Estimation of Uncertainty in Automated Heliostat Alignment 1
J. Jack Zhang , John D. Pye 1 2
1*
2
and Clifford K. Ho
Solar Thermal Group, Australian National University, Canberra, Australia.
National Solar Thermal Test Facility, Sandia National Laboratories, Albuquerque, USA. *Corresponding author:
[email protected]
Abstract Engineers seek to reduce the cost of solar-thermal central tower systems by reducing the cost of the heliostat field. Lower-cost heliostats can be achieved by accepting lower precision in manufacture and installation. Automated alignment methods can then be used to correct pointing errors due to misalignments after the heliostat is installed, to obtain and maintain the required pointing accuracy. The method under consideration here is the beam-target method, in which the location of the reflected spot on a target is measured repeatedly, and linear regression is applied to estimate various geometrically-based physical misalignment parameters to facilitate pointing error correction. In this study, a new statistical analysis of the automated alignment process including calculation of alignment uncertainty after the 'training' process is presented. In addition to correcting errors, this new analysis now allows evaluations to when there is enough training data gathered to be confident that, if the calculated corrections are applied, the beam and target will coincide within an allowable uncertainty. The relative value of training data gathered at different times of day and different times of year was also investigated. A large number of randomlygenerated misalignments are studied numerically. Worst-case misalignment combinations are found for an upper bound on the training data requirements for particular heliostats to be subsequently determined. 1. Introduction 1.1 Background
Toward the end of the 20th Century, solar power has become a widely adopted sustainable energy solution. The market competitiveness of solar devices such as parabolic dishes and heliostats is critically governed by effective conversion of sunlight to electricity, which may be optimised by maximising light absorption through precise alignment to the sun. Traditionally, solar tracking was facilitated by mounting live sun sensors on steering controllers equipped individually for each device. A slight misalignment to the sun was dynamically adjusted with the aid of sensory data. This type of control, known as closed loop control was expensive but accurate. Carden [1] was able to replace this configuration by a low cost, centralised command centre and using pre-stored steering trajectory calculated from ideal target location and sun position data. The system was termed open loop control. For on-axis tracking devices that face directly to the sun, open-loop control has been frequently adopted. Edwards [2] successfully incorporated such system to short focal length parabolic dishes in a master-slave configuration known as adaptive controllers. A notable case study by Maish [3] has demonstrated its feasibility in applications to on-axis tracking photovoltaic panels. However, installation misalignments and manufacturing defects may cause steering deviation from the ideal path. These deviations, known as tracking errors, are identified as a major drawback of open loop systems by both [2] and [3] as sun sensors were abolished. Open-loop control was concurrently developed for off-axis tracking devices, notably heliostats in a central receiver solar thermal power plant. These reflective mirrors point toward the bisector between the sun and a fixed target. Minimisation of tracking errors (referred to specifically as pointing errors) for heliostats is critical as long focal length exaggerates the effect of small misalignments on the position of the focal spot. The current study addresses this more troublesome issue. 1.2 Early work on automated heliostat alignment
Baheti and Scott [4] proposed two modes of operation for a heliostat under open loop control. In calibration mode, the heliostats were aimed directly at the sun and their pointing accuracy was measured using a sun sensor mounted on the heliostat. Baheti and Scott [5] were able to identify three major misalignment types. Firstly the Australian Solar Council Solar 2012 conference, Melbourne.
defective tilt of the heliostat pedestal from the ideal vertical. Secondly, the reference bias in the pointing angles from the ideal zero. Thirdly, the linear errors introduced by imperfections in the drive gears. A mathematical error model relating the pointing errors to 6 parameters that represent these physical misalignments was developed. An effective method to correct errors for an on-axis tracking, azimuth and elevation axes driven heliostat was demonstrated through a four-step method: 1. 2. 3. 4.
