## Abstract

The pointing calibration of a newly built heliostat field can take up to several years. The state-of-the-art method typically used is accurate but slow. A faster method could reduce the commissioning time and therefore increase the viability of a solar tower plant significantly. In this work, we present HelioPoint, a fast airborne heliostat calibration technique, and demonstrate its accuracy to be better than 0.3 mrad. The measurement is performed using a light-emitting diode (LED) and a camera fitted to a drone. The only infrastructure required is the roughly pre-aligned heliostats themselves. The method is independent of the sun position and can be performed at any time for arbitrary calibration points.

## Introduction

Heliostats are an integral part of solar tower plants. They consist of one or more mirror facets arranged in a way to concentrate solar radiation at a given distance. To focus the sunlight onto a central receiver, the heliostats track the angle bisector between the “moving” sun incident vector and the vector pointing from the heliostat to the receiver. The orientation needs to be updated every few seconds. However, inaccuracies in construction as well as sensor and actuator errors lead to tracking errors, i.e., deviations from the desired optical axis of the heliostat, with a strong impact on solar field performance. Calibrating the heliostats allows the operator to correct such errors, reducing the spillage losses and ensuring a well-controlled irradiance distribution on the receiver.

The parameters to precisely describe the heliostat movement with a kinematic model can be determined only if several orientations of a heliostat are measured, creating an overdetermined parameter system. The different orientations used for calibration are called calibration points. A heliostat counts as fully calibrated (calibration is valid for arbitrary positions) if all degrees of freedom can be determined precisely. Depending on the heliostat, this results in about 10–30 calibration points to be measured, covering the full heliostat movement over the course of the year.

The required pointing accuracy depends on the dimensions of the solar tower plant. However, as a general rule of thumb, in large commercial plants, the heliostat orientation should be known and controlled on a smaller-equal 1-mrad level.^{2} For a new solar tower plant, the heliostats are first roughly aligned during construction. Afterwards, a fine calibration with an accuracy of ≤0.3 mrad is needed for each calibration point [1]. The current state-of-the-art method for a fine calibration is a one-by-one calibration procedure, also known as a camera-target method, in which the sunlight is reflected by a heliostat onto a white diffusive reflective target plane attached to the central tower. A camera takes pictures of the target and from the centroid position of the reflected light spot the heliostat orientation is determined [2]. This method has three major drawbacks.

First, the measurement is dependent on the sun. A specific orientation can only be calibrated during a specific sun position, i.e., for a specific time of day and year, during sunlight. Furthermore, the sun shape and appearance influence the measurement.

Second, a heliostat field of a solar tower plant can consist of over a hundred thousand heliostats. A sequential calibration method like the camera-target method is very slow and can take up to several years for the complete calibration of an entire solar field.

Third, the tower and target must be available (i.e., its construction must be completed and safe to calibrate on).

The HelioCon roadmap [3] therefore identifies the improvement of the calibration procedure as a crucial step toward increased cost efficiency.

Although several other heliostat calibration and tracking control methods exist or have been proposed [1], no faster and still equally precise method has yet gained market acceptance. Building on our earlier work [4,5], we now present a fast airborne calibration method for heliostats and demonstrate its competitive accuracy. The method determines the actual orientation of the heliostat optical axis.^{3} It can be applied non-intrusively to larger groups of heliostats at once, independent of the time of day and year, significantly accelerating the calibration procedure. Furthermore, it does not rely on any other infrastructure compared to other promising methods [6]. The simple and fast procedure also increases the feasibility of performing a second calibration a couple of years after commissioning and has the potential to be further developed into an O&M tool.

