of the linear 2MCPV was only dependent on θ_p, and the directional intensity of the sky diffuse radiation on the cross-section of the east–west oriented 2MCPV was isotropic [6] for the isotropic sky diffuse radiation and was equal to 0.5I_d [13]. P_d in Equation (62) was the electricity generated by I_abs,d and should be calculated on the basis of the three-dimensional sky diffuse radiation, as follows [12]:

$$P_d = \frac{2C_g{I}_d}{\pi }{\int }_{{ɸ}_0}^{\pi }dɸ{\displaystyle {\int }_0^{0.5\pi }\sin \theta \cos \theta \eta d\theta } = C_{d,pv}{I}_d$$

(64)

as the PV efficiency of 2MCPV depended on θ_p and n_x, and the electricity generated by diffuse radiation from a finite element of the sky dome was dP_d = i_dC_gcosθηsinθ $d\theta dɸ$ with i_d = πI_d/3 [12]. The φ₀ in Equation (64) was related to θ by:

$$\left\{\begin{array}{ll}{ɸ}_0=0\hfill & (\theta \le 0.5\pi -\beta )\hfill \\ \cos {ɸ}_0=c\tan \beta c\tan \theta \hfill & (0.5\pi -ɸ<\theta \le 0.5\pi )\hfill \end{array}\right.$$

(65)

The I_d in Equations (63)–(64) was the sky diffuse radiation on the horizon. For a given 2MCPV, C_d in Equation (63) and C_d,pv in Equation (64) depended on β and could be numerically calculated. The radiation on the unit area of the solar cells of 2MCPV in a day could be estimated by integrating I_abs over the daytime of the day, as:

$$H_{day} = C_g{\displaystyle {\int }_{-t_0}^{t_0}{I}_{b}g({\theta }_{ap})n_{x}fdt}+C_d{H}_d$$

(66)

The daily electricity from unit area of the solar cells was estimated by:

$$P_{day} = C_g{\displaystyle {\int }_{-t_0}^{t_0}{I}_{b}g({\theta }_{ap})n_x\eta dt}+C_{d,pv}{H}_d$$

(67)

where the H_d in Equations (66)–(67) is the daily sky diffuse radiation on the horizon, t₀ is the sunset time on the horizon in a day.

At any time of a day, the position of the sun in terms of n_s could be determined, then f and η could be calculated. Therefore, given the time variation of I_b and H_d in a day, H_day, and P_day could be numerically calculated, then summing H_day and P_day and all days of a year yields the ACR on the solar cells (S_a) and AEG (P_a) of 2MCPV.

4. Methodology

To evaluate the mathematical model suggested in this work, the optical efficiency of the 2MCC calculated, based on the mathematical method in this work were compared with those from the ray-tracing analysis, with the aid of the commercial Tracepro software provided by Lambda Research, NASA of US [14].

To find the optimal geometry of 2MCPV for maximizing S_a and P_a, the monthly global radiation on the horizon in Beijing (λ = 39.95°, a site with abundant solar resources) and Chongqing (λ = 29.5°, a site with poor solar resources) were used for calculations [34]. Given the global radiation on the horizon for a month, the monthly average daily radiation on the horizon was estimated by dividing the monthly value over the days of the month, then, the monthly average H_d and the time variation of I_b in a day of the month were estimated, based on correlations proposed by Collares–Pereira and Rabl [35]. The time interval for calculating the daily radiation on the solar cells and the daily electricity from 2MCPV was set to be 1 min. The step of θ and φ for calculating C_d,pv in Equation (64) was set to be 0.1°, and the step of γ₁ and γ₂ for finding the optimal geometry of 2MCC for maximizing S_a and P_a was taken to be 0.1°. To fully investigate the performance of 2MCPV, 2MCPVs with the aperture’ tilt-angle being yearly fixed (1T-2MCPV) and yearly adjusted, four times, at three-tilts (3T-2MCPV) were considered. For 1T-2MCPV, the tilt-angle (β) was set to be site latitude (λ). For 3T-2MCPV, β = λ during the period of 22 days, before and after equinoxes, and was adjusted to be λ − α and λ + α in summers and winters, respectively [22,30]. To ensure solar rays within θ_a at the solar-noon in the days (δ = ±8.5°) when the tilt-angle was adjusted, the value of α for 3T-2MCPV was set to be 22° for θ_a ≥13.54° and α = θ_a + 8.5 for θ_a < 13.54° [30].

