Research ArticleEffect of Tracking Error of Double-Axis Tracking Device on theOptical Performance of Solar Dish Concentrator

Jian Yan , Zi-ran Cheng, and You-duo Peng

Hunan Provincial Key Laboratory of Health Maintenance for Mechanical Equipment, Hunan University of Science and Technology,Xiangtan, Hunan 411201, China

Correspondence should be addressed to You-duo Peng; ydpeng@hnust.edu.cn

Received 25 July 2017; Revised 9 November 2017; Accepted 29 November 2017; Published 14 March 2018

Academic Editor: Alberto Álvarez-Gallegos

Copyright © 2018 Jian Yan et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, a flux distribution model of the focal plane in dish concentrator system has been established based on ray trackingmethod. This model was adopted for researching the influence of the mirror slope error, solar direct normal irradiance, andtracking error of elevation-azimuth tracking device (EATD) on the focal spot characteristics (i.e., flux distribution, geometricalshape, centroid position, and intercept factor). The tracking error transmission law of the EATD transferred to dishconcentrator was also studied. The results show that the azimuth tracking error of the concentrator decreases with the increaseof the concentrator elevation angle and it decreases to 0mrad when the elevation angle is 90°. The centroid position of focal spotalong x-axis and y-axis has linear relationship with azimuth and elevation tracking error of EATD, respectively, which could beused to evaluate and calibrate the tracking error of the dish concentrator. Finally, the transmission law of the EATD azimuthtracking error in solar heliostats is analyzed, and a dish concentrator using a spin-elevation tracking device is proposed, whichcan reduce the effect of spin tracking error on the dish concentrator. This work could provide fundamental for manufacturingprecision allocation of tracking devices and developing a new type of tracking device.

1. Introduction

Solar energy is a clean and environment-friendly renewableenergy source, which is plentiful and can be widely distrib-uted. Developing and utilizing solar energy are importantways to solve energy shortages and environmental pollutionproblems, for example, the H2 production by solar thermo-chemical reactions [1], solar cooker [2], and solar energythermal electric power system [3]. Among all those typicalsolar energy applications in medium-high temperature fields,the solar concentrator was commonly used in the above-mentioned fields to collect the sunlight as heat source andprovide heat source to the heat receiver because the solarradiation density received by the earth surface was low. Thereare four main types of solar concentrators: Fresnel concentra-tor [4, 5], parabolic trough concentrator [6, 7], tower heliostat[8, 9], and parabolic dish concentrator [10–12]. The single-dish concentrator could realize the high concentration ratioand high temperature so it is superior to other concentrators.Typical application, such as the solar dish concentrator

photovoltaic system and the dish-Stirling solar power system,consists of a dish concentrator and a Stirling engine [10] suchas the 38 kW XEM-Dish system as shown in Figure 1.

The dish concentrator via an elevation-azimuth double-axis tracking device (EATD) was used to accurately trackthe sun’s position so it could focus the sunlight to the precal-culated position of the receiver [10–12]. The EATD consistsof an elevation tracking device and an azimuth trackingdevice (Figures 1(b) and 1(c)), which is widely used in dish-concentrating system. The azimuth tracking axis in EATDis vertical to the ground level, and the elevation tracking axisin EATD is parallel to the ground level and vertical to thefocal axis of a dish concentrator. The program trackingprocess of a dish concentrator system is as follows: calculatethe position information of the sun (the elevation angle andazimuth angle of the sun) by astronomical algorithm [13];according to the sun position information, the output rota-tion angle of an azimuth motor and elevation motor in theEATD is controlled and then the elevation tracking axis n1and azimuth tracking axis n2 in the EATD can rotate a

HindawiInternational Journal of PhotoenergyVolume 2018, Article ID 9046127, 13 pageshttps://doi.org/10.1155/2018/9046127

corresponding angle (Figure 2(b)), which will change the azi-muth and elevation angle of the dish concentrator for realiz-ing the focal axis aiming at the sun’s center. Therefore, thetracking accuracy of a dish concentrator could be influencedby astronomical algorithm, driving error of the motor andEATD, installation error of the EATD, and so on, where driv-ing error of the EATD is the main factor (i.e., output rotationangle error of the azimuth and elevation tracking axis in theEATD, called EATD tracking error). The EATD trackingerror could lead the dish concentrator to produce trackingerror so it would cause some adverse phenomenon, such asthe optical losses and nonuniform flux in the receiver. Thus,improving the tracking accuracy of a dish concentrator andreducing the manufacturing precision requirement of theEATD (i.e., reducing the manufacturing cost) have alwaysbeen the pursuit of commercial applications. Therefore,researching the influence of the EATD tracking error onthe tracking performance and optical performance of dishconcentrator system is very important.

Many works focus on the research for the influence oftracking error on the optical performance or flux distributionof dish concentrator system [14–21], but they do not considerthe influence of EATD tracking error. Hughes [14] used statis-tical methods to establish an expression for the expected valueof the intercept factor for various configurations and controllaw implementations. The analysis assumes that a radiallysymmetric flux distribution is generated at the focal plane.Badescu [15] researched the different tracking error distribu-tions and their effects on the long-term performances of para-bolic dish solar power systems; he adopted four differenttracking error distributions to evaluate the intercept factorand the thermal and engine efficiencies, three of them arebounded distributions and the last one is the widely usedclassical Gaussian distribution. Further, he also studied the

theoretical derivation of heliostat tracking error distributionusing methods of integral geometry and geometrical probabil-ities [16]. Hu and Yao usedMCRTmethod to research the fluxdistribution on the solar cell plane in a two-stage concentra-tion photovoltaic system with concentrator tracking error(dish concentrator is the first level) [17]. Shanks et al. [18]researched the flux distribution and optical efficiency in ahigh-concentrating photovoltaic system with concentratortracking error (using the Cassegrain concentrator). Xia et al.[19] researched the influence of the concentrator trackingerror on the intercept factor of a sixteen-dish concentrator.Li et al. [20] and Castellanos et al. [21] through the theoret-ical model research the influence of the total optical error onthe intercept factor in the dish concentrator system. In theexperimental measurement, the measured results of the fluxdistribution on the focal plane have been widely used to eval-uate the surface slope error, tracking error, and optical effi-ciency of the dish concentrator system [19, 22, 23], whichcould also be used for adjusting the tracking error of dishconcentrators. For example, Johnston [22] developed a fluxmapping equipment and measured the flux distribution onthe focal plane of a 20m2 dish concentrator. This concentra-tor mirror slope error has been determined is 2.0mrad. Xiaet al. [19] developed a type of concentrated solar flux distri-bution measurement formed by a thermal infrared imagerand water-cooled Lambert target and measured the flux dis-tribution on the focal plane of a sixteen-dish concentrator.Through the comparison of the measured results and raytracking calculation results of the flux distribution, theydetermined the concentrator mirror slope error is 2.2mradand concentrator total tracking error is 0.8mrad. But the sin-gle tracking error of the concentrator has not been analyzedin their paper. The measured results of the flux distributionon the focal plane of the dish concentrator could be used to

g

Upright column

Elevation-azimuth double-axis tracking device

Concentrator

S260 Stirling engine

Azimuth tracking axis

Azimuth tracking device

(a)

