Prediction of Solar Concentration Flux Distribution for a Heliostat Based on Lunar Concentration Image and Generative Adversarial Networks

ABSTRACT

The predictive analysis of solar flux distribution on the receiver surface is critical in optimizing the concentration processes of concentrating solar power (CSP) plants. Due to the difficulties of directly measuring the solar flux distribution of the heliostat field, tracking the Moon and measuring the lunar concentration ratio distribution become a promising option. However, many factors affect the flux distribution of a heliostat field. To obtain an accurate predictive model for the solar flux distribution, we propose a deep-learning method using conditional generative adversarial networks (cGAN) and lunar concentration images. The method can take account of tracking errors of individual heliostats, defects of reflecting surfaces, as well as atmospheric attenuation effects, and has the potential to give a reliable prediction of solar flux distribution. Mathematical relations between the solar flux distribution and the solar concentration ratio distribution are discussed in the paper. Experiments have been designed and carried out with an ordinary heliostat at the Beijing Badaling solar concentrating power station. Experimental results show that the AI-generated solar concentration ratio distributions are very close to the actual solar concentration ratio distributions, demonstrating the feasibility of AI models for the prediction of solar flux distribution.

Introduction

Energy is the basis for all human activities and industrial developments. In half a century, the application of renewable energy has been a significant issue attracting worldwide attention. Solar energy accounts for about 99.98% of all feasible renewable energy resources (Rekker et al. 2018). Concentrating solar power (CSP) technology has grown fast in the last two decades. It exhibits a considerable potential to supplement photovoltaic (PV) power systems or to be part of the hybrid renewable energy system (Ballestrín, Burgess, and Cumpston 2012). Currently, there are two major types of CSP systems: parabolic-trough concentrating systems and solar tower concentrating systems. For the solar tower system, many heliostats track the Sun and reflect the sunlight onto the receiver located on the tower top, forming a high-intensity solar flux distribution on the aperture of the receiver. The receiver absorbs the incoming light energy and heats the flowing medium, which can be used to drive a sterling machine to generate electrical power or stored for future use.

An accurate measurement of the solar flux distribution is considered crucial for improving the performance of the solar concentrator, as well as the overall efficiency of the concentration solar power system. It is also important for the maintenance and safe-running of the receiver. However, directly measuring the energy flow density on the surface of receiver presents many challenges, such as potential disturbances to normal operations of receiver, limitation of space, and high temperature (He et al. 2019). Researchers have tried the method of installing arrays of heat flux sensors on the receiver and directly measuring the heat flux distribution (Ballestrin 2002). Such method can only measure the flux densities at a limited number of points, and it is not applicable to most commercial CSP systems. Another way is using the CCD camera to take pictures of the concentrated solar spots on the target surface and use a small number of heat flux sensors as references to estimate the energy flux distribution on the surface of the receiver (Röger et al. 2014). This way has been used for many years in CSP due to the lower cost and relatively high spatial resolution at present (Lüpfert et al. 2000; Vontobel, Schelders, and Real 1982). This way requires a very high standard CCD imaging system and sophisticated cooling system; otherwise, the accuracy achieved with way is not satisfying.

Using the Moon as a light source to measure the energy flux distribution of a concentrating heliostat field is another option. The Moon is a cold light source, and no high temperature is generated even if many heliostats concentrate moonlight on the receiver surface of the central tower. Also, it can be done at night without influencing the daytime tracking tasks of the heliostats. Hisada first used the method of concentrating moonlight instead of sunlight to study the solar flux distribution (Hisada et al. 1957). Roosendaal et al. use a concentrated full-moon image on the target surface to investigate a new style of membrane dish concentrator (Roosendaal, Swanepoel, and Le Roux 2020). Lovegrove used the full Moon as a light source to study the concentrating performance of a large dish concentrator (Lovegrove, Burgess, and Pye 2011). Xiao et al. used the full Moon to measure the installation error of the dish concentrator (Xiao et al. 2017). Using the full Moon as a light source, the concentration ratio distribution of the Yanqing concentration solar power plant has been measured and the solar flux distribution on the receiver aperture is calculated (Guo et al. 2020). A lunar flux mapping model is proposed to estimate the expected solar concentration ratio distribution (CRD) based on the measured lunar CRDs (Guo et al. 2021). A calculation model for the moonlight CRD of a dish concentrator with various moon status is proposed by Hao (Hao, Minghuan, and Zhifeng 2022); however, the accuracy is not satisfying.

Besides the influence of light sources, the concentration flux distribution of a concentrator is also affected by the tracking inaccuracies of heliostats, the defects on the reflecting surface, as well as environmental factors, such as the atmosphere temperature and humidity. Therefore, directly using lunar concentration ratio distribution and the solar DNI to calculate the solar flux distribution cannot achieve good result. Here we propose a deep learning-based method for the prediction of solar concentration ratio distribution based on lunar concentration ratio distribution in this paper. Specifically, conditional generative adversarial networks are used to find the mapping functions between the lunar concentration distribution and solar concentration distribution of a same concentrating device.