To compare the performance of 2MCPV with CPC-based PV systems (CPV), the AEG, generated by similar CPV, which was identical in the C_g and θ_a to 2MCC optimized for maximizing C_g, was calculated based on the mathematical model proposed by Tang et al. [12].

5. Results and Discussion

5.1. Comparisons of Optical Efficiency between 2MCC and Similar CPC

It can be seen from Figure 9 that the optical efficiency of 2MCC and similar CPC expected by mathematical models developed in the present and previous works [12], were almost identical to those from the ray-tracing analysis, indicating that they could reasonably be used to predict the optical performance of 2MCC and CPC, respectively. As compared to similar truncated CPC, the optical efficiency of 2MCC was slightly lower as θ_p ≤ θ_a because α₂ of 2MCC was much less than the edge-ray angle (θ_t) of the truncated CPC (see Figure 10). Thus, more radiation directly irradiated on the solar cells, but it was higher when θ_p was slightly larger than θ_a, because a fraction of radiation irradiating on the right mirrors of the 2MCC arrived on the absorber after reflections but the radiation on the right parabola of CPC eventually escaped away from the aperture. It was also seen that when θ_p > θ_max, f = 0 for 2MCC but not for the truncated CPC, as θ_t of the similar truncated CPC was much larger than the θ_max of 2MCC (see Figure 10), indicating that CPC was more favorable for the collection of sky diffuse radiation, as compared to 2MCC.

5.2. Optimal Design of 2MCC

As aforementioned, given θ_a, the geometry of 2MCC was uniquely determined by γ₁ and γ₂. Thus, the performance of 2MCPV also depended on γ₁ and γ₂, and a set of γ₁ and γ₂ for maximizing ACR (S_a) and ARG (P_a) could be respectively found through iterative calculations. Figure 11 presents the optimal geometry of 1T-2MCPV optimized for maximizing S_a and P_a in Beijing (upper two sub-plots) and Chongqing (down two sub-plots). It was observed that the γ_i,Sa (i = 1, 2), the optimal γ_i for maximizing S_a, and γ_i,Pa (i = 1, 2), the optimal γ_i for maximizing P_a, increased with a decrease of ρ, and for a given ρ, γ_i,Sa > γ_i,Pa and γ_i,Sa > γ_i,g. Increasing γ_i, especially γ₁, resulted in more radiation directly irradiating on the solar cells, and decreasing γ_i reduced the IA of the solar rays on the solar cells, thus improving the PV efficiency of the solar cells. It was also seen that γ_1,Pa < γ_1,g and γ_2,Pa ≈ γ_2,g for high ρ (ρ = 0.9), whereas for low ρ (ρ = 0.7), γ_1,Pa ≈ γ_1,g and γ_2,Pa > γ_2,g. It was also shown that γ_1,Sa and γ_1,Pa were highly sensitive to θ_a but γ_2,Sa and γ_2,Pa were weakly sensitive to θ_a. Geometric concentrations of 1T-2MCC optimized for maximizing C_g (termed as C_g,max), S_a (termed as C_g,Sa), and P_a (termed as C_g,Pa) are presented in Figure 12. It shows that C_g,Sa and C_g,Pa were almost identical, and they were close to C_g,max for ρ = 0.9 but were about 1–1.5% and 1.2–1.8% lower for ρ = 0.7 in Beijing and Chongqing, respectively.