(b)

(c)

Screw mandrel

Elevation tracking device

Elevation motor

Elevation tracking axis

Azimuth motorn2

n1

Figure 1: A 38 kW dish-Stirling solar thermal power system (38 kW XEM-Dish system). The diameter is 17.70m and the focal length is9.49m of the parabolic dish concentrator and was built in Xiangtan Electric Manufacturing Group, China.

2 International Journal of Photoenergy

evaluate or calibrate the tracking error of the dish concentra-tor. But this process would be influenced by the concentratormirror slope error and solar direct normal irradiance (DNI),which would change the geometry shape or flux distributionof the focal spot on the focal plane [24, 25]. Therefore, it isnecessary to find a new focal spot characteristic to establishthe functional relationship between the focal spot characteris-tic and tracking error of dish concentrator to realize the eval-uation and calibration of concentrator tracking error.

The above-mentioned literatures all focus on the influ-ence of the total tracking error of dish concentrator systemson the flux distribution and optical performance, while theresearches of the influence of the tracking error of the EATDon the tracking performance and optical performance of dishconcentrator system are limited. In this paper, we focused onthe above two aspects; a flux distribution model of the focalplane in dish concentrator systems has been established basedon ray tracking method. The concentrator mirror and solarcone in nonuniform energy distribution are discreted bysquare grids in this model (in Section 2). This model wasadopted for researching the influence of the mirror slopeerror and DNI and EATD tracking error on the focal spotcharacteristics (i.e., flux distribution, geometrical shape, cen-troid position, and intercept factor), and the tracking errortransmission law of the EATD transferred to dish concentra-tor was studied (in Section 4).Moreover, the transmission lawof the EATD azimuth tracking error in solar heliostats is ana-lyzed, a dish concentrator system using a spin-elevationdouble-axis tracking device is proposed, and the transmissionlaw of the tracking error from the spin axis to the dish con-centrator was studied (in Section 4.4). This work could pro-vide fundamental for calibrating the tracking error of dishconcentrator system, manufacturing precision allocation ofa double-axis tracking device (appropriately decrease themanufacturing precision of the azimuth or spin trackingdevice), and developing a new type of tracking device.

2. Mathematical Model

The flux distribution model of the plane receiver in dishconcentrator systems is developed based on the ray trackingmethod. The ray tracking method needs to discrete the mirrorsurface of dish concentrators, plane receiver, and sun cone.There are two methods to discrete the mirror surface and suncone, a random MCRT method [4, 12] and a grid discretemethod [26]. In this work, we adopted the grid discrete methodto establish the sun cone discretization model considering anonuniform energy distribution and flux distribution modelof the focal plane in dish concentrator systems (i.e., opticalmodel). The mirror slope error of the dish concentrator andtracking error of the EATD are included in this optical model.

2.1. Optical Model. As shown in Figure 2, a coordinate systemO-xyz was built on the vertex point O of a parabolic mirrorsurface and the z-axis points to the focal point F. The mirrorsurface function of a dish concentrator can be defined asx2 + y2 = 4fz, where f is the focal length of the dish concentra-tor. In Figure 2(a), R1 is aperture radius of the dish concen-trator, and angle θm and radius R2 are no mirror area in thedish concentrator. The steps for optical discretization of thedish concentrator are as follows: (1) use a square with sidelength of 2R1 (blue wire frame in Figure 2(a)) to surroundthe dish concentrator mirror and then the discrete into alot of square grid with side length of 2R1/K (M=K, i.e., thedaylighting area Akm of each grid is equal, Akm= (2R1/K)

2);(2) according to the coordinate and distance Rkm of the pointpkm (i.e., grid center point in k line m column) to determinewhether the grid km is in the reflection mirror. If the pointpkm is in the reflection mirror, the identifier fkm=1 (i.e., thegrid km is effective); otherwise, fkm=0 (i.e., the grid km is anoneffective); (3) the coordinates and unit normal vectorNkm of the center point pkm of all effective grid are calculatedand stored for subsequent ray tracing simulation.

x

y

𝜃m

K

M

Rkm

Pkm

P11

R2 R1

O

(a)

2𝛿Iwn

Nkm

Pkm

1Nkm

x y

z

n2n3

n1

F

𝛽

𝛽

O

A

C

B

NF

𝜀3

𝜀1

𝜀2

𝜀4

S

g

(b)

Figure 2: (a) Geometry parameters and mirror discrete of the dish concentrator and (b) schematic diagram of the tracking error and mirrorslope error of the dish concentrator.

3International Journal of Photoenergy

Inpractical application, the reflectionmirror of adish con-centrator would have microslope error, which would impactthe normal vector of the reflection mirror (Figure 2(b)). Themirror slope error could be described as a radial angle ε3 andtangential angle ε4 [17, 24]:

ε3 = −2σ2 ln r1,ε4 = 2πr2,

1

where r1 and r2 are uniformly distributed random numberswithin the range of 0~1, σ is standard deviation of the mirrorslope error.

Considering the mirror slope error of the dish concentra-tor, the vector N1

km on point pkm is calculated by (2), whereRot n3, ε3 and Rot Nkm, ε4 are mirror slope error matrixwhich could all be determined by function Rot e, β asshown in (3). Rot e, β could be used to realize the vector P∈ R1×3 rotate β angle around the unit vector e = ex, ey, ez ;n3 = Nkm × ex / Nkm × ex ; ex= [1,0,0].