Materials and Methods

Solar and Lunar Concentration Ratio Distribution

The concentration flux distribution at the receiver can be seen as the product of the direct normal irradiance (DNI) of the light source and the concentration ratio distribution, which means that the solar concentration flux distribution at the receiver can be obtained with the concentration ratio distribution and the real-time DNI values of the Sun. Here, the concentration ratio refers to the relative density. Assuming the environmental factors stay unchanged, the measured lunar and solar concentration ratio distributions are formulated as Equations (1)(1) $\mathrm{CR}{\left(\mathrm{x},\mathrm{y}\right)}_{\mathrm{moon}}=\frac{{\mathit{I}}_{\mathit{moon}}}{\mathit{DN}{I}_{\mathit{moon}}}$(1) and (2(2) $\mathrm{CR}{\left(\mathrm{x},\mathrm{y}\right)}_{\mathrm{sun}}=\frac{\mathit{F}\left(\mathit{x},\mathit{y}\right)}{\mathit{DN}{\mathit{I}}_{\mathit{sun}}}$(2) ).

(1) $\mathrm{CR}{\left(\mathrm{x},\mathrm{y}\right)}_{\mathrm{moon}}=\frac{{\mathit{I}}_{\mathit{moon}}}{\mathit{DN}{I}_{\mathit{moon}}}$(1)

(2) $\mathrm{CR}{\left(\mathrm{x},\mathrm{y}\right)}_{\mathrm{sun}}=\frac{\mathit{F}\left(\mathit{x},\mathit{y}\right)}{\mathit{DN}{\mathit{I}}_{\mathit{sun}}}$(2)

where $\mathit{\hspace{0.17em}}{I}_{\mathit{moon}}$ is the discrete distribution of the lunar concentration illuminance in the x-y plane for the full Moon, $\mathit{DN}{I}_{\mathit{moon}}$ is the direct normal illuminance of the Moon, $\mathrm{CR}{\left(\mathrm{x},\mathrm{y}\right)}_{\mathrm{moon}}$ is the lunar concentration ratio distribution for the full Moon generated by a heliostat, $\mathit{F}\left(\mathit{x},\mathit{y}\right)$ is the solar flux distribution at the discrete points in the x-y plane of the target surface, $\mathit{DN}{I}_{\mathit{sun}}$ is sun direct normal irradiance, and $\mathrm{CR}{\left(\mathrm{x},\mathrm{y}\right)}_{\mathrm{sun}}$ is the solar concentration ratio distribution generated by the same heliostat.

If the incident light are ideal parallel beams with constant density, the concentration ratio distribution of a heliostat should depend on the curvature of the heliostat, the optical qualities of sub-mirrors, and the manufacturing precision of the whole heliostat system. For a heliostat on the Earth, the disc angles of Sun and Moon are very close. The solar and lunar light has a similar spectrum and traverse through the same atmosphere. As a consequence, researchers have used the lunar concentration ratio distribution to calculate the solar flux distribution of heliostats in the field tests of CSP plant (Hisada et al. 1957; Lovegrove, Burgess, and Pye 2011; Roosendaal, Swanepoel, and Le Roux 2020), as shown in Equation (3)(3) $\mathrm{F}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{CR}{\left(\mathrm{x},\mathrm{y}\right)}_{\mathrm{sun}}\ast \mathrm{DN}{\mathrm{I}}_{\mathrm{sun}}=\mathrm{CR}{\left(\mathrm{x},\mathrm{y}\right)}_{\mathrm{moon}}\ast \mathrm{DN}{\mathrm{I}}_{\mathrm{sun}}$(3) .

(3) $\mathrm{F}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{CR}{\left(\mathrm{x},\mathrm{y}\right)}_{\mathrm{sun}}\ast \mathrm{DN}{\mathrm{I}}_{\mathrm{sun}}=\mathrm{CR}{\left(\mathrm{x},\mathrm{y}\right)}_{\mathrm{moon}}\ast \mathrm{DN}{\mathrm{I}}_{\mathrm{sun}}$(3)

Equation (3)(3) $\mathrm{F}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{CR}{\left(\mathrm{x},\mathrm{y}\right)}_{\mathrm{sun}}\ast \mathrm{DN}{\mathrm{I}}_{\mathrm{sun}}=\mathrm{CR}{\left(\mathrm{x},\mathrm{y}\right)}_{\mathrm{moon}}\ast \mathrm{DN}{\mathrm{I}}_{\mathrm{sun}}$(3) make a hypothesis that CR(x,y)_moon = CR(x,y)_sun. In fact, both solar and lunar rays are not ideal parallel beams with constant density. As a consequence, the lunar concentration ratio distribution cannot be exactly the same as the solar concentration ratio. Considering the dynamic change of environmental factors and the fact that the moon light are sun rays reflected by the surface of Moon, a simple equalization of the solar CRD to the lunar CRD in Equation (3)(3) $\mathrm{F}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{CR}{\left(\mathrm{x},\mathrm{y}\right)}_{\mathrm{sun}}\ast \mathrm{DN}{\mathrm{I}}_{\mathrm{sun}}=\mathrm{CR}{\left(\mathrm{x},\mathrm{y}\right)}_{\mathrm{moon}}\ast \mathrm{DN}{\mathrm{I}}_{\mathrm{sun}}$(3) are not likely to give accurate predictions of the solar flux distribution. Therefore we propose an innovative method based on conditional generative adversarial networks (cGAN) to find the more accurate mapping function between the solar CRD and the lunar CRD for a heliostat.

