Performance Prediction of a Dual-axis Tracking Solar Trough Collector Based on Artificial Neural Network

A dual-axis tracking parabolic trough solar collector, using a certain straight-trough tube, was set up to experimentally investigate the heat collecting performance. An artificial neural network(ANN) model was developed. Experimental data were used to train and predict the mean temperature of Heat transfer fluid in the solar trough collector based on the developed model. The Levenberg-Marquardt (LM) method was also applied to optimize the weights and thresholds for the classic BP Newton algorithm, providing an ANN model with 9 hidden nodes and 30,000 training times. The predicted values are all in good agreement with the experimental data, with a mean relative error of 0.21% and a maximum error of 1.2%. In comparison, the mean relative error of the one-dimensional steady-state model reaches 1.07%. It indicates that the ANN exhibits excellent performance in predicting the export temperature of the solar collector with a specific flow rate of Heat transfer fluid. This ANN model is promising to predict the performance of solar trough collectors under different operating and environmental conditions.


Introduction
In the past few years, there has been significant and robust growth in the field of Concentrating Solar Power (CSP) energy systems [1][2][3][4].This innovative technology effectively harnesses solar radiation and converts it into high-temperature heat, which is subsequently transformed into mechanical and electrical energy using a turbine.Solar trough collector [5][6][7][8] is one of the ripest techniques and most commercialized solar collectors for middle-high temperatures.
In the present day, the utilization of single-axis tracking mode has become extensively prevalent in solar trough collectors.However, there is still room for improvement in terms of concentration efficiency and heat efficiency due to factors such as high cosine loss and blade-end loss.Composite tracking modes can improve the system efficiency but has difficulty in commercialization because of the relatively complex structures [9][10][11].The dual-axis tracking has been recognized as one of the most efficient and fundamental tracking methods for parabolic trough collector technology.It is particularly well-suited for small-scale applications that demand a higher level of heat collection efficiency.In their study, Qu et al. [12] successfully designed and implemented a 300-kWth solar parabolic trough collector with a combination of north-south and rotatable axis tracking.Through experimentation, they observed that the utilization of rotatable axis tracking significantly improved the daily average efficiency, increasing it from 43% to 48% during winter.These findings offer a promising solution for effectively mitigating cosine loss in scalable parabolic trough collector systems.Also, a dual-axis tracking heat collecting system was established by Ma et al. [13] to test the heating behavior based on a certain domestic straight-trough solar collector.The dual-axis tracking achieves an optical efficiency of 0.813, significantly surpassing the efficiencies of both east-west and north-south tracking modes.Nevertheless, to this day, the industrialization of dual-axis tracking systems has remained elusive, primarily because the implementation of a highly accurate tracking system incurs additional operational expenses.
The complexity of both single-axis and dual-axis tracking trough heat collecting systems is notable on account of their performance influenced by the concentration system, tracking system, coating performance, heat and mass transfer of the collector, etc.Some simplifications (such as ignoring the inhomogeneity of the concentration system, the circumferential temperature difference between the glass tube and metal tube, the heat loss of expansion joint and the support components, and supposing that each component of the system and heat transfer fluid is all in stable performance, averaging the physical property parameter and so on) should be conducted to build a mathematical model for theoretic research.That may cause far more differences leading to a decline in the simulation accuracy.According to a recent literature review, the majority of studies focused on optimizing parabolic trough power plants rely on physical or analytical models.These models encompass both steady-state and dynamic systems [14][15][16].Additionally, existing research on the transient thermal performance of parabolic trough technology can be broadly categorized into two major groups: studies that examine the overall solar thermal power plants and those that focus on detailed transient thermal modeling methodologies for solar collector tubes [17][18].Due to the complexity addressed above, the existing models for parabolic trough solar collectors cannot achieve a high reasonable accuracy for system design with a relatively lower calculation time.The symbol definitions involved in this study are shown in Table 1.