In ideal tracking condition, the focal axis of the dishconcentrator is parallel to the center line S (S= [0, 0,− 1.0])in the sun cone (sun half-apex angle δ = 4 65mrad). How-ever, when the EATD exists tracking error (elevation andazimuth tracking error of the EATD is ε1 and ε2, respectively,Figure 2(b)); the point pkm moving to the position p1km andthe corresponding normal vector is N2

km. As well as theincluded angle ωerr between the focal axis of the dish concen-trator and center line S in sun cone, this angle ωerr is the totaltracking error of the dish concentrator. This could be calcu-lated by (4), (5), (6), (7), where n1 is elevation tracking axisvector of the EATD, n1 = [1, 0, 0]; n2 is the azimuth trackingaxis vector of the EATD, n2 = [0, −cosβ, sinβ]; the angle β issolar elevation angle (i.e., working elevation angle of the dishconcentrator); g is a position vector of the crosspoint g of theelevation tracking axis and azimuth tracking axis in theEATD (Figures 1 and 2(b)), g= [0, gy,gz]. Noteworthiness,the tracking error of the dish concentrator originated in theEATD tracking error. This tracking error transmissionrelated to the working elevation angle β of the dish concen-trator, which could be clearly seen in (6).

p1km = pkm − g ⋅Rot n1, ε1 ⋅ Rot n2, ε2 + g, 4

N2km =N1

km ⋅ Rot n1, ε1 ⋅Rot n2, ε2 , 5

N1F =NF ⋅ Rot n1, ε1 ⋅ Rot n2, ε2 , 6

ωerr ε1, ε2, β = arcos sin ε1 + β ⋅ sin β ⋅ 1 − cos ε2+ cos ε1 ⋅ cos ε2

7

When the dish concentrator system exists tracking error,the plane receiver (located at the focal plane of dish concen-trators) would rotate with the dish concentrator. This time,the focal point Fmoves to position F1 and the correspondingplane receiver normal vector is N1

F , which can be calculated

by (8) and (6), respectively, where F= [0, 0, f]. These two vec-tors can determine the space equation of the plane receiver.

F1 = F − g ⋅Rot n1, ε1 ⋅ Rot n2, ε2 + g 8

In this work, the plane receiver is a square with sidelength is L, which can be discreted into a lot of square grid(discrete parameter H=U). The calculation process of theflux distribution on the plane receiver as is as follows: (1)based on the law of specular reflection, the reflected ray equa-tion of the incident sunlight Iwn on the mirror point p1km iscalculated by (4) and (5); (2) the intersection point q of thereflected ray and plane receiver is calculated, q= [x1, y1, z1];(3) the grid number hu containing point q is found and thenincreasing the sunlight Iwn carrying energy (Section 2.2) tothe grid hu; (4) following these steps, calculating every effec-tive grid in the dish concentrator whole sunlight transmis-sion process could get the flux distribution results on theplane receiver.

2.2. Discretization Model of the Sun Cone. The solar radiationintensity in a sun cone gradually decreases from the center tothe border. The nonuniform energy distribution model isgiven as [27]:

I γ =I1 1 − 0 5138 γ

δ

4, 0 ≤ γ ≤ δ,

0, γ > δ,9

where γ is the intersection angle of the discrete ray and centerray S (Figure 3); I1 is the radiation intensity of the center ray.

The sun cone received by different positions in the dishconcentrator mirror was assumed to be exactly identical.Thus, the sun cone was established on the pointO and discre-tization as shown in Figure 3(b). In details, the solar disk onthe focal planewas established (the radius isRs,Rs = f× tan(δ))

N1km =Nkm ⋅ Rot n3, ε3 ⋅ Rot Nkm, ε4 , 2

Rot e, β =

cos β + e2x 1 − cos β exey 1 − cos β + ez ⋅ sin β exez 1 − cos β − ey ⋅ sin β

exey 1 − cos β − ez ⋅ sin β cos β + e2y 1 − cos β eyez 1 − cos β + ex ⋅ sin β

exez 1 − cos β + ey ⋅ sin β eyez 1 − cos β − ex ⋅ sin β cos β + e2z 1 − cos β

3

4 International Journal of Photoenergy

and it adopts the square envelope and then discrete to squaregrid (W=N). Thus, the position vector of the pointQwn is

Qwn = w − 0 5 2Rs

W− Rs, Rs − n − 0 5 2Rs

N, f 10

Based on the principle of energy conservation equation,the dish concentrator and sun cone before and after thediscretization should satisfy the following relationship:

Adish ⋅W0 = 〠K

k=1〠M

m=1f km ⋅ Akm ⋅ e0

= 〠K

k=1〠M

m=1f km ⋅ Akm ⋅ 〠

W

w=1〠N

n=1f wn ⋅ I1

⋅ 1 − 0 5138 γwnδ

4,

11

where e0 is the radiation intensity of a single sun cone;γwn = arctan dwn/f is the intersection angle of the sun-light ray wn and central sunlight; dwn = Qwn − F ; W0 isthe DNI value (W/m2); Adish =∑K

k=1∑Mm=1 f km ⋅ Akm is aper-

ture total area of the dish concentrator. If dwn ≤ Rs, fwn=1;if dwn > Rs, fwn=0. Thus, I1 can be written as follows:

I1 =W0

〠Ww=1〠

Nn=1 f wn 1 − 0 5138 γwn/δ 4 12

Finally, the sunlight vector Iwn = −Qwn/ Qwn and theenergycarriedbythesunlightcanbedeterminedby(9)and(12).

2.3. Model Validation. Based on the optical model and suncone model in this paper, the ray tracing codes werecompiled in Visual C++ 6.0 software for calculating the fluxdistribution on the focal plane of the dish concentrator.The Jeter theoretical results [28] were adopted to check thevalidity of the optical model and ray tracing codes and thecomparison results as shown in Figure 4(a). We can see that

the ray tracing calculation results in good agreement to Jeter’stheoretical results, which verifies the validity of the flux dis-tribution model in this paper. The flux distribution contourmap calculated by ray tracing method in this paper is shownin Figure 4(b).

3. Focal Spot Characterization

Due to the reason that the tracking error of the EATD wouldcause the tracking error of the dish concentrator, the positionand geometry of the focal spot on the plane receiver would allchange. The focal spot moves and extends to the upper rightdirection, and the flux contour is approximately ellipseshaped but the center position of the ellipse is different, suchas in Figure 5. In this paper, using a flux distributionmap, fluxdistribution curve along the x-axis and y-axis, intercept factor,centroid position, and ellipse geometry (a flux contour curve)are used to describe the focal spot. Where centroid positionand ellipse geometry could describe the movement directionand geometrical extent of the focal spot, respectively.