Conditional Generative Adversarial Network

With the rapid progress of deep learning technology, more and more AI tools have been invented to solve tough problems in science and engineering recently. Among various neural network structures, generative adversarial networks (GAN) (Goodfellow et al. 2014) are distinguished with their unique idea of the competitive learning process with a pair of neural networks (Generator and Discriminator) to obtain a result. If extra information is provided to the Generator and Discriminator, the GAN becomes conditional generative adversarial network (CGAN). Since the invention, CGAN have achieved great successes on image synthesis, image style transfer, image translation, and among others.

The core task of predicting solar concentration ratio distribution of a heliostat is actually to find out the optical modulation function of the heliostat with certain incidence angle. Conditional generative adversarial neural networks are suitable for the modeling of complex systems with certain constraints. Two different GAN frameworks, CycleGAN and Pix2Pix, have been explored in this paper.

CylcleGAN

The schematic diagram of Cycle Generative Adversarial Network (CycleGAN) is shown in Figure 1. There are two generators (G, F) and two discriminators (${\mathrm{D}}_{\mathrm{x}}$, ${\mathrm{D}}_{\mathrm{y}}$) in CycleGAN.

Figure 1. Illustration of CycleGAN.

In Figure 1, G stands for the generator network which generates new image from x-domain data, and Dy stands for Discriminator, which discriminates the generated image from true images, The Generator and the Discriminator are adversarial to each other and are trained recursively to reach stable by mini-maximizing the loss function, shown in Equations (5)(5) $L_{\mathit{cyc}}\left(\mathit{G},\text{F}\right)=n_{\mathit{y}~{\mathit{p}}_{\mathit{data}}\left(\mathit{y}\right)}\text{}left[}\left|\right|\text{{}\mathit{G}\left(\mathit{F}\left(\mathit{y}\right)\right)-\mathit{y}\left|\right|1+n_{\mathit{x}~{\mathit{p}}_{\mathit{data}}\left(\mathit{x}\right)}\left|\right|\mathit{F}\left(\mathit{G}\left(\mathit{x}\right)\right)-\mathit{x}\left|\right|1$(5) .

(4) $L_{\mathit{GAN}}\left(\mathit{G},D_y,\mathit{X},\mathit{Y}\right)=n_{\mathit{y}~{\mathit{p}}_{\mathit{data}}\left(\mathit{y}\right)}\left[\mathit{log}{D}_y\left(\mathit{y}\right)\right]+n_{\mathit{x}~{\mathit{p}}_{\mathit{data}}\left(\mathit{x}\right)}\left[\mathit{log}(1-D_y\left(\mathit{G}\left(\mathit{x}\right)\right)\right]$(4)

The generator F is a reversed generator, which generates X-domain image from Y-domain image. The generator F and discriminator Dx forms another pair of adversarial players to train each other, with the goal of minimizing the loss function $L_{\mathit{GAN}}\left(\mathit{F},D_x,\mathit{X},\mathit{Y}\right)$, which takes a similar form as Equation (4)(4) $L_{\mathit{GAN}}\left(\mathit{G},D_y,\mathit{X},\mathit{Y}\right)=n_{\mathit{y}~{\mathit{p}}_{\mathit{data}}\left(\mathit{y}\right)}\left[\mathit{log}{D}_y\left(\mathit{y}\right)\right]+n_{\mathit{x}~{\mathit{p}}_{\mathit{data}}\left(\mathit{x}\right)}\left[\mathit{log}(1-D_y\left(\mathit{G}\left(\mathit{x}\right)\right)\right]$(4) . In addition, cycle consistency loss function $L_{\mathit{cyc}}\left(\mathit{G},\mathrm{F}\right)$ is defined to ensure the consistency of a real sample through the cycle, as shown in Equation (5)(5) $L_{\mathit{cyc}}\left(\mathit{G},\text{F}\right)=n_{\mathit{y}~{\mathit{p}}_{\mathit{data}}\left(\mathit{y}\right)}\text{}left[}\left|\right|\text{{}\mathit{G}\left(\mathit{F}\left(\mathit{y}\right)\right)-\mathit{y}\left|\right|1+n_{\mathit{x}~{\mathit{p}}_{\mathit{data}}\left(\mathit{x}\right)}\left|\right|\mathit{F}\left(\mathit{G}\left(\mathit{x}\right)\right)-\mathit{x}\left|\right|1$(5)