Collecting pointing error data over a short period of time in calibration mode Using error data to estimate for the misalignment parameters Feeding parameters back to the model to predict errors in the future Adjust pointing direction with predicted errors to compensate misalignments in mirror steering mode
A linear regression was conducted using least squares approach to estimate the parameters. Statistically the estimation process is referred to as ‘training’ (pointing errors as ‘training data’). The method allowed for compensations to misalignments without physical repair and established what is known as automated heliostat alignment. 1.3 Review of other related research
Pointing errors caused by misalignment of non-constant nature, notably wind loading were studied by Mavis [6] on the Solar One test facility. The author concluded that wind loading is of random nature and should not be captured in the parameters. Training was recommended for an entire day to nullify its effect. Following Solar One, experiments on Solar Two were conducted by Jones and Stone [7]. Reference bias was found to produce a constant beam shift. A temporary correction was applied by artificially manipulating the heliostat location coordinates in the controller, in order to shift the beam back to the desired spot. This temporary correction was applied until a more accurate correction could be calculated and applied. An approach to correct errors by combining the advantages of close and open control tracking was developed by Kribus et al [8]. The system incorporated live optical sensors evenly surrounding the target as opposed to on individual controllers. Presence of target offset was notified if non-symmetrical brightness distribution was found in the projected image. An alternative drive mechanism referred to as the spinning and elevation method (Chen et al [9]) was developed recently. A general tracking formula was created (Chong et al [10]) and refined (Guo et al [11]) to accommodate both methods. Error correction algorithm was subsequently integrated to the general formula [10]. The derivation process relied on earth-centre and earth-surface coordinate frames as reference. The true hour angle and inclination angles were calculated first using local time. The misalignment was then estimated by comparing ideal and actual pointing angles at regular intervals. Further studies (Zang et al [12]) (Wei et al [13]) analysed the two drive mechanism comparatively and argues that the latter could achieve better pointing precision. In recent study by Sandia National Laboratory researchers Khalsa, Ho and Andraka [14], the mathematical error model established by [5] was extended from direct on-axis sun tracking to evaluate the pointing error in the reflected beam. Two additional parameters were introduced to represent non-orthogonally and bore sight misalignments. These were found when the two drive axes of the heliostat were not precisely in 90 degrees, and when the normal of the mirror surface deviates from the ideal optical axis, respectively. Training data was sampled by aiming a single test heliostat at a large white target on the receiver tower. The offset between the centroid of the beam and the centre of the target was determined using optical image analysis to calculate pointing errors. The four-step method outlined previously was then applied. This technique effectively corrected errors over the duration of a single summer day, and is the basis for the current study. However, observations of reduced prediction accuracy in winter have left a desire for further study of the confidence of the training data set. In summary, previous work on heliostat error correction demonstrated that geometric misalignments could be determined from pointing-error training data and used to apply steering corrections, without the need for mechanical adjustments. However, the lack of investigation into the confidence of training data was identified. This information is critical to asses
number of training data-points required for accurate parameter estimation dates and times preferred for training-data collection, and projected accuracy of beam location after training is completed.
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1.4 The current work
Previous work has applied training data without a strong emphasis on determining the minimal amount of data required for accurate correction. In practice, the initial calibration stage requires each heliostat to be taken offline and tuned sequentially. During training, a heliostat is offline and not generating energy; also, with a single target, the whole field can take months to calibrate. This leads to the question: how much training data is sufficient to achieve a desired pointing precision in the reflected beam, and when should these data be best sampled? In the current work, a statistical method is adopted to represent the uncertainty of the training data based on a 95.5% confidence interval. The four-step method by Baheti and Scott and the error model developed by Khalsa, Ho and Andraka are implemented in a numerical model in the Python programming language. Pointing error data, for a range of misalignment combinations, is simulated, then random ‘wind noise’ added, to replicate statistically meaningful observations, allowing a wider range of effects to be simulated than would be possible with field-based experimental work. A case study of the Sandia heliostat is presented. The optimal sampling time is determined by comparing the uncertainty in the reflected spot location as a result of parameter estimation from training data which has been sampled at particular training times. A general recommended training schedule for a heliostat is then deduced after considering a large number of simulated misalignment cases. 2. Python code 2.1 The Solar Pointer (SP) module
Pysolar [15] is used to determine the position of the sun in azimuth (θ) and elevation (α) angles.