The intercept factor is defined as the ratio of the energyon the plane receiver (radius is Rin) to the total receivedenergy [14, 19]. The centroid coordinates of the focal spotcould be calculated by (13), where f(h, u) is the flux densityof the grid hu on the plane receive. Ellipse equation as shownin (14), including five parameters: center point coordinates(xs, ys), minor semiaxis b, major semiaxis a, and intersectionangle ϕ2 with the major axis and axis x1. The grid of the fluxdensity equal or approaching to Ct×W0 was found on thereceiver and then the center point coordinates of the gridwere extracted (as the ellipse boundary point) for fitting(16) to get the ellipse parameters using the least squaremethod (Ct is a flux density extraction threshold value).The searching process of the ellipse boundary point is thataccording to the plane receiver discrete grid, from top to bot-tom (y-axis direction), to find the two adjacent grid for thefirst time, satisfy f h, u ≤ Ct ⋅W0 ≤ f h, u + 1 . Then weselected one grid center as the boundary point (i.e., ellipse

SIwn

2𝛿

Rs

f

F

Ox

z

𝜆wn𝛾wn

(a)

N

W

F

x

y

Rs

Qwn

dwn

𝜆wn

(b)

Figure 3: Discretization of the (a) sun cone and (b) solar disk.

5International Journal of Photoenergy

up boundary point) which is the flux density closest to theCt×W0. In the same way, from bottom to the top to findthe ellipse bottom boundary point.

xt =〠H

h=1〠Uu=1 f h, u ⋅ u − 0 5 L/U〠H

h=1〠Uu=1 f h, u

,

yt =〠H

h=1〠Uu=1 f h, u ⋅ h − 0 5 L/U〠H

h=1〠Uu=1 f h, u

,13

x − xs cos ϕ2 + y − ys sin ϕ22

a2

+ y − ys cos ϕ2 − x − xs sin ϕ22

b2= 1

14

4. Result and Discussion

4.1. Concentrator Tracking Error. The tracking error of theEATD would cause the tracking error of the dish concentra-tor, but the tracking error of the dish concentrator may notbe equal to the tracking error of the EATD. This trackingerror transmission (i.e., tracking error of the EATD transmit-ted to dish concentrator) is related to the working elevationangle β of the dish concentrator. In Figure 6, when the EATDonly has azimuth tracking error, the total tracking error ωerrof the dish concentrator decreases with the increase of eleva-tion angle β and the error ωerr decreases to 0 rad when theangle β is 90°(curve 1~5 in Figure 6(a)). When the EATDhas both elevation and azimuth tracking error, the error ωerrdecreases with the increase of the elevation angle β and theerror ωerr is equal to the elevation tracking error ε1 of theEATD when the angle β is 90°, which could be clearly seenin Figure 6(a) (curve 7~11) and Figure 6(b). That is to say,the increase of the working elevation angle β of the dish con-centrator would vanish or reduce the influence from the azi-muth tracking error of the EATD to the dish concentrator.This phenomenon is called vanish-reduction effect in track-ing error transmission by us. However, the elevation trackingerror of the EATD transmit to the dish concentrator has novanish-reduction effect, that is, the elevation tracking errorof the dish concentrator equal to the elevation tracking errorof the EATD. Thus, the concentrator total tracking error in(6) can be rewritten as follows:

ωerr ε1, ε2, β

= ε2r2 + ε1

2

= arcos cos 90 − β 2 ⋅ 1 − cos ε2 + cos ε2 + ε12,

15

where ε2r is azimuth tracking error of the dish concentra-tor from the azimuth tracking error ε2 of the EATD.

x1

y1

xt

yt

F1

t 𝜙2

2b 2a

L/2

s

Figure 5: Geometrical parameters of the focal spot on the planereceive.

0

5

10

15

20

25

30

35

40H

eat fl

ux (W

/m2 )

−0.015 −0.010 −0.005 0.000 0.005 0.010 0.015Radial distance from focal center (m)

Jeter’sThis paper

(a)

−10 0 10−15

−10

−5

0

5

10

15

0.5

1

1.5

2

2.5

3

3.5×107

x (mm)

y (m

m)

(b)

Figure 4: (a) The comparison of the flux distribution result on the focal plane and (b) flux distribution contour map using ray tracing methodin this paper (W/m2).

6 International Journal of Photoenergy

Obviously, only when the angle β = 0, then ωerr ε1, ε2, 0 =

ε22 + ε1

2, which is worth to be noticed.

The vanish-reduction effect produce essence is becausethe focal axis of the dish concentrator and the tracking axisof the EATD are not vertical. The intersection angle betweenthe focal axis of the dish concentrator and azimuth trackingaxis of the EATD is (90°−β) (Figure 2(b)). The vanish-reduction effect is more obvious with the decrease of theangle (90°−β). Meanwhile, the influence of the elevationtracking error of the EATD on the dish concentrator wasreduced. Due to the elevation tracking axis of the EATDalways vertical to focal axis of the dish concentrator, thereis no vanish-reduction effect in the elevation tracking errorof the EATD transmission to the dish concentrator. There-fore, we could appropriately decrease the manufacture preci-sion of the azimuth tracking device in the EATD to reduceproduction costs. Besides, it also could develop a new typeof double-axis tracking device to increase the vanish-reduction effect, such as a spin-elevation double-axis trackingdevice for dish concentrator system in Section 4.4. On theother hand, it is critical to evaluate or predict the long-termperformance of the concentrator system [14, 15]. If tracking

error of the dish concentrator all comes from output angleerror of the EATD, then according to the probability distri-bution of EATD output angle error model (based on statisti-cal methods) and tracking error transmission law by 15, wecan evaluate the long-term performance of dish concentratorsystem, which will be our future research work.

4.2. Flux Distribution on the Focal Plane. In this paper, wetake the typical 38 kW XEM-Dish system (Figure 1) as anexample to study the effect of the tracking error of the EATDon the focal spot characteristics of the focal plane. The calcu-lation parameters include concentrator radius R1 = 8 85m,focal length f = 9 49m, θm=30°, R2 = 0 9m, mirror reflec-tivity is 0.90, absorption rate of the plane receiver is 100%,DNI valueW0 = 1000W/m2, Ct=50, mirror discrete parame-ter K = 275, sun cone discrete parameter W = 201, and totaltracking rays are 1.46× 109.

The influence of the tracking error of EATD on the fluxcurve and flux contour map of the focal spot are shown inFigures 7 and 8, respectively. From Figure 7(a), the move-ment and extent of the focal spot along the y-axis negativedirection intensify with increase of the elevation trackingerror of the EATD, but the flux peak value changes small.