Re
Reynolds number α A constant value between 0~10 In order to bridge the divide between physical and artificial intelligence models, novel modeling techniques have been embraced to effectively forecast solar thermal performance with precision.ANN [19], with self-learning ability and superior ability of nonlinear mapping, has been successfully applied in the field of information, automation, and engineering.It can effectively solve complex nonlinear problems merely on the basis of experimental data.In the realm of solar collectors, Benghanem et al. [20] devised an ANN model to estimate and simulate daily global solar radiation.This model utilized various input variables such as air temperature, relative humidity, sunshine duration, and the day of the year, while the output was the daily global solar radiation.The model, incorporating sunshine duration and air temperature as input parameters, demonstrated exceptional precision and accuracy, as evidenced by a remarkable correlation coefficient of 97.65%.Ceylan et al. [21] developed and constructed an experimental system that comprised a photovoltaic module, heating and cooling subsystems, as well as a Proportional Integral Derivative (PID) control unit.The ANN model was used to estimate the temperature, efficiency, and power of a photovoltaic module, providing a valuable evaluation for photovoltaic panels.In [22], an ANN model was developed to predict the performance of a solar thermal energy system (STES), achieving a high level of accuracy with errors within ±3% for preheat water tank stratification temperatures and ±10% for solar fractions.Zaaoumi et al. [23] developed three models to estimate the hourly electric production of a PTSTPP situated in Ain Beni-Mathar, Eastern Morocco.The findings revealed that the ANN model outperformed the analytical models significantly.The results obtained from the ANN model indicated an estimated annual electrical energy of approximately 42.6 GWh/year, with operating energy of around 44.7 GWh/year.The ANN model accurately estimated 86.77% of hourly values within a deviation of less than 3 MW h, making it a reliable and readily usable tool for estimating energy production in a PTSTPP.Heng et al. [24] utilized an ANN to predict the temperature rise at the exit caused by a single heat flux pulse in the first step of their methodology.Subsequently, they employed superposition to predict the cumulative effects resulting from multiple heat flux pulses in the second step.By employing the methodology described, it is possible to evaluate the optimal performance of collector tubes under various radiation conditions at an early design stage of parabolic solar trough systems.The predicted results obtained from this approach can be utilized for initial system planning, conducting heat balance analysis, and facilitating the overall system design process.Boukelia et al. [25] developed a novel ANN model for predicting the levelized cost of electricity (LCOE) in two distinct PTSTPPs that were integrated with thermal energy storage and a fuel backup system.The ANN model provided the most accurate approach for estimating the LCOE.Using the obtained weights and biases of the optimal ANN topology, the researchers were able to determine the optimal designs of the two PTSTPPs in the LCOE analysis.
However, up to now, little research concerning ANN has been carried out in the field of middle-high temperature solar trough collectors, especially for dual-axis tracking [26].ANN can be applied to predict solar collectors of different types and sizes.Thus, the difficulty and complexity of the heat collecting performance prediction will be highly decreased.
This paper employed ANN to forecast the performance of dual-axis tracking trough solar collector system.Initially, an ANN model was constructed using data obtained from thermal performance testing of the solar collector.The optimum parameters of the ANN model were achieved as training and learning to the experiment data.Finally, thermal performance was predicted by this ANN model.This work will provide fundamentals for the optimal design and large-scale application of middle-high temperature heat collecting systems.

Experiment
The study constructed a dual-axis tracking trough solar heat collector system, as shown in Figure 1 and Figure 2 (which depict the schematic diagram and the photo of the collector, respectively).Table 2 presents the key parameters of the parabolic trough solar receiver utilized in the system.As shown in Figure 1, the import and export of the collector are connected to the storage tank by tubes, and the working fluid circulates in the tubes through the centrifugal pump.The sunlight is concentrated in the inner pipe of the solar collector through the concentrator, and the heat will be absorbed by selective coating from the outside wall of the inner pipe.Thus, the thermal energy can be transmitted to the circulating working fluid with the temperature rising up via thermal conductivity.The temperature variation of the heat transfer fluid presents the heat collecting performance of this system, so the import and export temperature of the collector will be measured.Factors such as the intensity of solar radiation, flow rate of heat transfer fluid and the initial temperature of heat transfer fluid, and so on, will be tested to estimate the thermal capability of the system.

The choice of model
Among the various types of artificial neural networks, ANN is widely recognized as one of the most commonly employed.In the field of science and engineering, a single hidden layer neural network, which comprises an input layer, a hidden layer, and an output layer, is often favored.For the purposes of this study, a single hidden layer neural network was utilized as the computational model.
Performance indicators that can show the quality of the model need to be established to accurately evaluate the prediction performance of the neural network model.The relative error (RE), the mean relative error (MRE), the sum squared error (SSE), the statistical coefficient of multiple determination or correlation of variance (R 2 ), and the mean square error (MSE) are normally used to evaluate the performance of one system, which can be calculated by following equations: (1) ) 1 where ei is the experimental value, pi is the network predicted value, and n is the number of output data.