Figure 1: Pointing direction calculation from position of sun and location of fixed target
The heliostat points in the direction of the bisector vector (P) between the position of the sun (S) and the fixed target (C) relative to the heliostat (Fig. 1). It is calculated by finding the vector sum according to Eq. 1. ̂
̂
̂
̂
̂
(1)
(2)
√
The Solar Pointer (SP) code generates a training data schedule based on user-supplied times and dates. The sun position is calculated for that schedule using the Pysolar library. Finally, the required heliostat pointing directions are calculated using the vector Eq 1, via NumPy [16], and converted to azimuth and elevation angles according to Eq. 2. 2.2 The Pointing Error Simulation (PES) module
The purpose of the PES module is to replicate experimental training data.
εi qi
Hi
Figure 2: Error model published by Khalsa, Ho and Andraka [14] relating pointing errors (q) to the 8 parameters (ε) using a matrix function of pointing directions (H).
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̂
̂
(3)
The error model established by Khalsa, Ho and Andraka (Fig. 2) is implemented in Python. By taking a predefined set of 8 parameters and the pointing direction data (also in θ and α) determined by the SP module, pointing errors could be simulated. The uncertainty in the simulated pointing errors (δθ and δα) could be found by replacing the parameters (εi) by the uncertainties in the each of the parameters (δεi) in Fig. 2 i.e. . This could be converted to the uncertainty of the spot location according to Eq. 3, where ̂ is the uncertainty in predicted errors , expressed in unit vector form. The receiver target lie on the X-Z plane, and represent the horizontal and vertical offset distance from the target respectively. 2.3 The Parameter Estimation (PE) module
The PE module aims to estimate for the parameters from pointing errors simulate by the PES module. The mathematical model shown in Fig. 2 is used with a set of Δθ and Δα (generated from the user desired training time) as input and returns with the parameters fitted from those training data using a linear least squares regression approach outlined in Eq. 4:
{ε}
(4)
The uncertainties in the parameter estimates are determined according to Eq. 5 in 3 steps. The variance of the parameters and the covariance matrix is calculated from Eq. 5a and 5b; note the constant ‘8’ term results from the 8 parameters to be fitted. ( )
(
)
[
]
( )
√
( )
(5)
The diagonal terms of the covariance matrix represent the coefficients of confidence (Cii). The uncertainty in each parameter ( ) is defined as two standard deviations Eq. 5c (a 95% confidence interval). In theory the uncertainty in the estimation is reduced when the number of data points, n, increases or if the coefficients of confidence Cii are lower. The factors Cii are derived from the elements of the H matrix, and hence reflect the geometric configuration of the collector at the times of measurement. This means that uncertainty in parameter estimates can be increased or decreased depending on the configuration of the sun and heliostat. If training data are sampled from a more ‘sensitive’ time, less data points will be required. 3. Simulation and findings 3.1 Case study of the Sandia heliostat
The first objective of the study is to replicate experimental data of Khalsa, Ho and Andraka for the Sandia heliostat no. 12E13, located 145 meters north and 121 meters east of the target (Fig. 4).
Figure 4: Simulated pointing errors vs. experimentally determined pointing errors .
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Figure 3: Location of the Sandia Heliostat 12E13 and the reference sign convention of the azimuth angle
The pointing directions are generated by the SP for the 15th of June from 9:00 to 17:00 and passed to the PES (Fig. 5). Fig. 3 shows simulated pointing errors are within 3% of the provided experimental data, and hence verify the accuracy of the new simulation code. Define Training Time
Pointing Direction (SP)
Pointing Directions
Pointing Error (PES)
Simulated Errors
Parameters (PEM)
Original Parameters
Figure 5: Workflow to generate pointing error training data and estimate parameters from data. 3.1.1 Day training analysis
To investigate the optimal training time in a day, the effect of additional evening data is studied by extending sampling time upper bound incrementally from 17:00 to 21:00 (sunset). The resultant change in the spot location uncertainty and convergence between the correct beam and the target is observed as more data was added. The effect of extra morning data is then investigated by extending the sampling time from 9:00 to 5:00 (sunrise). The degree of accuracy in which the original parameters are recovered from estimation are observed. Pointing errors were predicted for the 15th of each month in 2011, at hourly interval using these estimated parameters, and are subtracted from the errors simulated using the original parameters, thereby simulating a correction procedure. The remaining error (ideally negligible if the estimation is very close to the original) is used to plot the beamtarget offset on the receiver with uncertainties shown as rings around the corrected spot locations. From these plots, the maximum possible deviation in spot location can be readily observed. The workflow is shown in Fig. 6.