0.0000.0020.0040.0060.0080.0100.0120.0140.016

w err

(rad

) 76

8

10

5

9432

111

0 10 20 30 40 50 60 70 80 90𝛽 (°)

(a)

w err

(rad

)

00.005

0.010

0.0050.01

0

0.005

0.01

0.015𝛽 = 0°

𝛽 = 45°

𝛽 = 90°

𝜀 2 (rad)

𝜀1 (rad)

(b)

Figure 6: Influence of the tracking error of the EATD on the dish concentrator. 1: ε1 = 0mrad, ε2 = 2mrad; 2: ε1 = 0mrad, ε2 = 4mrad; 3:ε1 = 0mrad, ε2 = 6mrad; 4: ε1 = 0mrad, ε2 = 8mrad; 5: ε1 = 0mrad, ε2 = 10mrad; 6: ε1 = 10mrad, ε2 = 0mrad; 7: ε1 = 10mrad, ε2 = 10mrad;8: ε1 = 8mrad, ε2 = 8mrad; 9: ε1 = 6mrad, ε2 = 6mrad; 10: ε1 = 4mrad, ε2 = 10mrad; 11: ε1 = 2mrad, ε2 = 2mrad.

0

5

10

15

20

25

30

20015010050−50−100−150−200−250−300−350

0 mrad

0 mrad

10 mrad

y (mm)

Hea

t flux

(W/m

2 ) 𝜎 = 2 mrad, DATDelevation trackingerror is 0, 2, 4, 6, 8and 10 mrad in turns

𝜎 = 0 mrad, DATDelevation trackingerror is 0, 2, 4, 6, 8and 10 mrad in turns

(a)

Hea

t flux

(W/m

2 )

0

5

10

20

25

30

15

−250 −200 −150 −100 −50 0 50 100x (mm)

𝛽 = 0°, 𝜀2 = 8.7 mrad

𝜀2 = 0 mrad

𝛽 = 0°, 𝜀2 = 6.9 mrad

𝛽 = 30°, 𝜀2 = 8 mrad

𝛽 = 30°, 𝜀2 = 10 mrad

𝛽 = 30°, 𝜀2 = 6 mrad

𝛽 = 60°, 𝜀2 = 10 mrad

(b)

Figure 7: Influence of the EATD tracking error on the flux distribution of the focal plane.

7International Journal of Photoenergy

x: 0y: 0

x: −112y: −31.69

x: −83.54y: −23.45

−250 −200 −150 −100 −50 0−150

−100

−50

0

50

100

0.5

1

1.5

2

2.5×107

s

t

x (mm)

y (m

m)

(a)

x: −83.81y: −23.29

x: 0y: 0

−200 −100 0 100

−150

−100

−50

0

50

100

2

4

6

8

10×106

x (mm)

y (m

m)

(b)

x: 0y: 0

x: −32.7y: −110

x: −24.1y: −81.47

−150 −100 −50 0 50 100

−200

−150

−100

−50

0

0.5

1

1.5

2

2.5×107

x (mm)

y (m

m)

(c)

x: 0y: 0

x: −83.2y: −82.2

x: −108.2y: −106.9

−250 −200 −150 −100 −50 0 50

−200

−150

−100

−50

0

0.5

1

1.5

2

2.5

x (mm)

y (m

m)

×107

(d)

x: −83.2y: −82.2

x: −108.2y: −106.9

−200 −100 0

−200

−150

−100

−50

0

0.5

1

1.5

2

x (mm)

y (m

m)

×107

(e)

x: −83.48y: −81.38

x: 0y: 0

−300 −200 −100 0 100 200−300

−200

−100

0

100

2

4

6

8

x (mm)

y (m

m)

×106

(f)

x: 0y: −23.62

−100 −50 0 50 100

−100

−50

0

50

0.5

1

1.5

2

2.5

3

t

s

s (0,−34.25) t (0,−26.62)

x (mm)

y (m

m)

W/m2

×107

(g)

x: 0y: 0

x: −83.2y: 0

x: −111.9y: 0

−250 −200 −150 −100 −50 0

−100

−50

0

50

100

0.5

1

1.5

2

2.5

x (mm)

y (m

m)

×107

(h)

Figure 8: The flux distribution of the focal plane under the EATD tracking error (W/m2). (a) σ = 0, β = 30°, ε1 = 2mrad, and ε2 = 8mrad; (b)σ = 2 mrad, β = 30°, ε1 = 2mrad, and ε2 = 8mrad; (c) σ = 0, β = 0°, ε1 = 6.9mrad, and ε2 = 2mrad; (d) σ = 0, β = 0°, ε1 = 6.9mrad, andε2 = 6.9mrad; (e) σ = 0, β = 0°, ε1 = 6.9mrad, ε2 = 6.9mrad, and W0 = 800W/m2; (f) σ = 2 mrad, β = 0°, ε1 = 6.9mrad, ε2 = 6.9mrad, andW0 = 800W/m2; (g) σ = 0, β = 0°, ε1 = 2mrad, and ε2 = 0; (h) σ = 0, β = 0°, ε1 = 0, ε2 = 6.9mrad or β = 30°, ε1 = 0, and ε2 = 8mrad. Note:only Figures 8(e) and 8(f) are with W0 = 800W/m2 and the other figures are with W0 = 1000W/m2.

8 International Journal of Photoenergy

Although the mirror slope error of the dish concentratorwould increase the geometry size of the focal spot anddecrease the flux peak value of the focal spot [24], it wouldnot influence the centroid position of the focal spot. Fromthe comparison of Figures 8(a) and 8(b) and Figures 8(e)and 8(f), respectively, we could clearly see that. The DNIvalue would only influence the flux density value of the focalspot [25] but would not influence the centroid position of thefocal spot (comparison of Figures 8(d) and 8(e)). In conclu-sion, the centroid position of the focal spot on the focal planedepends on the tracking error of the dish concentrator buthas nothing to do with the mirror slope error and DNI.Therefore, the centroid position of the focal spot on the focalplane could be used for estimate and calibration of the track-ing error of the dish concentrator (Section 4.3).