Optimal design of 2MCPV with the tilt-angle of the aperture being yearly adjusted four times at three tilts (3T-2MCPV) is presented in Figure 13. Similar to 1T-2MCPV, for a given θ_a, the γ_i,Sa and γ_i,Pa increased with a decrease of ρ, and given ρ, γ_i,Sa > γ_i,Pa. It showed that for 3T-2MCPV, optimized for maximizing S_a, the γ_1,Sa was close to γ_1,g, while γ_2,Sa ≈ γ_2,g for high ρ but γ_2,Sa > γ_2,g for low ρ; whereas for 3T-2MCPV optimized for maximizing P_a, γ_1,Pa < γ_1,g, except in Chongqing for the case of θ_a > 24° and ρ = 0.7, while γ_2,Pa ≈ γ_2,g for high ρ and γ_2,Pa > γ_2,g for low ρ. It was also found that C_g,Sa and C_g,Pa were almost identical, and they were close to C_g,max for ρ = 0.9 but slightly lower for ρ = 0.7, as shown in Figure 14.

5.3. Performance of 2MCPV

Figure 15 presents the ACR of 1T-2MCC optimized for maximizing S_a, C_g, and that of similar 1T-CPC. It can be seen that the S_a collected by 1T-2MCC optimized for maximizing S_a (termed as S_a,_max) and that collected by 1T-2MCC optimized for maximizing C_g (termed as S_a,g) almost linearly decreased, with an increase of θ_a. For ρ = 0.9, S_a,g was almost identical to S_a,max, and for ρ = 0.7, S_a,g were respectively about 99% and 98% of S_a,max (see lines S_a,g/S_a,max), in Beijing and Chongqing. This meant that 1T-2MCC optimized for maximizing C_g, could be regarded as the optimal geometry for maximizing S_a, when ρ was high. It also showed that, in the site with abundant solar resources such as Beijing, the S_a,cpc, collected by similar 1T-CPC, was slightly lower than S_a,g and S_g,max for ρ = 0.9 but was higher for ρ = 0.7; whereas in the sites with poor solar resources such as in Chongqing, the S_a,cpc was slightly higher than S_a,max, except when θ_a < 28° and ρ = 0.9. This indicated that as compared to similar 1T-CPC, the 1T-2MCC even annually collected more radiation in the sites with abundant solar sources, because a fraction of radiation incident on the mirrors of 2MCC at θ_p > θ_a arrived on the absorber, but all radiation incident on the parabola of CPCs at θ_p > θ_a escaped away from the aperture of CPCs.

The AEG from 1T-2MCPV optimized for maximizing P_a and C_g as well similar 1T-CPC based PV system (1T-CPV) is presented in Figure 16. Similar to ACR of 1T-2MCC, the P_a,max (AEG from 1T-2MCPV optimized for maximizing P_a) and P_a,_g (AEG from 1T-2MCPV optimized for maximizing C_g) linearly deceased with an increase of θ_a. For ρ = 0.9, P_a,g was almost identical to P_a,max, while for ρ = 0.7, P_a,g were, respectively, about 99% and 98.3% of P_a,max (see lines P_a,g/P_a,max) in Beijing and Chongqing. It showed that for ρ = 0.9, the P_a,g was slightly larger than P_a,cpv (AEG from similar 1T-CPV), whereas, for ρ = 0.7, P_a,g was close to P_a,cpv in Beijing but was slightly less in Chongqing. These results indicated that 1T-2MCPV optimized for maximizing C_g could also be regarded as the one for maximizing P_a for high ρ, and 1T-2MCPV with a high ρ generated more electricity annually, as compared to similar 1T-CPV.