## Methods

The HelioPoint method takes advantage of the law of reflection to determine the heliostat orientation using a strong light-emitting diode (LED) and a camera fitted to a drone (see Fig. 1, left). Both LED and camera are pointed toward the heliostats to be calibrated. Calibration in the case of HelioPoint means that we assume a *fixed* kinematic model, i.e., all parameters describing the kinematic model of the heliostat are assumed to be correct as provided by the operator. Accordingly, only the tracking angles are calibrated. If the drone is positioned near the optical axis of a heliostat, a reflection of its LED can be detected in the image (see Fig. 1, right). The orientation of the heliostat surface at the detected reflection point is then determined by means of the known heliostat position as well as the drone position, i.e., the position of camera and LED.

Even though the calibration procedure can be performed on one individual heliostat, it is typically applied to clusters of several heliostats pointing at a defined camera position. This allows to obtain data on several heliostats from one single image as shown in Fig. 2. To account for surface and tracking errors, it is necessary to collect multiple sampling points from different drone positions by flying along a predefined pattern. This pattern preferably lies in a plane approximately perpendicular to the optical axes of the measured heliostats. The pattern should be defined in such a way that the sampling points are homogeneously distributed over the entire heliostat aperture area, covering all facets. In addition, depending on the heliostat geometry and the quality of the mirror shape, a certain minimum number of sampling points per heliostat should be achieved, which can be determined empirically or from simulations.

The complete measurement procedure from flight route planning to data analysis involves several steps which are explained below. Since the method was tested at the DLR solar tower test facility in Jülich, Germany, some of the following figures exemplarily show its heliostat field. However, the procedure is prepared to cover large industry-sized power plants.

### Calibration Points and Grouping Heliostats.

Typically, the operator of a power plant uses a set of specific orientations for each heliostat that are calibrated against a reference to ensure a given pointing accuracy for general tracking angles during operation. The orientations in this set are called calibration points. The number of calibration points needs to be larger than the degrees of freedom in the kinematic model of the heliostat, including not only design parameters such as the tracking angles azimuth and elevation, but also parameters such as the imprecise installation of the foundation, the pylon, etc. These calibration points are the same orientations that need to be measured by the method presented here. For that, the solar field is divided into sections of equal area yielding multiple clusters of heliostats. The number of heliostats per cluster varies with the heliostat density depending on the heliostat size and the location within the solar field. Each cluster is calibrated for the same calibration points, i.e., all heliostats in one cluster point to one camera position. This results in a slightly different orientation for each heliostat within a cluster. A threshold for this variation is defined, setting the maximum area size of the clusters for a given camera distance. The cluster area has a roundish shape of typically about 100 m in diameter.

### Flight Route Planning and Data Acquisition.

For each heliostat cluster and each calibration point, camera and LED fitted to a drone move along a defined pattern on a plane roughly perpendicular to the heliostat orientations at a distance of up to several hundred meters from the heliostat cluster, depending on various, mostly geometric and technical, constraints. The size of the flight pattern is derived from the expected heliostat pointing accuracy and the distance to the heliostats. While the drone is moving along this pattern an image series is recorded. In these images, the LED should be seen reflected by those heliostats where the drone is close to the optical axis. After the drone has finished the flight pattern, it moves on to cover a calibration point of the next cluster. Then, a new image series is recorded during the next flight pattern for the second cluster, and so on. While the drone measures a calibration point in one cluster, the heliostats in another cluster can be adjusted. In this way, all the clusters and calibration points can be measured successively. If more drones are used, the data acquisition can also be parallelized.

### Data Analysis.

Each LED reflection detected in an image can be used to derive the normal vector of the corresponding mirror at the specific location of the reflection. For the determination of the precise direction of that normal vector, the positions of heliostat, camera, and LED have to be known. Typically, the heliostat positions can be provided by the operator of the plant with sufficiently high accuracy. If not, one can for example perform a photogrammetric measurement to obtain the heliostat positions. The initial task is to identify which heliostat identification (ID) corresponds to which heliostat in the image. Using an image processing algorithm based on classical computer vision methods, all the heliostats and their mirror corners on the images can be detected. By means of a real-time kinematics (RTK) capable drone, the offset-corrected global positioning system (GPS) positions (accuracy about 10 mm, see Appendix A) of the drone, as well as the camera gimbal angles for each image, are obtained. Using a calibrated camera model, the heliostat positions can be projected onto the image allowing the identification of the individual heliostats. In case the projection is not accurate enough for unambiguous identification, the external camera orientation of the camera can be improved – depending on the gimbal stability – using automatic or manual feature matching.