The azimuth and elevation tracking error of the EATDonly have influence on the centroid coordinates of the focalspot along x-axis and y-axis, respectively. For example, thecoordinate of the focal spot centroid along y-axis is−23.45mm in both Figure 8(a) (β = 30°, ε1 = 2mrad, andε2 = 8mrad) and Figure 8(g) (β = 0°, ε1 = 2mrad, andε2 = 0mrad); the coordinate of the focal spot centroid alongx-axis is −83.20mm in Figure 8(a) (double tracking error)and Figure 8(h) (single tracking error); the azimuth and ele-vation tracking error of the EATD are all 6.9mrad (β = 0) inFigures 8(d)–8(f), so the coordinate of the focal spot cen-troid along x-axis and y-axis are the same −83.2mm. Theseshow that the influence of the azimuth and elevation track-ing error of the EATD on the centroid position of the focalspot is independent. Moreover, the position of the point g(Figure 2(b)) has no influence for the focal spot characteris-tics, that is, to say the EATD could be installed on everyposition in a dish concentration system and does not needto modify the tracking program of the EATD.

4.3. Centroid Moving Function and Intercept Factor. Thefunctional relationship between the centroid movement dis-tance of the focal spot on focal plane and EATD trackingerror is called centroid movement function; this functioncould be used for evaluating and quantitatively calibratingthe tracking error of the EATD. The influence curve ofthe azimuth tracking error of the EATD on the centroid

movement distance is given in Figure 9. We can see that themovement distance of the focal spot centroid has linear rela-tionship with the azimuth tracking error of the EATD. Thecentroid moving function is x=−12.06ε2 and y=−12.06ε1when the elevation angle of the dish concentrator is 0°.Considering the vanish-reduction effect, the centroid move-ment function of the focal spot in 38 kW XEM-Dish systemis as follows:

x = −12 06arcos cos2 90 − β ⋅ 1 − cos ε2 + cos ε2 ,y = −12 06ε1

16

Noteworthiness, the simplified formula y=−f× tan(ε1) isusually not equal to (16). For example, when the ε1 = 10mrad,the centroid movement distance is −120.60mm calculated by(16), while y=−f× tan(ε1) =−9490× tan(0.010) =−94.90mmand there is 25.70mm difference between them. There-fore, the centroid movement function of the focal spoton focal plane needs to be accurately determined by raytracing method.

The centroid movement function is used to guide the cal-ibration of the tracking error of the dish concentrator systemas the following: measuring the focal spot on the focal planein the dish concentrator system and extracting the centroidcoordinates of the focal spot; adopting the centroid coordi-nates to calculate the double-axis tracking error of the EATDby (16) and then calibrating the tracking error of the dishconcentrator. Of note, the assessment and calibration of thetracking error of the dish concentrator system are not suit-able for the large working elevation angle occasion (especiallyclose to the elevation angle is 90°). Because the vanish-reduction effect is becoming intense during this time, itwould lead the centroid movement distance of the focal spotalong the x-axis to be too small to be discovered. For exam-ple, the centroid movement distance of the focal spot equals0mm when the working elevation β equals 90° (Figure 9),so the effect of the azimuth tracking error of the EATD onthe dish concentrator is completely covered, but this effectappears again after the working elevation angle β decreasessuch as β = 30°.

Figure 9: The relationship between the centroid movement distance of the focal spot and azimuth tracking error of the EATD.

9International Journal of Photoenergy

The double-axis tracking error of the EATD and mirrorslope error has significant influence on the intercept factor(Figure 10). When the influence of the EATD tracking erroronly exists, the intercept factor curve would be the same aslong as the total tracking error ωerr of the dish concentratoris the same. Although the single tracking error of the EATDis different, such as curve 5, 6, and 7 in Figure 10, the inter-cept factor curves are the same. Another example is curve 2and 3 which are the same.

4.4. Application of the Vanish-Reduction Effect. The solarheliostat in tower power plant also adopts the EATD [9] asshown in Figure 11. The characteristic axis of the heliostat isthe mirror normal vector N, and azimuth tracking axis ofthe EATD is vertical to the ground level. Thus, the intersectionangle between the characteristic axis N and azimuth trackingaxis of the EATD attains the maximum and minimum when

the solar elevation angle β is 0° and 90°, respectively, givenby (17), where the intersection angle β1 between the reflectionray (aimed at target point on the receiver tower) and horizon-tal plane satisfied tan(β1) =H/L1; H is the height between thetarget point and heliostat center; L1 is horizontal distancebetween the target point and heliostat center. The angle β1increases with the decrease of the distance between the helio-stat and the receive tower, and the angle β1

min and β1max will

decrease at the same time, which could increase the vanish-reduction effect in the azimuth tracking error of the EATDtransmit to heliostat for improving the tracking performanceof the heliostat. According to this regulation, on the basis ofguarantying the tracking accuracy of the heliostat, this coulddecrease the manufacturing cost of the heliostat according togradual or gradient increase of the precision requirements ofthe azimuth tracking device in the EATD near or far fromthe receiver tower.

0.00

0.2

24 48 72 96 120 144 168 192 216 240 264 288

0.4

0.6

0.8

1.0

1.2

Inte

rcep

t fac

tor

Intercept radius Rin (mm)

2, 3

14

5, 6, 7

89 10

Figure 10: Influence of the EATD tracking error and mirror slope error on the intercept factor. 1: werr = 0; 2: β= 0°, ε1 = 3.6, ε2 = 0, werr = 3.6;

3: β= 0°, ε1 = 3, ε2 = 2, werr = 3.6; 4: β= 0°, ε1 = 6, ε2 = 0; 5: β= 30

°, ε1 = 2, ε2 = 8, werr = 7.184; 6: β= 0°, ε1 = 3.95, ε2 = 6, werr = 7.184; 7: β= 0

°,ε1 = 7.184, ε2 = 0; 8: β= 0

°, ε1 = 6, ε2 = 0, σ= 2; 9: β= 0°, ε1 = 10, ε2 = 0; 10: β= 0°, ε1 = 10, ε2 = 0, σ= 2. Note: only curve 10 and 14 have themirror slope error of σ = 2mrad, other curves have the mirror slope error of σ= 0mrad.

H

L1

N

T

𝛽1

𝜀1

𝜀2

n1

n2

Horizontal

Heliostat

𝛽max1

𝛽min1

Figure 11: A schematic diagram of the elevation-azimuth double-axis tracking of a heliostat in a tower plant.