Parameters (PEM)
Estimated Parameters
Pointing Error (PES)
Parameter Uncertainties
Simulated Error Correction Uncertainty in Corrected beam
Training Data Confidence Analysis
Figure 6: Workflow to determine confidence in the training data.
The uncertainty of the spot location at each test hour appear as a ring around the target, with purple representing hours in a winter days and blue representing hours in summer days. The effect of wind is not considered in the predicted spot locations as the random scatter generated will form a ring around the target with constant radius for all the plots. A smaller ring indicates better training and greater confidence in the heliostat accuracy.
Figure 7: Uncertainty result from data sampled starting from (left) 8 am (right) 6 am: a notable increase in accuracy is seen.
Large errors in the azimuth direction are observed in the early morning hours (Fig. 3). This means multiple error parameters are making a significant contribution. In theory, data sampled from such times are more valuable, but were previously left out [14]. Indeed, by including training data from the very early morning hours, around 5 to 6 am, the uncertainty of the spot location can be effectively reduced to within 0.5 m of the target, for this case of
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single-day summer training. Fig. 7 shows the reduction in tracking correction uncertainty as more early-morning data is included. Similarly, additional evening data are helpful to further reduce the uncertainty. Though better parameter estimation is achieved through extending the observation hours through the entire daylight period, it is not yet clear whether the improvement comes from morning data, evening data, or just more data. To assess this, Fig. 8 shows a comparison of full-day and morning-only data (15 to 18 June).
Figure 8: Uncertainty result from (left) one full summer day or (right) three consecutive summer mornings, of training.
On the left, a full day of data gives more accurate training than with slightly more data taken only from morning times. Uncertainty is comparable, but the full-day of data is preferably since less training data-points are required. A similar observation can be made from evening-only data. The conclusion is that a full day of summer training data is valuable, and is more effective than even three half-days at the same time of year.
Figure 9: Individual contributions of parameters in azimuth (left) and elevation (right) directions.
Further understanding of the sensitivities in the parameters is gained by breaking the overall tracking errors down to the contributions from the four misalignments (excluding the constant reference bias) and studying how these contributions vary with time (Fig. 9). We observe that for the particular geometric configuration shown here, bore sight misalignment partly cancels the effect of non-orthogonality, in the azimuth direction. There was little or no contribution for these same misalignments in the elevation direction. Accurate estimation of these misalignment parameters depends on including data from different periods (such as later in the evening) where the cancellation effect is less apparent. In general, a full day of training is found to be most effective as it allows for variation in sensitivity and best captures the more sensitive periods for all of the parameters. 3.1.2 Cross-season training analysis
In the initial cross-season study, two days of data are selected from winter (Fig. 10 left, 20-21 December) and one day from summer (Fig. 8 left, 15 June). Even though the two winter days have more total training data points, the uncertainty remains comparable: summer data is considerably better than winter data. The dates
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chosen correspond to solar elevation maxima and minima during an Albuquerque year. The influence of sun position on pointing accuracy is more apparent at these extremes.
Figure 10: Uncertainty results for (left) 2 winter days and (right) one winter and one summer day, of data.
Next, the benefit of combining both winter and summer data was considered. Fig 10 (right) shows the use of one winter and one summer day of training data. Compared with two winter days, a very significant improvement is seen; it is more than could be explained only by the small increase in number of data points (compare also Fig. 8 left). It is interesting to observe the uncorrected spot locations throughout the year for these training data cases (Fig 11, compare with Fig 8 left and Fig 10 right). As before, the right portion benefits from winter and summer training data, and greatly spot location uncertainty is seen, although in uncorrected form, the hourly/seasonal variation in uncertainty is not particularly apparent
Figure 11: Uncorrected beam locations, with uncertainties calculated from (left) one summer day and (right) one summer and one winter day, of data.