Model parameters selection
Fewer hidden nodes will lead to a poor ability to obtain information from samples and reflect concentrated sample rules.However, if the hidden nodes are too many, the irregular content of the samples will be remembered.Thus, the problem of overfitting will appear to reduce the generalization ability.Therefore, there may be optimal hidden nodes.The iterative trial is the prevailing approach used to determine the optimal number of hidden nodes.This method involves initially starting with a small number of hidden nodes and incrementally increasing it until the desired outcome is achieved.The optimal number of hidden nodes is identified based on the minimum network error achieved while utilizing the same sample set.In addition to this iterative trial method, certain empirical equations exist that can provide an initial estimate for the number of hidden nodes.Equations like Equation (6) are employed to roughly estimate the starting point for the iterative trial process: where m is the number of hidden nodes, k is the number of input layer nodes, l is the number of output layer nodes, and α is a constant value between 0 and 10.The standard BP algorithm adjusts weights based only on the current time-step error gradient, ignoring past gradients, leading to oscillation and slow convergence.Thus, including a momentum factor in the weight adjustment equation can enhance training speed and minimize oscillation amplitudes.
A small learning rate, which is also known as the step length, may slow down the convergence speed, while a large one may also cause violent oscillation.This paper used the fixed learning rate algorithm.
The learning rate and the momentum factor remained at 0.9 and 0.7, respectively.In this model, the training times were kept at 10000 at the preliminary debugging network model to investigate how the structural effects of the network on model predictive ability.

IO parameters
The imported parameters chosen to represent the main factors affecting the thermal performance of the collector are flow rate (f), global solar radiation intensity (I), the inlet temperature of tube 1 (T1), and ambient temperature (Ta).These factors are recognized as crucial in shaping the system's overall effectiveness and efficiency.
In this study, the output parameter chosen to analyze the thermal performance of the solar collector is the mean temperature of the heat transfer fluid.This parameter, which is calculated as the median of the import and export temperature, is closely related to the properties of heat transfer on the collector.By predicting and analyzing this output parameter, the research aims to gain insights into the thermal performance of the solar collector.
Figure 3 depicts the structure of the ANN model used for predicting the effectiveness and efficiency of the dual-axis tracking straight-trough solar collector.

Normalization pre-processing
Normalization processing needs to be carried out for imported and exported data of the network to maintain the equality of the input components in consideration of the varying physical meaning and dimensions of different input data.Thus, the training time can be reduced, and training accuracy will be improved, and all the data will be kept in the interval [0.1,0.9].The equation used is the following: min max min (0.9 0.1) 0.
Renormalization should be performed after the neural network prediction using the following Equation ( 8): max min min ( 0.1)/(0.90.1) ( ) where xi is input or output data, xmin is the minimum value of the data, xmax is the maximum value of the data, and xi is the data under normalization.

Training and predicting experimental data
113 groups of experimental data were chosen as the training groups, and part of the training data is shown in Table 3.The remaining 21 groups are the predicting groups, as shown in Table 4.

Training of ANN model
Once the ANN model is constructed, the solution to the model involves determining the optimal values for the adjustable parameters: the number of hidden layers and the training iterations.These parameters play a crucial role in the solving process.

Hidden nodes optimization
Hidden nodes were selected as 3-12 to test the solar collector performance network model, and the optimum number of hidden nodes would be chosen according to the MSE, R2, and MRE%.The training results of different hidden nodes are shown in Table 5.The training errors change with the different hidden nodes.Nevertheless, more hidden nodes do not mean better network performance.Considering the fitting precision and training error of the neural network, 9 is selected as the number of hidden nodes, which means the structure of the BP neural network is 4-9-1.

The choice of training times
The generalization of the neural network mainly refers to the right response ability of the data beyond the training groups to the network after learning, that is, the mapping (predicting) ability to the training samples.An overfitting phenomenon will sometimes happen during the training process of neural networks.With the increasing training times, the error decreases, and the training stops when reaching the preset training minimum error.However, sometimes the predicted error would increase with the rise of the training times, which may lead to a worse result of the prediction model.Therefore, not reducing the training error but improving the generalization of the neural network will make a better prediction model.There exists the best training time under a given number of hidden nodes.To investigate the effects of the training time, the simulation was trained and forecasted alternately.The training MSE was recorded after every training session.Then, this network ran with test data at the network weights kept unchanged, and the resulting prediction error was recorded.Figure 4 shows these two kinds of error curves with the number of hidden layer nodes 5.The training error decreases rapidly before 30, 000 iterations, but the downward trend slows down as training progresses and eventually stabilizes at around 0.41.However, there appeared to be a very different trend as to the predicted error.The error decreased quickly before 30000 times while thereafter gradually rose until it remained at about 0.82.Therefore, 30, 000 times is preferred for the best generalization, on careful consideration of the training and the predicted errors.