10 International Journal of Photoenergy

β1min = 0 5 90° − β1  if β = 90°,

β1max = 90° − 0 5β1 if β = 0°

17

In order to increase the vanish-reduction effect to reducethe effect of the tracking error of the double-axis device onthe dish concentrator, a dish concentrator system using aspin-elevation double-axis tracking device (SETD) is pro-posed (Figure 12). The spin-elevation double-axis trackingwas put forward by Chen et al. [8] and they defined the spinaxis is a line between the heliostat center and fixed target andthe elevation tracking axis in the SETD vertical to the spinaxis. The SETD is widely used in the solar concentrator fieldwhere the solar concentrator can move but the heat absorberreceiver is fixed [29]. However, the dish concentrator andreceiver move together in the dish concentrator system andthere is no fixed target in this system. Thus, we definedany direction as the spin axis. In Figure 12, angle β2 is theinclination between the azimuth tracking axis of the SETDand horizontal plane. The spin tracking error ε2r of the dishconcentrator caused by spin tracking error of the SETDtransmission to the dish concentrator could be calculatedby (18). From Figure 13, we could clearly see that the errorε2r reduced to 0mrad first and then increased with theincrease of the working elevation angle β of the dish concen-trator, and the error ε2r equals 0mrad when the angle βequals to the inclination angle β2. Therefore, selecting theappropriate angle β2 could make the vanish-reduction effectadopt the SETD’s obvious intensity than the EATD duringthe tracking error transmission, in order to increase theinfluence of the tracking error of the tracking device on thedish concentrator. Such as the angle β2 = 60° in Figure 13,within the elevation angle β range of 0°~75°, the spin track-ing error ε2r of the dish concentrator adopted the SETDform obviously less than the EATD (i.e., β2 = 90°). Thus,appropriately reducing the manufacture precision require-ments of the spin tracking device in SETD could decreasethe manufacturing cost.

ε2r β2, β = arcos cos2 β − β2 ⋅ 1 − cos ε2 + cos ε2 18

5. Conclusion

In this paper, a flux distribution model of the focal plane of adish concentrator system has been developed based on raytracking method for researching the effect of the mirror sur-face slope error, DNI and EATD tracking error on the focalspot characteristics, and the error transmission law of thetracking error of the double-axis tracking device transmis-sion to dish concentrator system. The main conclusions areas follows:

(1) The azimuth tracking error ε2r of the dish concen-trator is transmitted by the azimuth tracking errorε2 of the EATD. When the error ε2 is constant, theerror ε2r decreases with the increase of the workingelevation angle β of the dish concentrator and theerror ε2r equals to 0mrad when β = 90°. We call thisphenomenon the called vanish-reduction effect intracking error transmission. The vanish-reduction

effect-produced essence is caused by the nonverticalof the focal axis being nonvertical to the dish con-centrator and tracking axis of the EATD, and thevanish-reduction effect is increased with thedecrease of the nonvertical angle, which couldimprove the tracking accuracy of the dish concentra-tor. Due to the elevation tracking axis of the EATDalways vertical to the focal axis, the elevation track-ing error of the dish concentrator equals to the ele-vation tracking error of the EATD. In addition, theintercept factor only depends on the total trackingerror of the dish concentrator when only consideringthe tracking error.

𝛽2

𝜀1

𝜀2

NF

n1

n2

𝛽

F

Horizontal

Figure 12: A schematic diagram of the spin-elevation double-axistracking in dish concentrator system.

0.000.501.001.502.002.503.003.504.004.50

0

𝜀 2r (

mra

d)

88

𝛽2 = 0°

𝛽 (°)

𝛽2 = 30°𝛽2 = 5°

𝛽2 = 60°𝛽2 = 90°

8072645648403224168

Figure 13: Error transmission law of the tracking error of the spinaxis transmission to dish concentrator (spin axis tracking error is4.0mrad).

11International Journal of Photoenergy

(2) The centroid position of the focal spot on the focalplane depends on the tracking error of the dish con-centrator without the effects of the mirror slopeerror and DNI. Only the azimuth tracking errorand elevation tracking error of the EATD haveinfluence on the centroid’s movement distance ofthe focal spot along x-axis and y-axis, respectively,and the centroid’s moving distance has linear func-tion relationship with the EATD tracking error. For38 kW XEM-Dish system, the centroid’s movingfunction is x = −12 06ε2r and y = −12 06ε1. This func-tion could beused to evaluate and quantitative calibra-tion of the tracking error of the dish concentratorsystem but is not suitable for the large working eleva-tion angle occasion.

(3) The vanish-reduction effect also exists in the azimuthtracking error of the EATD transmission to the solarheliostat. Through decreasing the distance betweenthe heliostat and receiver tower, increasing the heightof the receiver tower and solar elevation angle couldincrease the vanish-reduction effect in order toimprove the tracking performance of the heliostat.Besides, a dish concentrator system using a spin-elevation double-axis tracking device (SETD) is pro-posed, which could increase the influence of the spintracking error of the SETD on the dish concentratorsystem and improve the tracking performance ofthe dish concentrator system.

Nomenclature

Akm: Daylighting area of the mirror grid km (m2)f: Focal length of the dish concentrator (m)K, M: Discretion numbers of the dish concentrator mirrorN, W: Discretion numbers of the sun coneR1: Radius of the dish concentrator (m)R2: Void field radius of the dish concentrator (m)W0: DNI value (Wm−2)werr: Total tracking error of the dish concentrator (mrad),

werr = ε2r2 + ε1

2

Greek symbols

δ: Solar half angle (mrad)σ: Standard deviation of the mirror slope error (mrad)ε1: Elevation tracking error of the EATD (mrad)ε2: Azimuth tracking error of the EATD (mrad)ε2r: Azimuth angle or spin angle tracking error of the dish

concentrator (mrad)β: Concentrator working elevation angle (degrees)β1: The intersection angle between the reflection ray (at

heliostat center) and the horizontal plane (degrees)β2: Angle between spin axis and horizontal plane (degrees)θm: Lack angle of the dish concentrator (degrees).

Abbreviations

EATD: Elevation-azimuth double-axis tracking deviceDNI: Direct normal irradiation.

Subscripts

km: Concentrator mirror grid at position (k, m)wn: Sun cone grid at position (w, n).

Conflicts of Interest

The authors declare that there is no conflicts of interestregarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural ScienceFoundation of People’s Republic of China (nos. 51576061and 51641504), Hunan Provincial Natural Science Founda-tion of China (no. 2016JJ2052), and Hunan Province Post-graduate Innovation Project Foundation of People’s Republicof China (nos. CX2016B549 and CX2017B628).

References

[1] H. Sayyaadi and M. S. Boroujeni, “Conceptual design, processintegration, and optimization of a solar Cu-Cl thermochemicalhydrogen production plant,” International Journal of Hydro-gen Energy, vol. 42, no. 5, pp. 2771–2789, 2017.