In the next step, again using an image processing algorithm, the LED reflections on each heliostat are detected and the positions of the centroids of the reflections are transformed from image coordinates into the concentrator coordinate system (CCS) of the respective heliostat. Note that due to facet canting and shape deviations of real heliostats, several reflections of the LED may appear on the heliostat surface in the same image. The data can be treated in the same way contributing to the statistics.

*measured*normal vectors $nmeas$ are calculated using the corresponding camera position and the

*expected*heliostat orientation (see Fig. 3) by

The *measured* normal vectors obtained by Eqs. (1) and (2) are then compared to the *expected* normal vectors (see Fig. 4). The expected normal vectors are calculated at all sampling point positions using the fixed kinematic model, the reference surface (by design or measured), and our best guess for the tracking angles (e.g., provided by the operator). Note that the typically limited knowledge about the kinematic model and the reference surface leads to errors in the expected normal vectors. A discussion of how the uncertainties of measured and expected normal vectors affect the result follows in the “Results and Discussion” section.

The resulting angular differences between measured and expected normal vectors can be represented as projections into the concentrator's *y*–*z* and *x*–*z* planes (rotation around *x*- and *y*-axis, respectively). The mean of the distribution of differences for *y*–*z* and *x*–*z* planes is typically not zero, meaning that the expected heliostat orientation differs from the real one. The expected tracking angles of the heliostat model are then adjusted by the value of the mean, effectively centering the angular difference distributions around zero (Fig. 5).

With this improved *expected* heliostat orientation, the normal vectors are updated and this iterative process is repeated. After usually only two iterations the changes are negligible. In the next step, a generalized extreme studentized deviate test is applied to determine outliers, and the iterative procedure is done a last time. Eventually, the final heliostat orientation is obtained. Note that this orientation corresponds to the optical axis of the heliostat relevant for operation, similar to the one measured by the conventional camera-target method. The required amount of sampled positions on the mirror surface for a robust and reliable estimation of the heliostat orientation depends strongly on the distribution of the angular differences of the heliostat mirror surface.

This calculation is done for all heliostats in the cluster, for all clusters, and for all calibration points, yielding the information to fully calibrate an entire heliostat field.

## Results and Discussion

In a test campaign at the DLR solar tower plant in Jülich, the orientations of about 500 heliostats were measured. In this section, the measurement results for five of those heliostats are compared to reference data. We carry out an uncertainty analysis and demonstrate the agreement of measurement results and reference data up to the derived uncertainty. The reference data used are reference orientation vectors representing the optical axes of heliostats which are provided directly by the deflectometry-based QDec-H system [7]. Note that the alternative approach of using the camera-target method to create reference data seems obvious but is disadvantageous, since the conversion of the measured beam profile and its position into the optical axis of the heliostat leads to additional uncertainties.

In addition to a precise (<0.1 mrad) heliostat orientation reference, the QDec-H method yields surface slope deviation maps with a precision of <0.1 mrad. Using these reference slope deviation maps allows us to exclude the uncertainty due to slope deviations from our measurement method, resulting in a reduced standard deviation of 0.2 mrad (see Fig. 6), compared to up to 1 mrad when surface slope deviation maps are not considered, as shown in Fig. 5. Given the QDec-H reference data, these surface deviations are expected to have a standard deviation of about 0.8 mrad, which agrees with our observation.