(5) $L_{\mathit{cyc}}\left(\mathit{G},\text{F}\right)=n_{\mathit{y}~{\mathit{p}}_{\mathit{data}}\left(\mathit{y}\right)}\text{}left[}\left|\right|\text{{}\mathit{G}\left(\mathit{F}\left(\mathit{y}\right)\right)-\mathit{y}\left|\right|1+n_{\mathit{x}~{\mathit{p}}_{\mathit{data}}\left(\mathit{x}\right)}\left|\right|\mathit{F}\left(\mathit{G}\left(\mathit{x}\right)\right)-\mathit{x}\left|\right|1$(5)

The total loss function of CycleGAN therefore consists of$L_{\mathit{GAN}}\left(\mathit{G},D_y,\mathit{X},\mathit{Y}\right)$, $L_{\mathit{GAN}}\left(\mathit{G},D_y,\mathit{X},\mathit{Y}\right)$ and the cycle-consistency loss $L_{\mathit{cyc}}\left(\mathit{G},\mathrm{F}\right)$, as is shown in Equations (6)(6) $\mathcal{L}\left(\mathrm{G},\mathrm{F},{\mathrm{D}}_{\mathrm{X}},{\mathrm{D}}_{\mathrm{Y}}\right)={\mathcal{L}}_{\mathrm{GAN}}\left(\mathrm{G},{\mathrm{D}}_{\mathrm{Y}},\mathrm{X},\mathrm{Y}\right)+{\mathcal{L}}_{\mathrm{GAN}}\left(\mathrm{F},{\mathrm{D}}_{\mathrm{X}},\mathrm{Y},\mathrm{X}\right)+{\mathcal{L}}_{\mathrm{CYC}}\left(\mathrm{G},\mathrm{F}\right)$(6) .

(6) $\mathcal{L}\left(\mathrm{G},\mathrm{F},{\mathrm{D}}_{\mathrm{X}},{\mathrm{D}}_{\mathrm{Y}}\right)={\mathcal{L}}_{\mathrm{GAN}}\left(\mathrm{G},{\mathrm{D}}_{\mathrm{Y}},\mathrm{X},\mathrm{Y}\right)+{\mathcal{L}}_{\mathrm{GAN}}\left(\mathrm{F},{\mathrm{D}}_{\mathrm{X}},\mathrm{Y},\mathrm{X}\right)+{\mathcal{L}}_{\mathrm{CYC}}\left(\mathrm{G},\mathrm{F}\right)$(6)

Pix2Pix

Conditional GAN differs from traditional GAN with additional inputs to the Generator besides random noise. The additional inputs can be ground-truth image, labels, or other dimensional information. The Generator in CGAN normally adopt the encoder-decoder model. Some adopt U-Net, which uses skip connections between mirrored layers in the encoder and decoder stacks, as shown in Figure 2.

Figure 2. The structure of U-Net.

The model of Pix2Pix was first presented in CVPR 2017 and has achieved successes in many areas, such as transformation from aerial image to map, or vice versa, and semantic segmentation. The discriminator in Pix2Pix uses a network called PatchGAN, which judges the structure at the scale of patchs (Isola et al. 2017).

The loss function of a conditional GAN can be defined as in Equation (7)(7) ${\mathcal{L}}_{\mathrm{cGAN}}\left(\mathrm{G},\mathrm{D}\right)={\mathbb{E}}_{\mathrm{x},\mathrm{y}}\left[log\mathrm{D}\left(\mathrm{x},\mathrm{y}\right)\right]+{\mathbb{E}}_{\mathrm{x},\mathrm{y}}\left[log\left(1-\mathrm{D}\left(\mathrm{x},\mathrm{G}\left(\mathrm{x},\mathrm{z}\right)\right)\right)\right]$(7) :

(7) ${\mathcal{L}}_{\mathrm{cGAN}}\left(\mathrm{G},\mathrm{D}\right)={\mathbb{E}}_{\mathrm{x},\mathrm{y}}\left[log\mathrm{D}\left(\mathrm{x},\mathrm{y}\right)\right]+{\mathbb{E}}_{\mathrm{x},\mathrm{y}}\left[log\left(1-\mathrm{D}\left(\mathrm{x},\mathrm{G}\left(\mathrm{x},\mathrm{z}\right)\right)\right)\right]$(7)

where D(x,y) indicates the probability that a ground-truth image is discriminated as true, $\mathit{G}\left(\mathit{x},\mathit{z}\right)$ is the image generated by G with input x and a given noise vector z, D(x, G(x,z)) indicates the probability that a image generated by G is discriminated as true.

To ensure the output of G (x, z) approximating the target image y, D(x,y) should be as great as possible and D(x, G(x,z)) should be as small as possible. To satisfy the requirement of generating outputs as close as possible to the true image, L1 distance between the true image and the generated image can be measured and adopted in the objective function too. Therefore, the final learning objective function for the generator G in Pix2Pix is formulated as Equation (8)(8) $G^{\text{*}}=\mathit{arg}\underset{G}{\mathit{min}}\underset{D}{\mathit{max}}{n}_{\mathit{cGAN}}\left(\mathit{G},\mathit{D}\right)+\mathit{\lambda }{E}_{\mathit{x},\mathit{y},\mathit{z}}\left|\right|\mathit{y}-\mathit{G}\left(\mathit{x},\mathit{z}\right)\left|\right|1$(8) .

