In this paper, a flux distribution model of the focal plane in dish concentrator system has been established based on ray tracking method. This model was adopted for researching the influence of the mirror slope error, solar direct normal irradiance, and tracking error of elevation-azimuth tracking device (EATD) on the focal spot characteristics (i.e., flux distribution, geometrical shape, centroid position, and intercept factor). The tracking error transmission law of the EATD transferred to dish concentrator was also studied. The results show that the azimuth tracking error of the concentrator decreases with the increase of the concentrator elevation angle and it decreases to 0 mrad when the elevation angle is 90°. The centroid position of focal spot along

Solar energy is a clean and environment-friendly renewable energy source, which is plentiful and can be widely distributed. Developing and utilizing solar energy are important ways to solve energy shortages and environmental pollution problems, for example, the H_{2} production by solar thermochemical reactions [

A 38 kW dish-Stirling solar thermal power system (38 kW XEM-Dish system). The diameter is 17.70 m and the focal length is 9.49 m of the parabolic dish concentrator and was built in Xiangtan Electric Manufacturing Group, China.

The dish concentrator via an elevation-azimuth double-axis tracking device (EATD) was used to accurately track the sun’s position so it could focus the sunlight to the precalculated position of the receiver [_{1} and azimuth tracking axis _{2} in the EATD can rotate a corresponding angle (Figure

(a) Geometry parameters and mirror discrete of the dish concentrator and (b) schematic diagram of the tracking error and mirror slope error of the dish concentrator.

Many works focus on the research for the influence of tracking error on the optical performance or flux distribution of dish concentrator system [^{2} dish concentrator. This concentrator mirror slope error has been determined is 2.0 mrad. Xia et al. [

The above-mentioned literatures all focus on the influence of the total tracking error of dish concentrator systems on the flux distribution and optical performance, while the researches of the influence of the tracking error of the EATD on the tracking performance and optical performance of dish concentrator system are limited. In this paper, we focused on the above two aspects; a flux distribution model of the focal plane in dish concentrator systems has been established based on ray tracking method. The concentrator mirror and solar cone in nonuniform energy distribution are discreted by square grids in this model (in Section

The flux distribution model of the plane receiver in dish concentrator systems is developed based on the ray tracking method. The ray tracking method needs to discrete the mirror surface of dish concentrators, plane receiver, and sun cone. There are two methods to discrete the mirror surface and sun cone, a random MCRT method [

As shown in Figure ^{2} + ^{2} = 4_{1} is aperture radius of the dish concentrator, and angle _{m} and radius _{2} are no mirror area in the dish concentrator. The steps for optical discretization of the dish concentrator are as follows: (1) use a square with side length of 2_{1} (blue wire frame in Figure _{1}/_{km}_{km}_{1}/^{2}); (2) according to the coordinate and distance _{km}_{km}_{km}_{km}_{km}_{km}_{km}

In practical application, the reflection mirror of a dish concentrator would have microslope error, which would impact the normal vector of the reflection mirror (Figure _{3} and tangential angle _{4} [_{1} and _{2} are uniformly distributed random numbers within the range of 0~1,

Considering the mirror slope error of the dish concentrator, the vector _{km}_{x}

In ideal tracking condition, the focal axis of the dish concentrator is parallel to the center line _{1} and _{2}, respectively, Figure _{km}_{err} between the focal axis of the dish concentrator and center line _{err} is the total tracking error of the dish concentrator. This could be calculated by (_{1} is elevation tracking axis vector of the EATD, _{1} = [1, 0, 0]; _{2} is the azimuth tracking axis vector of the EATD, _{2} = [0, −cos

When the dish concentrator system exists tracking error, the plane receiver (located at the focal plane of dish concentrators) would rotate with the dish concentrator. This time, the focal point _{1} and the corresponding plane receiver normal vector is

In this work, the plane receiver is a square with side length is _{wn} on the mirror point _{1}, _{1}, _{1}]; (3) the grid number _{wn} carrying energy (Section

The solar radiation intensity in a sun cone gradually decreases from the center to the border. The nonuniform energy distribution model is given as [

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

The sun cone received by different positions in the dish concentrator mirror was assumed to be exactly identical. Thus, the sun cone was established on the point _{s}, _{s} = _{wn} is

Based on the principle of energy conservation equation, the dish concentrator and sun cone before and after the discretization should satisfy the following relationship:
_{0} is the DNI value (W/m^{2}); _{wn}_{wn}

Finally, the sunlight vector

Based on the optical model and sun cone model in this paper, the ray tracing codes were compiled in Visual C++ 6.0 software for calculating the flux distribution on the focal plane of the dish concentrator. The Jeter theoretical results [

(a) The comparison of the flux distribution result on the focal plane and (b) flux distribution contour map using ray tracing method in this paper (W/m^{2}).