Verification of BP model
According to the above results, the ANN model with 9 hidden nodes and 30000 training times was selected as the best network.The experimental data were trained and predicted using this ANN model, and the results are shown in Figure 5 and Figure 6.Most of the relative errors of training data are below 1%, with the largest one lower than 1.3%.As shown in Figure 6, the relative errors of predicted value are within ±1%, and the mean relative error is 0.21%.The ANN model proves to be successful and reliable on account of predicting the solar trough collectors.7 shows the comparison between the neural network model prediction results and the simulated value by a one-dimensional steady-state heat transfer model established by the authors' group [20].The predicted results obtained from the ANN model exhibit a significantly higher level of agreement with the experimental data when compared to the predictions made by the one-dimensional steady-state heat transfer model.The ANN model has a better-predicted ability as it can offset the shortage of theory models and improve the accuracy of the simulation.

Prediction of ANN model
For further evaluation of the predicted performance of the ANN, the influence of heat transfer fluid flow rate, which is one of the input parameters, on the prediction was also investigated.Figure 8 shows the changes in the predicted results of Tm with respect to the flow rate, with the other three input parameters (I, Ta, T1) fixed constant.The mean temperature of heat transfer fluid decrease with the rise of the flow rate while leveling off and remaining stable around 152.7℃ when the flow rate is higher than 0.8 m 3 /h.The heat quantity transferred to the heat transfer fluid changes little due to steady thermal collecting conditions.The fluid velocity of heat transfer fluid increases when the flow rate rises, as will improve the heat transfer coefficient.Therefore, the mean temperature of Heat transfer fluid will decrease under the given heat transfer quantity.Considering of Reynolds number (Re), the flow pattern of heat transfer fluid changes from laminar flow to transitional flow gradually when the flow rate reaches 0.6 m 3 /h.The turbulence aggravates when the flow rate increases higher than 0.8 m 3 /h further, leading to the heat transfer coefficient rapidly rising to a relatively stable value.Therefore, the temperature of heat transfer fluid finally reaches a steady level at a flow rate higher than 0.8 m 3 /h.Thus, ANN can accurately predict the performance and trend of the dual-axis tracking trough solar collector with the changes of operation and environment parameters.The results mentioned above provide a strong confirmation of the accuracy of the prediction achieved using the ANN model.The efficiency of heat collection of the collector can be directly achieved from the input information by the ANN.The model provides a convenient way to predict the efficiency of heat collection with given experimental data such as flow rate, environment temperature, solar radiation intensity, and the inlet temperature of tube 1.It can be applied without requiring an understanding of normal heat transfer and flow characteristics, which require complex mathematical formulations to express.By employing learning and associative memory techniques with the training data, it becomes possible to accurately predict the thermal performance of the solar collector.This approach allows for the identification and understanding of the intricate nonlinear relationship between the input and output data.As a result, a deeper comprehension of the system's behavior can be obtained, leading to more precise performance predictions.

Conclusions
1) The ANN model with 9 hidden nodes and 30,000 training times was selected as the best network to predict the dual-axis tracking solar trough heat collecting system.
2) The predicted values from the ANN model are in good agreement with the experimental data, with a mean relative error of 0.21% and a maximum error of 1.2%.In contrast, the one-dimensional steadystate model of the heat transfer procedure has a mean relative error of 1.07%.
3) The developed ANN model can precisely predict the performance of solar trough collectors under different operating and environmental conditions.Solar collectors of different types and sizes can also be predicted by ANN.Thus, the difficulty and complexity of the solar heat collecting performance prediction will be highly decreased.

Figure 1 .
Figure 1.Schematic diagram of the dual-axis tracking trough concentrating solar collector system.

Figure 2 .
Figure 2. A parabolic trough collector with dual-axis sun tracking.

Figure 3 .
Figure 3. ANN model for predicting the capability of the solar collector.

Figure 4 .
Figure 4. Mean squared error (MSE) of different training times.

Figure 7 .
Figure 7.The comparison of one-dimensional steady-state model calculated value, ANN prediction value and the experimental data.

Figure 8 .
Figure 8. ANN model predictions for the mean temperature of heat transfer fluid with respect to flow rate.

Table 1 .
Symbol nomenclature table

Table 2 .
Main parameters of the solar collector.

Table 5 .
The training result of different numbers of hidden nodes