As the final heliostat orientation is derived from the angular difference distribution mean, its accuracy can be estimated from the *standard deviation of the mean estimator*.^{4} In the case of normally distributed data, this quantity is given by $\sigma mean=\sigma ad/#samples$, where $\sigma ad$ denotes the standard deviation of the angular difference distribution. However, this is not necessarily the case for each measured heliostat since facet canting effects or systematic surface deviations will disturb the normal distribution. Whether the assumption of a normal distribution is reasonable is decided using the Shapiro–Wilk test with a threshold of 0.05 for the resulting *p*-value. Datasets with a *p*-value larger than 0.05 are therefore considered to sample a normal distribution. For those datasets, we obtain a good value for $\sigma mean$ using the formula described above. If the *p*-value is smaller than 0.05, the normal distribution assumption has to be discarded and the calculation of the standard deviation of the mean cannot be done in the same way. Nevertheless, the order of magnitude of those values is considered to be valid.

Note that the mean standard deviation only accounts for uncertainties reflected in the width of the angular difference distribution, while additional “hidden” uncertainties $\sigma sys,i$ due to systematic shifts of this distribution have to be considered separately. Assuming statistical independence for the error sources, the overall uncertainty of the presented method is given as $\sigma tot=\sigma mean2+\Sigma i\sigma sys,i2$. Table 1 lists potential error sources and the effect through which each error source affects the measurement uncertainty.

As the angular difference distribution compares *measured* with *expected* normal vectors, broadening or shifting in this distribution may occur due to errors in both quantities:

The *measured* normal vectors can be falsely computed because of inaccurate assumptions of camera position and/or reflex position. On the “drone side,” the uncertainty of the camera position $pcam$ is determined by the accuracy of the RTK positioning, as discussed in Appendix A. While relative RTK errors of zero-mean only increase the standard deviation (i.e., broadening), an absolute RTK error leads to a systematic shift in all vectors and hence in their angles. However, we mitigate this effect by feature matching for each image. The measured normal vector is also subject to inaccuracies “on the heliostat side,” i.e., in the reflex position in the global coordinate system (GCS). For the reflex detection $(xCCS,yCCS)$, we assume a zero-mean error, broadening but not shifting the angular difference distribution. An error in the heliostat position $phelio$, however, leads to a systematic shift in all measured normal vectors. Depending on the heliostat kinematics used in the field, an inaccurate fixed kinematic model $Khelio$ can result in rotational and translational errors for the assumed reflex positions in the GCS, leading to both broadening and shifting.

On the other hand, the *expected* normal vectors can be subject to inaccuracies in the assumed mirror surface: While a zero-mean error distribution for the assumed mirror surface only results in broadening, a systematic error potentially impairs the method's accuracy through shifting: as an example, a falsely canted mirror facet would lead to a shift of all expected normal vectors on this facet. If the sampling distribution happened to cover only this facet, the heliostat orientation would be fitted to a falsely canted facet. To avoid this effect, a symmetric sampling point distribution should be aimed for. Such shifts are then detectable in the resulting angular difference distribution. Based on this, the operator can decide how to define the optical axis of a heliostat, depending on the specific application.

The uncertainties of the mean and the *p*-value for the corresponding distribution for all five heliostats are summarized in Table 2. From these numbers, we derive a conservative mean estimation uncertainty of $\sigma mean=0.2mrad$ for measurements using the ideal mirror surface. This uncertainty is reduced to below 0.1 mrad when using the precise surface maps from QDec-H.

Additionally, we consider the uncertainty due to systematic shifts based on the following considerations: for large distances *d* between camera and heliostats, the standard deviations $\sigma x,i$ of assumed camera position and reflex position translate to angle uncertainties according to $\sigma sys,i=\sigma x,i/d$. Assuming no absolute error in the RTK positioning, uncertainties of 10 mm for $phelio$, 30 mm for shifts due to errors in $Khelio$, and no systematic errors in the mirror surface, the additional systematic uncertainty amounts to $\sigma sys=0.2mrad$ at a distance of 150 m.