Due to the reason that the tracking error of the EATD would cause the tracking error of the dish concentrator, the position and geometry of the focal spot on the plane receiver would all change. The focal spot moves and extends to the upper right direction, and the flux contour is approximately ellipse shaped but the center position of the ellipse is different, such as in Figure

Geometrical parameters of the focal spot on the plane receive.

The intercept factor is defined as the ratio of the energy on the plane receiver (radius is _{in}) to the total received energy [_{s}_{s}_{1}. The grid of the flux density equal or approaching to _{t} × _{0} was found on the receiver and then the center point coordinates of the grid were extracted (as the ellipse boundary point) for fitting (_{t} is a flux density extraction threshold value). The searching process of the ellipse boundary point is that according to the plane receiver discrete grid, from top to bottom (_{t} × _{0}. In the same way, from bottom to the top to find the ellipse bottom boundary point.

The tracking error of the EATD would cause the tracking error of the dish concentrator, but the tracking error of the dish concentrator may not be equal to the tracking error of the EATD. This tracking error transmission (i.e., tracking error of the EATD transmitted to dish concentrator) is related to the working elevation angle _{err} of the dish concentrator decreases with the increase of elevation angle _{err} decreases to 0 rad when the angle _{err} decreases with the increase of the elevation angle _{err} is equal to the elevation tracking error

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

The vanish-reduction effect produce essence is because the focal axis of the dish concentrator and the tracking axis of the EATD are not vertical. The intersection angle between the focal axis of the dish concentrator and azimuth tracking axis of the EATD is (90° −

In this paper, we take the typical 38 kW XEM-Dish system (Figure _{m} = 30°, _{0} = 1000 W/m^{2}, _{t} = 50, mirror discrete parameter ^{9}.

The influence of the tracking error of EATD on the flux curve and flux contour map of the focal spot are shown in Figures

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

The flux distribution of the focal plane under the EATD tracking error (W/m^{2}). (a) _{1} = 2 mrad, and _{2} = 8 mrad; (b) _{1} = 2 mrad, and _{2} = 8 mrad; (c) _{1} = 6.9 mrad, and _{2} = 2 mrad; (d) _{1} = 6.9 mrad, and _{2} = 6.9 mrad; (e) _{1} = 6.9 mrad, _{2} = 6.9 mrad, and _{0} = 800 W/m^{2}; (f) _{1} = 6.9 mrad, _{2} = 6.9 mrad, and _{0} = 800 W/m^{2}; (g) _{1} = 2 mrad, and _{2} = 0; (h) _{1} = 0, _{2} = 6.9 mrad or _{1} = 0, and _{2} = 8 mrad. Note: only Figures _{0} = 800 W/m^{2} and the other figures are with _{0} = 1000 W/m^{2}.

The azimuth and elevation tracking error of the EATD only have influence on the centroid coordinates of the focal spot along _{1} = 2 mrad, and _{2} = 8 mrad) and Figure _{1} = 2 mrad, and _{2} = 0 mrad); the coordinate of the focal spot centroid along

The functional relationship between the centroid movement distance of the focal spot on focal plane and EATD tracking error is called centroid movement function; this function could be used for evaluating and quantitatively calibrating the tracking error of the EATD. The influence curve of the azimuth tracking error of the EATD on the centroid movement distance is given in Figure _{2} and _{1} when the elevation angle of the dish concentrator is 0°. Considering the vanish-reduction effect, the centroid movement function of the focal spot in 38 kW XEM-Dish system is as follows:

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

Noteworthiness, the simplified formula _{1}) is usually not equal to (_{1} = 10 mrad, the centroid movement distance is −120.60 mm calculated by (_{1}) = −9490 × tan(0.010) = −94.90 mm and there is 25.70 mm difference between them. Therefore, the centroid movement function of the focal spot on focal plane needs to be accurately determined by ray tracing method.