The optical performance of 3T-2MCC is shown in Figure 17. It was seen that for ρ = 0.9, the S_a,g was almost identical to S_a,max, while for ρ = 0.7, S_a,g were respectively about 99.6% and 99% of S_a,max in Beijing and Chongqing. This meant that the 2MCC optimized for maximizing C_g could be regarded as the optimal ρ design of 3T-2MCC for maximizing S_a when ρ > 0.7. For ρ = 0.9, S_a,g and S_a,cpc were almost identical, whereas for ρ = 0.7, S_a,g was slightly less than S_a,cpc. Figure 18 shows the PV performance of 3T-2MCC and similar 3T-CPC-based PV systems. It can be seen that P_a,g was very close to P_a,cpv, even for ρ = 0.7. For ρ = 0.9, P_a,g > P_a,cpv especially for small θ_a, whereas for ρ = 0.7, P_a,g > P_a,cpv in Beijing as θ_a < 35° and in Chongqing when θ_a < 25°. These results indicate that the 2MCC optimized for maximizing C_g could also be regarded as the optimal design of 3T-2MCPV for maximizing P_a for ρ > 0.7, and compared to 3T-CPV, 3T-2MCPV was more favorable in the AEG for θ_a < 25° and ρ > 0.7.

In practical applications, θ_a should be larger than 35.5° for 1T-2MCC and 1T-CPC and 13.5° for 3T-2MCC and 3T-CPC, to ensure that they efficiently operate for at least 7 h, in all days of a year. Therefore, as compared to similar CPV, the 1T-2MCPV was more favorable in the AEG and ACR, when ρ was high, such as ρ = 0.9, but the situation was reversed when ρ was low, such as ρ = 0.7, while 3T-2MCPV annually collected almost identical radiation but generated more electricity as θ_a < 25° and ρ > 0.7.

6. Conclusions

In this work, the design of multi-mirror composite concentrator with flat-plate absorber was addressed. The analysis indicated that, given the acceptance angle (θ_a), the geometry of n-MCC was uniquely determined by the tilt-angle (γ_i, i = 1, 2, …, n) of all mirrors, and a set of γ_i (i = 1, 2, …, n) for maximizing the geometric concentration (C_g) could be found through multi-loop iterative calculations. Results showed that, given θ_a, the C_g increased with an increase of the mirror number (n), and the optimal tilt angle (γ_1,g) of the bottom mirror for maximizing C_g was larger than n times that of the top mirror (i.e., γ_1,g > nγ_n,g). Results also indicated that, given n, the maximum geometric concentration (C_g,max) of n-MCC decreased with an increase of θ_a, and the θ_a of 2MCC and 3MCC must be, respectively, less than 17.7° and 21.3°, to ensure C_g > 2.

To investigate the performance of a 2MCC-based PV system (2MCPV) and an optimal design for maximizing ACR and AEG, a three-dimensional radiation transfer model was developed by means of imaging principle of mirrors and vector algebra. Results showed that the optical efficiencies of 2MCC obtained by theoretical calculations and those from the ray-tracing analysis were in complete agreement, thus, it allowed to accurately predict the optical performance of 2MCC.

Analysis indicated that the optimal and photovoltaic performance of 2MCPV was dependent on the geometry of 2MCC, and the reflectivity of the mirrors (ρ) and solar resources in a site. Thus, given ρ, an optimal geometry of 2MCC for maximizing S_a and P_a in a site can be respectively found through iterative calculations. Calculations showed that when ρ was high, such as ρ = 0.9, the C_g of 1T- and 3T-2MCPV for maximizing S_a and P_a were almost identical to that of 2MCC for maximizing C_g. Hence, 2MCC optimized for maximizing C_g could be regarded as the one for maximizing S_a and P_a of 1T- and 3T-2MCPV. For 1T-2MCPV, the ACR and AEG linearly decreased with an increase of θ_a, as compared to similar 1T-CPV, it yearly collected more radiation and generated more electricity when the ρ was high. Whereas for 3T-2MCPV, as compared to similar 3T-CPV, it annually collected more radiation when the ρ was high and generated more electricity when θ_a < 25° and ρ > 0.7, even in the sites with poor solar resources.

The transfer of radiation within 3MCC was extremely complex, and the theoretical investigation in this work was only focused on 2MCC. It was believed that, the results obtained in this work were also helpful for design and application of 3MCC and even n-MCC.