(8) $G^{\text{*}}=\mathit{arg}\underset{G}{\mathit{min}}\underset{D}{\mathit{max}}{n}_{\mathit{cGAN}}\left(\mathit{G},\mathit{D}\right)+\mathit{\lambda }{E}_{\mathit{x},\mathit{y},\mathit{z}}\left|\right|\mathit{y}-\mathit{G}\left(\mathit{x},\mathit{z}\right)\left|\right|1$(8)

Experimental Set-Up and Data Acquisition

Experimental Set-Up

A heliostat (#7) of the DaHan solar power plant, which is located in Yanqing, Beijing, is chosen and selected for the experiment. A monochrome industrial camera (MV-EM120M) with resolution of 1280 × 960 is used to capture the target spots concentrated by the heliostat. The camera is equipped with an off-the-shelf NIKKOR 80–400 mm lenses, and is connected to a laptop via a GigEhernet Cable.

Data Acquisition

The orbiting trajectories of the Moon, Earth, and Sun have been studied. Specific time periods when the orientation of the Moon is close to that of the Sun are picked out. For an example, the daytime trajectories of the Sun from March 1 to March 4 were highly similar to the Moon trajectory on April 6 (the daytime period is 6:00–19:00; the night period is 18:00–7:00 on the next day). The trajectories of Sun (red line) and Moon (blue line) are shown in Figure 3. The concentrated solar and lunar images formed by the chosen heliostat at these specific time have been captured and stored in the computer.

Figure 3. (a) the trajectory of moon on April 6 (blue) and that of sun on April 1 (red line); (b) the trajectory of moon on April 6 (blue) and that of sun on March 2 (red).

The tracking system of the heliostat is based on the relative movement of Earth and Sun. To track the Moon at night, the time and date of the system have to be changed according to the matched date of Sun. For an example, the date of the tracking system at night of April 6 was changed to March 1. The experimental principle is shown in Figure 4. The small deviations between the Sun and Moon trajectories were compensated by manually modifying the aimpoint so that the heliostat could concentrate the Moon light onto the target plane.

Figure 4. Illustration of experimental set-up.

During the image capturing processes, the camera position and the focal length of lens were kept constant. The heliostats surrounding the #7.0 heliostat were all set to the stowed state during the experiment to prevent the interference. The camera settings are unchanged except that the exposure time is shortened during daytime to prevent overexposure. Figure 5 is a picture taken at the moon-tracking site on April 6.

Figure 5. A picture taken at the site of experiment.

Data Processing

Data Matching

The concentrated lunar and solar spots of the #7.0 heliostat are captured by a same camera with different exposure time and stored in the computer as images in BMP format. Figure 6 is an image of the concentrated solar spot. Based on mathematical analysis, the relative trajectories between the moon and the sun with respect to the earth are established. Time when the relative trajectories coincide with is found. Specifically, the relative trajectory of the moon at night of April 6, 2023 (Beijing Local Time) matches well with the trajectory of the sun in the morning of March 2, 2023 (Beijing Local Time), as shown in Figure 7. Moon-tracking experiments with #7 heliostat have been arranged and conducted through the whole night from April 6 to April 7. The images of concentrated lunar spot were processed and normalized. Images of the concentrated solar spot of #7 heliostat at matched time are collected. The lunar and solar concentrated spots are paired to constitute the moon_sun data set, providing the data base of AI learning. The specific periods, when the inclination angles of the moon match well with that of the sun, are as shown in Table 1.

Figure 6. The concentrated solar spot on the target plane of the CSP tower.

Figure 7. The lunar and solar trajectory comparison at (a) Time slot 1; (b) Time slot 2; (c) Time slot 3; (d) Time slot 4.

Table 1. Time slots when the paired lunar and solar concentration images are acquired.

Time of acquisition	Lunar concentration	Solar concentration
Time slot 1	21:12 ~ 22:12, April 6	9:00 ~ 10:00 AM, March 2
Time slot 2	3:00 ~ 4:00 AM, April 7	14:54 ~ 15:54, March 1
Time slot 3	23:18，April 6 ~ 0:18, April 7	11:00 ~ 12:00, March 2,
Time slot 4	00:54, April 7 ~ 01:54, April 7	12:45 ~ 13:45, March 1

For each time slot, over 1000 lunar and solar concentration images have been collected. Some images are not valid due to the missing of light speckles or the disturbance of calibrating marks. The images are checked, labeled, and validated. Finally, there are 207 pairs of matched lunar and solar concentration images for Time slot 1, 173 pairs for Time slot 2, 249 pairs for Time slot 3, and 236 pairs for Time slot 4.