The centroid movement function is used to guide the calibration of the tracking error of the dish concentrator system as the following: measuring the focal spot on the focal plane in the dish concentrator system and extracting the centroid coordinates of the focal spot; adopting the centroid coordinates to calculate the double-axis tracking error of the EATD by (

The double-axis tracking error of the EATD and mirror slope error has significant influence on the intercept factor (Figure _{err} of the dish concentrator is the same. Although the single tracking error of the EATD is different, such as curve 5, 6, and 7 in Figure

Influence of the EATD tracking error and mirror slope error on the intercept factor. 1: _{err} = 0; 2: _{1} = 3.6, _{2} = 0, _{err} = 3.6; 3: _{1} = 3, _{2} = 2, _{err} = 3.6; 4: _{1} = 6, _{2} = 0; 5: _{1} = 2, _{2} = 8, _{err} = 7.184; 6: _{1} = 3.95, _{2} = 6, _{err} = 7.184; 7: _{1} = 7.184, _{2} = 0; 8: _{1} = 6, _{2} = 0, σ = 2; 9: _{1} = 10, _{2} = 0; 10: _{1} = 10, _{2} = 0, σ = 2. Note: only curve 10 and 14 have the mirror slope error of

The solar heliostat in tower power plant also adopts the EATD [_{1} between the reflection ray (aimed at target point on the receiver tower) and horizontal plane satisfied tan(_{1}) = _{1}; _{1} is horizontal distance between the target point and heliostat center. The angle _{1} increases with the decrease of the distance between the heliostat and the receive tower, and the angle

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

In order to increase the vanish-reduction effect to reduce the effect of the tracking error of the double-axis device on the dish concentrator, a dish concentrator system using a spin-elevation double-axis tracking device (SETD) is proposed (Figure _{2} is the inclination between the azimuth tracking axis of the SETD and horizontal plane. The spin tracking error _{2r} of the dish concentrator caused by spin tracking error of the SETD transmission to the dish concentrator could be calculated by (_{2r} reduced to 0 mrad first and then increased with the increase of the working elevation angle _{2r} equals 0 mrad when the angle _{2}. Therefore, selecting the appropriate angle _{2} could make the vanish-reduction effect adopt the SETD’s obvious intensity than the EATD during the tracking error transmission, in order to increase the influence of the tracking error of the tracking device on the dish concentrator. Such as the angle _{2r} of the dish concentrator adopted the SETD form obviously less than the EATD (i.e.,

A schematic diagram of the spin-elevation double-axis tracking in dish concentrator system.

Error transmission law of the tracking error of the spin axis transmission to dish concentrator (spin axis tracking error is 4.0 mrad).

In this paper, a flux distribution model of the focal plane of a dish concentrator system has been developed based on ray tracking method for researching the effect of the mirror surface slope error, DNI and EATD tracking error on the focal spot characteristics, and the error transmission law of the tracking error of the double-axis tracking device transmission to dish concentrator system. The main conclusions are as follows:

The azimuth tracking error _{2r} of the dish concentrator is transmitted by the azimuth tracking error _{2} of the EATD. When the error _{2} is constant, the error _{2r} decreases with the increase of the working elevation angle _{2r} equals to 0 mrad when

The centroid position of the focal spot on the focal plane depends on the tracking error of the dish concentrator without the effects of the mirror slope error and DNI. Only the azimuth tracking error and elevation tracking error of the EATD have influence on the centroid’s movement distance of the focal spot along

The vanish-reduction effect also exists in the azimuth tracking error of the EATD transmission to the solar heliostat. Through decreasing the distance between the heliostat and receiver tower, increasing the height of the receiver tower and solar elevation angle could increase the vanish-reduction effect in order to improve the tracking performance of the heliostat. Besides, a dish concentrator system using a spin-elevation double-axis tracking device (SETD) is proposed, which could increase the influence of the spin tracking error of the SETD on the dish concentrator system and improve the tracking performance of the dish concentrator system.

_{km}

Daylighting area of the mirror grid ^{2})

Focal length of the dish concentrator (m)

Discretion numbers of the dish concentrator mirror

Discretion numbers of the sun cone

_{1}:

Radius of the dish concentrator (m)

_{2}:

Void field radius of the dish concentrator (m)

_{0}:

DNI value (Wm^{−2})

_{err}:

Total tracking error of the dish concentrator (mrad),

Solar half angle (mrad)

Standard deviation of the mirror slope error (mrad)

_{1}:

Elevation tracking error of the EATD (mrad)

_{2}:

Azimuth tracking error of the EATD (mrad)

_{2r}:

Azimuth angle or spin angle tracking error of the dish concentrator (mrad)

Concentrator working elevation angle (degrees)

_{1}:

The intersection angle between the reflection ray (at heliostat center) and the horizontal plane (degrees)

_{2}:

Angle between spin axis and horizontal plane (degrees)

_{m}:

Lack angle of the dish concentrator (degrees).

Elevation-azimuth double-axis tracking device

Direct normal irradiation.

Concentrator mirror grid at position (

Sun cone grid at position (

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

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