Figure 12: Beam locations with (left) one week of summer and (right) two days of winter and two days of summer, data. Australian Solar Council Solar 2012 conference, Melbourne.
To further confirm the benefit of training data spanning the full range of solar angles, extended data was compared. Fig 12 shows one week of summer data (left) compared to two days of winter and two days of summer data (right), clearly significantly less training data points. Reduced uncertainty is apparent, confirming that winter training is valuable if used in combination. These case studies clearly demonstrate that with careful selection of training schedules, the accuracy of parameter estimation can be greatly improved, and heliostat spot location can be kept within specified limits. Fig. 13 shows the annual variation in the contributions to pointing error from the various misalignment types, for a particular geometry configuration. The sum of the error contributions is higher in summer (middle of the year) and lower in winter. However, the individual sensitivities vary significantly; at times there can be strong cancellation, or times of year when parameter contributions can only be weakly detected. For example, the relative contribution of pedestal tilt exceeds bore sight during summer, whereas in winter, their relative magnitudes are similar with minor cancellation introduced by non-orthogonality. The parameter ε8 is therefore more sensitive in mid-December, as a result, the uncertainty of ε8 (bore sight) was significantly reduced by adding winter data. However if data are only sampled from winter, the estimation of ε2 (pedestal tilt) would have been very poor. If a subsequent error correction were implemented in summer, both the azimuth and elevation error would be seriously under-predicted. For this reason, data from summer should always be included if possible. Overall, cross-season sampling compensates for variation in parameter sensitivity, and therefore allows for much more efficient use of available training time.
3.2 General optimal training time for any particualr helisotat
`
(a) Pedestal tilt
(a)
(c) Linear errors
(b) Non-orthogonality
(b)
(d) Bore sight
Figure 13: Individual azimuth error variation for daylight hours throughout the year; the errors exibit varying charateristics and cancellation at different season. The blue region encompasses daily variations, upper and lower bounds give maximum daily limits. 3.2 General optimal training time for any particular heliostat
The preceding case study was completed for a particular geometric configuration of misalignment parameters, based on measurements of the Sandia heliostat ‘12E13’, and gave insights into the effects of training-schedule decisions on the resulting uncertainties in spot location, but it does not rigorously address the issue for the entire field of heliostats, which have many different potential misalignment combinations, and also varied heliostat Australian Solar Council Solar 2012 conference, Melbourne.
location. To address this, a set of 1000 randomised misalignment combinations were generated, and simulated training and performance evaluation was conducted. Misalignment parameters were varied between -10 mrad to 10 mrad; the location was kept constant, set to the location of 12E13. For each misalignment combination, training was conducted at hourly intervals on a single day, 15 June. With limited training data like this, the accuracy of predicted spot location is not particularly high, but identifiable variation in spot-location uncertainty can be observed, arising from the effects of different parameter combinations.
Figure 14: Individual error contributions for the problematic case (left) Combined summer and winter day data sampled at 30 min intervals (right).
The results of these experiments were data for the maximum spot-location error for a predicted year of simulated performance, including the added uncertainty interval. Results for one worst-case scenario are shown in Fig. 14 (left). Plotted here are the contributions in azimuth error, which shown cancellation as discussed earlier. These simulations can be run with varied amounts of training data. With reduced data, forecast maximum spotlocation error increased up as much as 8 m away from the target. Training schedule was also assessed using the randomised 1000-combination misalignment set. With a single day of summer training data gathered at 20 minute intervals, the maximum forecast spot-location error was 5 m. One day of summer training and one day of winter training, together, at a data-point interval of 30 minutes yield a maximum error of 4 m with the same misalignment set (Fig. 14 right). This observation confirms and generalises the earlier conclusion about the value of cross-season training data. 4. Conclusion The cost-effectiveness of heliostats could be enhanced by using software to achieve high pointing precision, reducing the need for mechanical adjustments. A range of geometric-based misalignment parameters can be identified for heliostats and can be compensated by gathering ‘training data’ of reflected spot location relative to a desired target; statistical estimation of misalignments can then be determined, together with an uncertainty in those parameter values. Using these uncertainties, future spot-location data can be calculated and worst-case spot-location error calculated. Incorporation of uncertainties into this process was not found in the literature, but has been implemented as a software simulation in the present study. Using this simulation, the value of training data from both summer and winter was shown to be advantageous as it greatly improves the accuracy (including reducing the size of the confidence interval) of future spot locations once the corrections are applied. For the Sandia ‘12E13’ heliostat, it was shown that with two days of winter data and two days of summer data, the heliostat’s focal spot can be corrected to remain within 30 cm of the desired target location with 95% confidence over a full year of forecast operation. For a set of randomly-misaligned heliostats sharing the same location as the real 12E13 heliostat, an adequate heliostat accuracy can be assured using 30-minute training data from one day in winter and one day in summer. Further investigation is required generalise this to all possible heliostat locations. Future work will also test the simulation model with field data from the Sandia site and, it is hoped, will be incorporated into a new system allowing strategic scheduling of heliostat training to ensure continued high-accuracy alignment of the field.