Image Pre-Processing

To facilitate the learning process, same pre-processing operations have been carried out for all lunar and solar concentrating images. A median filter is applied to remove the pepper noise in the image at first. Then the gray values of all pixels are reviewed and the pixel which has the maximum gray value is defined as the center of concentration spot. Around the centered pixel, a region of 400 × 400 is cropped and stored as the area of interest (AOI) of concentrating spot. Using AOI as the training data can minimize the influence of the four calibration patterns located in the corner of the image. With this strategy, the lunar and solar spots can be correctly identified and extracted from the original image. A sample pair of lunar and solar images after the image pre-processing are shown in Figure 8.

Figure 8. A pair of lunar (left) and solar (right) concentrated spots in the data set.

Construction of Dataset

After processing and extracting the corresponding lunar and solar concentration images within the four periods, we obtain a total of 865 pairs of the lunar and solar concentrated spots. Of the 865 pairs, 50 pairs were selected as the test set and the remaining 815 pairs were used as the training set.

Results and Analysis

Having constructed the dataset, two generative models, one based on the framework of CycleGAN and another based on the framework of Pix2Pix, have been trained and evaluated respectively. The solar concentration images generated by the AI models are compared with the ground-truth solar concentration images in three criteria.

Evaluation Criterion

Given a fixed and consistent background, the concentration ratio distribution of solar spot can be reflected by the histogram of the image to certain extent (Cha and Srihari 2002). In a histogram, the horizontal axis indicates the scales of gray values (e.g. 0 ~ 255) of a digital image, the value of vertical axis shows the number of pixels at each scale of gray values. Hence, the shape of a histogram can reflect the concentration ratio distribution of illustration within a specified region, given the object having a consistent optical quality.

Structural Similarity Index Measure (SSIM) is another criterion for the evaluation of similarity between two images (Wang et al. 2004). SSIM compares the brightness, contrast and structure of two images separately, then weights the three elements and make a synthesis. SSIM has three characteristics: symmetry, i.e. $\mathrm{SSIM}\left(\mathrm{x},\mathrm{y}\right)\mathit{\hspace{0.17em}}$=$\mathrm{SSIM}\left(\mathrm{y},\mathrm{x}\right)$; bounded, $\mathrm{SSIM}\left(\mathrm{x},\mathrm{y}\right)$ ≤1; and maximum uniqueness, if and only if x = y, $\mathrm{SSIM}\left(\mathrm{x},\mathrm{y}\right)$ = 1. The calculating formula of SSIM is as shown in Equation (9)(9) $\mathrm{SSIM}\left(\mathrm{x},\mathrm{y}\right)=\frac{\left(2{\mathit{\mu }}_{\mathrm{x}}{\mathit{\mu }}_{\mathrm{y}}+\mathrm{C}1\right)\left(2{\mathit{\sigma }}_{\mathrm{xy}}+\mathrm{C}2\right)}{\left({{\mathit{\mu }}_{\mathrm{x}}}^2+{{\mathit{\mu }}_{\mathrm{y}}}^2+\mathrm{C}1\right)\left({{\mathit{\sigma }}_{\mathrm{x}}}^2+{{\mathit{\sigma }}_{\mathrm{y}}}^2+\mathrm{C}2\right)}$(9) .

(9) $\mathrm{SSIM}\left(\mathrm{x},\mathrm{y}\right)=\frac{\left(2{\mathit{\mu }}_{\mathrm{x}}{\mathit{\mu }}_{\mathrm{y}}+\mathrm{C}1\right)\left(2{\mathit{\sigma }}_{\mathrm{xy}}+\mathrm{C}2\right)}{\left({{\mathit{\mu }}_{\mathrm{x}}}^2+{{\mathit{\mu }}_{\mathrm{y}}}^2+\mathrm{C}1\right)\left({{\mathit{\sigma }}_{\mathrm{x}}}^2+{{\mathit{\sigma }}_{\mathrm{y}}}^2+\mathrm{C}2\right)}$(9)

Considering the significance of middle image blocks, Gaussian weighting is used to calculate each pair of image blocks’ mean, variance, and covariance. Combined with the threshold segmentation and edge extraction operations in image processing, the SSIM index can give a better description on similarity of two images compared to the histogram.

Spectrum analysis of an image can give a quantified description of the spatial distribution of optical flux with respect to the imaging object as well. As a consequence, Fourier spectrum of the lunar and solar spots and AI-generated solar spots are calculated using FFT and used as the third criterion for the evaluation of AI generative models.

Experiment Result with CycleGAN

The solar concentration images generated by CycleGAN model is compared with the true solar concentration spots in the test set. There are 50 pairs of image in the test set. All images were processed similarly and their histograms are calculated. Analysis show that the model generated solar spot share a similar histogram with the true solar spot. The average of histogram similarity between the original lunar and solar spots is 0.679, while the average of histogram similarity between the CycleGAN-generated solar spot and the true solar spot is 0.843, with an increase of 16.4%. The variance of histogram similarity of the lunar and solar spots is 0.009, while the variance of histogram similarity of the generated solar spot and the true solar spot is 0.006.