This gives us a conservative estimate for the overall measurement uncertainty of $\sigma tot=0.3mrad$ using an ideal mirror surface. Note that the numerical values in this computation may change depending on the available prior knowledge on heliostat positions, kinematic model, and surface slope deviations.

The offset of our final orientation results to the QDec-H reference data is shown in Table 3. Here, we find very good agreement between our measurement and QDec-H data, considering the uncertainties as described above and the ones from the reference of <0.1 mrad. The complete sampling data for each heliostat are presented in Appendix B.

## Conclusion

We present a fast and accurate airborne calibration method for heliostats, prepared for commercial application, and demonstrate its accuracy on the tracking offset to be better than 0.3 mrad, which qualifies this method as a so-called fine-calibration method. The method only requires an LED and a camera fitted to a drone. Since this method can be applied to bigger groups of heliostats at once, the projected calibration time of a commercial plant can be reduced to a few weeks including data acquisition and evaluation instead of many months or even years using the state-of-the-art methodology. This significantly shortens the commissioning time and performance guarantee period of new plants, reducing their costs and increasing their viability. Compared to competitive alternative techniques, our method does not require any other infrastructure apart from the heliostats installed at the plant and can therefore be applied early on during commissioning. Furthermore, it works independently of the sun position and the time of the year.

## Footnotes

Large plants can reach heliostat-to-tower distances of more than 1 km; assuming a distance of 1 km and a tracking error of 1 mrad, an offset of around 2 m between the desired and the target aim point would be observed.

We define the optical axis as the mean axis of all normal vectors of the entire mirror surface of a heliostat.

Note that the standard deviation of the mean estimator is not the standard deviation of the distribution. The latter corresponds to the accuracy of the individual sampling points.

## Acknowledgment

Financial support from the German Federal Ministry for Economic Affairs and Climate Action (HelioPoint-II, contract 3028684) is gratefully acknowledged. We also thank our DLR colleagues Oliver Kaufhold and Felix Göhring at the solar tower in Jülich, Germany.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the article.

## Nomenclature

### Greek Symbols

### Subscripts

### Abbreviations

## Appendix A: Measuring Real-Time Kinematics Positioning Precision

For a precise evaluation of our method, it is essential that next to the heliostat position and kinematic model, also the drone position is well known. We use a drone equipped with real-time kinematics (RTK) to obtain an offset-corrected GPS position for each image. To measure the accuracy of our system, we performed a dome-shaped flight as illustrated in Fig. 7. For each image, we obtain the RTK data, and for comparison, we evaluate all images photogrammetrically using the commercial software aicon 3d studio.

The photogrammetric evaluation yields sub-mm precise camera positions. For the evaluation, the RTK data, as well as the camera positions from aicon 3d studio, are represented as point clouds. The photogrammetry data are then fitted into the RTK data to evaluate the relative precision of the latter. The results as shown in Fig. 8 indicate a standard deviation of the RTK precision for the Cartesian components of about 6–8 mm with a maximum for the *z*-component. The covariance of the data shows an approximate linear independence of the components. As a conservative upper limit for the standard deviation of any arbitrary component (which would be chosen according to the camera orientation of a specific image), we can use 10 mm. For typical camera-to-heliostat distances of more than 100 m, this results in an error of less than 0.1 mrad for each individual sampling point. The effect is linearly reduced for larger distances and averaged out for several sampling points as used to evaluate the heliostat orientation. We therefore conclude that the relative RTK uncertainty is negligible for this method. In contrast, a potential systematic global positioning offset (with respect to the solar field) has a direct effect on the calibration results. However, the uncertainty of such a global positioning offset can be reduced by reference markers in the field, which, e.g., can be represented by the heliostats. Due to the good relative precision, all images can be used to improve the global position, which should result in global accuracies of similar magnitude (∼10 mm) as the relative accuracies.