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5. Bibliography [1] P O Carden, 1975. Steering a field of mirrors using a shared computer-based controller. Solar Energy 20(4), 343-355. [2] B Edwards, 1978, Computer based sun following system. Solar Energy 21(6), 491-496 [3] A B Maish, 1988. A self-aligning photovoltaic array tracking controller. Conference Record of the Twentieth IEEE Photovoltaic Specialists Conference, 2, 1309-1314 [4] R S Baheti and P F Scott, 1980. Design of Self-Calibrating Controllers for Heliostats in a Solar Power Plant. IEEE Trans. Autom. Control, 25(6), 1091-1097. [5] R S Baheti, and P F Scott, 1980. Adaptive control and calibration of heliostats. Proceedings of 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes, December. [6] C L Mavis, 1988. 10 MWe Solar Thermal Central Receiver Pilot Plant Heliostat and Beam Characterization System Evaluation November 1981 – December 1986, Report SAND87-8003, Sandia National Laboratories, Livermore, CA. [7] S A Jones and K W Stone, 1999. Analysis of Strategies to Improve Heliostat Tracking at Solar Two. Report SAND99-0092C, Sandia National Laboratories, Albuquerque, NM [8] A Kribus, I Vishnevetsky, A Yogev and T Rubinov, 2003. Closed loop control of heliostats. Solar Energy, 29, 905-913 [9] Y T Chen, A Krisbus, A H Lim, C S Lim, K K Chong, J Karni, R R., A Pfahl, and T P Bligh, 2004. Comparison of two sun tracking methods in the application of a heliostat field. J Sol. Energy Eng. Vol. 126 pp638 -644 [10] K K Chong, C W Wong, F L Siaw, T K Yew, S S Ng, M S Liang, Y S Lim and S L Lau, 2009. Integration of an On-Axis General Sun-Tracking Formula in the Algorithm of an Open-Loop Sun-Tracking System, Sensors 9, 7849-7865. [11] M Guo, Z F Wang, W F Liang, X Zhang, C Zang, Z W Lu and X Wei, 2010. Tracking formulas and strategies for a receiver oriented dual-axis tracking toroidal heliostat. Solar Energy 84, 939-947 [12] C Zang, Z Wang, and X Liu, 2009. Design and Analysis of a Novel Heliostat Structure. IEEE International Conference on Sustainable Power Generation and Supply. [13] X Wei, Z Lu, W Yu, H Zhang and Z Wang. 2011. Tracking and ray tracing equations for the target-aligned heliostat for solar tower power plants. Renewable Energy 36, 2687-2693 [14] S S Khalsa, C K Ho and C E Andraka, 2011, An Automated Method to Correct Heliostat Tracking Errors. Proceedings of SolarPACES, Granada, Spain, Sept 20-23. [15] Brandon Stafford, 2009*. Sun position library. http://pysolar.org/. Last accessed on 26 of Nov 2012. [16] NumPy developers, 2012. Scientific Computing Tools for Python, http://numpy.scipy.org/. Last accessed on 26 Nov 2012. *The Pysolar library claims to be based on the NREL sun position algorithm (SPA), but a noticeable but small discrepancy between Pysolar and the online NREL Solar Position Calculator (SPC) were observed. This discrepancy may be due to more recently updated versions of SPA online.
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