It is well known that the histogram of an image is highly related to the textile of the imaging surface. In the experiment, the plane used for the concentration of the heliostat is an ordinary surface on the central tower rather than a Lambert-quality surface. The uncontrolled surface quality deteriorate the result of histogram similarity analysis. The histograms of a lunar spot, paired solar spot, and the CycleGAN-generated solar spot are shown in Figure 10. Figure 9 shows the corresponding images.

Figure 9. (a) Lunar spot; (b) solar spot; (c) CycleGAN-generated solar spot.

From Figure 10, one can see that the number of pixels with gray value over 130 is near zero for the lunar spot, while for the solar spot, the number of pixels with gray value over 130 is apparently not zero. The CycleGAN-generated solar spot also has non-zero number of pixels in this range. The ratio distribution of generated image is apparently closer to that of the true solar spot. In addition, in Figure 10 the number of pixels at the grayscale of 60 in the original image is abnormal; this is caused by a known defect of the CCD camera used. The CycleGAN-generated model can fix such problem caused by external factors.

Figure 10. (a) Histogram comparison of lunar spot (blue) and solar spot (red); (b) histogram comparison of the generated solar spot (blue) and solar spot (red).

The similarities between the AI-generated solar spot and the true solar spot using the measure of SSIM are also calculated. The average SSIM similarities of the orignal lunar spot and the solar spot are around 0.76, while the average SSIM similarities between the generated solar spot and the true solar spot reach 0.95. The average value of the similarity has an increase of 19%. It is obvious that the CycleGAN-generated solar spots are quite close to the true solar spots in terms of SSIM. A lunar spot, the paired solar spot, and their structural differences are shown in Figure 11. The CycleGAN-generated solar spot, the true solar spot, and their structural differences are shown in Figure 12 instead.

Figure 11. (a) lunar spot; (b) solar spot; (c) visualization of the structural difference between the solar and lunar spots; (d) binary image of the structural similarity after adaptive thresholding; (e) different area marked in the solar spot; (f) different area marked in the lunar spot.

Figure 12. (a) Solar spot generated by CycleGAN model; (b) solar spot; (c) visualization of structural similarity between the CycleGAN generated solar spot and the true solar spot; (d) binary image of structural similarity after thresholding; (e) different areas marked in the CycleGAN-generated solar spot; (f) different areas marked in the solar spot.

It is observed that the structural differences in the solar spot generated by the CycleGAN model were significantly reduced comparing to that of the solar spot and the lunar spot. Notably, these structural disparities predominantly locate in the central region and the four corners of the image. The disparity in the corner, in our opinion, are mainly related to the four calibration markers located on the target surface.

The spectrum analysis of the lunar spot, the corresponding solar spot, and the CycleGAN-generated solar spot are also carried out. The Fourier spectrum of a lunar spot, its matched solar spot, and the CycleGAN-generated solar spot are shown in Figure 13.

Figure 13. The spectrum of lunar spot (left), solar spot (middle) and CycleGAN-generated spot (right).

Experiment Result with Pix2Pix

The solar concentration images generated by Pix2Pix model are compared with the ground-truth solar spots in the same way as done with CycleGAN. The average similarity between the lunar spot and the true solar spot is 0.679 for the whole test set. The average similarity between the solar spot generated by Pix2Pix model and the true solar spot is 0.884, with an increase of 20.5%. The variance of histogram similarity of the lunar and solar spots is 0.009, while the variance of histogram similarity of the solar spot generated by the Pix2Pix model and the true solar spot is 0.002, decreased by 0.7%.

Same as CycleGAN, the histogram similarity of generated solar spot has been significantly improved. The same pair of lunar and solar spot in the test set was used again for illustration. Corresponding images are shown in Figure 14. The comparison of their histogram is shown in Figure 15.

Figure 14. (a) lunar spot; (b) solar spot; (c) solar spot generated by Pix2Pix model.

Figure 15. (a) Histograms of the lunar and solar spot; (b) histograms of the Pix2Pix generated solar spot and the true solar spot.

All images from the test set were processed similarly for the calculation of SSIM. The average SSIM value between the lunar and the solar spots is 0.76, with a variance of 0.004. On the other hand, the average SSIM between the Pix2Pix generated solar spot and the true solar spot is 0.97, with an variance of 0.001. The improvement of SSIM is significant. Obviously, the generated solar concentration image shares a high similarity with the ground-truth image from the perspective of SSIM criterion. A paired lunar and solar concentration spots, their structural differences are shown in Figure 16.

Figure 16. (a) Solar spot generated by Pix2Pix model; (b) solar spot; (c) structural difference between the Pix2Pix generated solar spot and the true solar spot; (d) binarized structural differences; (e) different areas marked in the Pix2Pix generated solar spot; (f) different areas marked in the solar spot.

From Figure 16, one can find out that the area of the structural differences between the generated solar spot and the ground-truth solar spot is dramatically reduced. The structural differences are mainly located in the center and the four corners of the image. The central area of structural differences is found significantly smaller than that of the CyclgGAN-generated solar spot. The SSIM values between the original lunar spot, the AI-generated solar spot, and the true solar spot for all samples in the test set are shown in Figure 18.