[2] L. Nkhonjera, T. Bello-Ochende, G. John, and C. K. King’ondu,“A review of thermal energy storage designs, heat storagematerials and cooking performance of solar cookers with heatstorage,” Renewable and Sustainable Energy Reviews, vol. 75,pp. 157–167, 2017.

[3] D. A. Baharoon, H. A. Rahman, W. Z. W. Omar, and S. O.Fadhl, “Historical development of concentrating solar powertechnologies to generate clean electricity efficiently – a review,”Renewable and Sustainable Energy Reviews, vol. 41, pp. 996–1027, 2015.

[4] Y. Qiu, Y. L. He, M. Wu, and Z. J. Zheng, “A comprehensivemodel for optical and thermal characterization of a linearFresnel solar reflector with a trapezoidal cavity receiver,”Renewable Energy, vol. 97, pp. 129–144, 2016.

[5] G. Li and Y. Jin, “Optical simulation and experimental verifica-tion of a Fresnel solar concentrator with a new hybrid secondoptical element,” International Journal of Photoenergy,vol. 2016, Article ID 4970256, 8 pages, 2016.

[6] F. Chen, M. Li, R. Hassanien EmamHassanien et al., “Study onthe optical properties of triangular cavity absorber for para-bolic trough solar concentrator,” International Journal ofPhotoenergy, vol. 2015, Article ID 895946, 9 pages, 2015.

[7] J. Wang, J. Wang, X. Bi, and X. Wang, “Performance simula-tion comparison for parabolic trough solar collectors inChina,” International Journal of Photoenergy, vol. 2016, ArticleID 9260943, 16 pages, 2016.

[8] Y. T. Chen, K. K. Chong, T. P. Bligh et al., “Non-imaging,focusing heliostat,” Solar Energy, vol. 71, no. 3, pp. 155–164,2001.

[9] O. Behar, A. Khellaf, and K. Mohammedi, “A review of studieson central receiver solar thermal power plants,” Renewableand Sustainable Energy Reviews, vol. 23, pp. 12–39, 2013.

[10] J. Coventry and C. Andraka, “Dish systems for CSP,” SolarEnergy, vol. 152, pp. 140–170, 2017.

[11] H. Ma, G. Jin, X. Zhong, K. Xu, and Y. Li, “Optical design of asolar dish concentrator based on triangular membrane facets,”

12 International Journal of Photoenergy

International Journal of Photoenergy, vol. 2012, Article ID391921, 5 pages, 2012.

[12] Q. Mao, L. Zhang, and H. Wu, “Study on concentratingcharacteristics of a solar parabolic dish concentrator withinhigh radiation flux,” International Journal of Photoenergy,vol. 2015, Article ID 787280, 7 pages, 2015.

[13] I. Reda and A. Andreas, “Solar position algorithm for solarradiation applications,” Solar Energy, vol. 76, no. 5, pp. 577–589, 2004.

[14] R. O. Hughes, “Effects of tracking errors on the performance ofpoint focusing solar collectors,” Solar Energy, vol. 24, no. 1,pp. 83–92, 1980.

[15] V. Badescu, “Different tracking error distributions and theireffects on the long-term performances of parabolic dishsolar power systems,” International Journal of Solar Energy,vol. 14, no. 4, pp. 203–216, 1994.

[16] V. Badescu, “Theoretical derivation of heliostat tracking errorsdistribution,” Solar Energy, vol. 82, no. 12, pp. 1192–1197,2008.

[17] Y. Hu and Y. Yao, “Optical analysis and output evaluation fora two-stage concentration photovoltaic system by usingMonteCarlo ray-tracing method,” Optik-International Journal forLight and Electron Optics, vol. 131, pp. 713–723, 2017.

[18] K. Shanks, N. Sarmah, J. P. Ferrer-Rodriguez et al., “Theoret-ical investigation considering manufacturing errors of a highconcentrating photovoltaic of cassegrain design and itsexperimental validation,” Solar Energy, vol. 131, pp. 235–245, 2016.

[19] X. L. Xia, G. L. Dai, and Y. Shuai, “Experimental and numericalinvestigation on solar concentrating characteristics of asixteen-dish concentrator,” International Journal of HydrogenEnergy, vol. 37, no. 24, pp. 18694–18703, 2012.

[20] H. Li, W. Huang, F. Huang, P. Hu, and Z. Chen, “Optical anal-ysis and optimization of parabolic dish solar concentrator witha cavity receiver,” Solar Energy, vol. 92, pp. 288–297, 2013.

[21] L. S. M. Castellanos, G. E. C. Caballero, V. R. M. Cobas et al.,“Mathematical modeling of the geometrical sizing and thermalperformance of a dish/Stirling system for power generation,”Renewable Energy, vol. 107, pp. 23–35, 2017.

[22] G. Johnston, “Focal region measurements of the 20m 2 tileddish at the Australian National University,” Solar Energy,vol. 63, no. 2, pp. 117–124, 1998.

[23] S. Ulmer, W. Reinalter, P. Heller et al., “Beam characterizationand improvement with a flux mapping system for dish concen-trators,” in ASME Solar 2002: International Solar EnergyConference, pp. 285–292, Reno, NV, USA, 2002.

[24] H. Lee, “The geometric-optics relation between surface slopeerror and reflected ray error in solar concentrators,” SolarEnergy, vol. 101, pp. 299–307, 2014.

[25] Q. Mao, Y. Shuai, and Y. Yuan, “Study on radiation flux of thereceiver with a parabolic solar concentrator system,” EnergyConversion and Management, vol. 84, pp. 1–6, 2014.

[26] M. H. Tan and K. K. Chong, “Influence of self-weight on elec-trical power conversion of dense-array concentrator photovol-taic system,” Renewable Energy, vol. 87, pp. 445–457, 2016.

[27] M. M. Elsayed, K. A. Fathalah, and O. M. Al-Rabghi, “Mea-surements of solar flux density distribution on a plane receiverdue to a flat heliostat,” Solar Energy, vol. 54, no. 6, pp. 403–411,1995.

[28] S. M. Jeter, “The distribution of concentrated solar radiation inparaboloidal collectors,” Journal of Solar Energy Engineering,vol. 108, no. 3, pp. 219–225, 1986.

[29] B. H. Lim, K. K. Chong, C. S. Lim, and A. C. Lai, “Latitude-orientated mode of non-imaging focusing heliostat usingspinning-elevation tracking method,” Solar Energy, vol. 135,pp. 253–264, 2016.

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