Figure 17. Spectrum of Pix2Pix-generated solar spot (left) and ground-truth solar spot (right).

Figure 18. Comparison of SSIM values for all samples in the test set.

To compare the spectrum of AI-generated spots and ground-truth solar spots, all images in the test set were processed and the Fourier spectrum of the lunar spots, solar spots as well as AI-generated spots are computed using the fft2 function in Matlab. The spectrum of the Pix2Pix-generated solar spot and the true solar spot are shown in Figure 17. The cosine similarity between the spectrums is calculated. It is found that the cosine similarities between the spectrum of AI-generated solar spots and the ground truth solar spots are both improved, compared to that of the spectrum of the lunar spots and the solar spots. The average cosine similarity between the spectrum of the lunar spots and the solar spots is 0.87, while the similarity between the spectrum of Pix2Pix generated solar spots and the true solar spots is around 0.94, the similarity between the spectrum of CycleGAN-generated solar spots and the true solar spots is around 0.95. The cosine similarity values of the spectrum for all samples in the test set are shown in Figure 19.

Figure 19. Comparison of spectrum similarities for all samples in the test set.

Conclusions

A method based on artificial intelligence and lunar concentration ratio distribution for the prediction of solar flux distribution is proposed in this paper. Conditional generative adversarial neural networks (cGAN) are explored, and two generative models are built for the prediction of solar concentration flux distribution of a heliostat. Simulation and experimental results with a heliostat at BADALING CSP station demonstrate the effectiveness of this AI method. Without extra measurements on the parameters of the reflecting surface and the environmental factors, the solar concentrating images generated with the predictive model achieved a high degree of similarities, compared with the ground-truth solar concentrating images of the same heliostat, in either aspect of histogram, SSIM, and spatial Fourier spectrum. Obviously, this method can be extended to the prediction of concentrating flux distribution of a heliostat field in the future and bring opportunities on the optimization of heliostat tracking strategy at a lower cost.

Two different GAN models have been trained and validated in the paper. Both models achieve good performance on the prediction of solar concentration ratio distribution. In general, the model based on the framework of Pix2Pix gives more robust and accurate predictions in the central region of the concentrating spots, compared to the model based on CycleGAN model.

Nomenclature

Abbreviation	=	Description
CSP	=	Concentrating solar power
PV	=	Photovoltaic
CCD	=	Charge-Coupled Device
CRD	=	Concentration Ratio Distribution
CGAN	=	Conditional Generative Adversarial Networks
DNI	=	Direct Normal Irradiance
FFT	=	Fast Fourier Transform
GAN	=	Generative Adversarial Networks
SSIM	=	Structural Similarity Index Measure
Symbol	=	Description
$\mathit{CR}{\left(\mathit{x},\mathit{y}\right)}_{\mathit{moon}}$	=	Lunar concentration ratio distribution on the target surface
$I_{\mathit{moon}}$	=	Illuminance distribution of lunar spots on the target surface
$\mathit{DN}{I}_{\mathit{moon}}$	=	Direct normal irradiance of the Moon
$\mathit{CR}{\left(\mathit{x},\mathit{y}\right)}_{\mathit{sun}}$	=	Solar concentration ratio distribution on the target surface
$\mathit{F}\left(\mathit{x},\mathit{y}\right)$	=	Solar flux distribution on the target surface
$\mathit{DN}{I}_{\mathit{sun}}$	=	Direct normal irradiance of the Sun
$\mathit{G}$	=	Generator
$\mathit{D}$	=	Discriminator
$L_{\mathit{GAN}}\left(\mathit{G},D_y,\mathit{X},\mathit{Y}\right)$	=	The GAN loss function between generator $\mathbf{G}$ and discriminator ${\mathbf{D}}_{\mathbf{y}}$
$L_{\mathit{cyc}}\left(\mathit{G},\mathit{F}\right)$	=	Cycle Consistency Loss
${\mathbb{E}}_{\mathit{y}\text{~}{\mathit{p}}_{\mathit{data}}\left(\mathit{y}\right)}$	=	Expectation of y with the distribution ${\mathbf{p}}_{\mathbf{data}}\left(\mathbf{y}\right)$
${\mathbb{E}}_{\mathit{x}\text{~}{\mathit{p}}_{\mathit{data}}\left(\mathit{x}\right)}$	=	Expectation of x with the distribution ${\mathbf{p}}_{\mathbf{data}}\left(\mathbf{x}\right)$
${\mu }_x$	=	The average gray value of image $\mathbf{x}$
${\mu }_y$	=	The average gray value of image $\mathbf{y}$
$\mathit{C}$	=	constant
${\mathit{\sigma }}_{\mathit{x}}$	=	variance of image $\mathbf{x}$
${\mathit{\sigma }}_{\mathit{y}}$	=	variance of image $\mathbf{y}$
${\mathit{\sigma }}_{\mathit{xy}}$	=	covariance between image $\mathbf{x}$ and $\mathbf{y}$

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by National Natural Science Foundation of China (No. 61671429).