Photovoltaic power forecasting using wavelet Neuro-Fuzzy for active solar trackers

Abstract

The generation of electric energy by photovoltaic (PV) panels depends on many parameters, one of them is the sun’s angle of incidence. By using solar active trackers, it is possible to maximize generation capacity through real-time positioning. However, if the engines that update the position of the panels use more energy than the difference in efficiency, the solar tracker system becomes ineffective. In this way, a time series forecasting method can be assumed to determine the generation capacity in a pre-established horizon prediction to evaluate if a position update would provide efficient results. Among a wide range of algorithms that can be used in forecasting, this work considered a Neuro-Fuzzy Inference System due to its combined advantages such as smoothness property from Fuzzy systems and adaptability property from neural networks structures. Focusing on time series forecasting, this article presents a model and evaluates the solar prediction capacity using the Wavelet Neuro-Fuzzy algorithm, where Wavelets were included in the model for feature extraction. In this sense, this paper aims to evaluate whether it is possible to obtain reasonable accuracy using a hybrid model for electric power generation forecasting considering solar trackers. The main contributions of this work are related to the efficiency improvement of PV panels. By assuming a hybrid computational model, it is possible to make a forecast and determine if the use of solar tracking is interesting during certain periods. Finally, the proposed model showed promising results when compared to traditional Nonlinear autoregressive model structures.

Keywords

Photovoltaic panels Neuro-Fuzzy inference system time series forecasting wavelets solar trackers

1 Introduction

There exist different types of solar trackers, from sensor-based to sensorless systems. A sensor-based solar tracker considers a closed-loop system where photosensors are used for tracking the sun direction using a feedback control scheme [1]. Frequently, light-dependent resistors (LDRs) are assumed to provide feedback signals to obtain the correct azimuth angle showing the daily path of the sun. In applications involving photovoltaic (PV) panels, solar trackers can maximize the amount of energy to be captured by changing panels position. To obtain the best performance, motors are used for position adjustment, however, if the motors are not correctly sized they can make the design inefficient [2]. One of the great benefits regarding the use of PVs is the advantage of being a renewable energy source [3].

The generation forecast in PV systems is an interesting task, mainly because when the generation capacity is properly estimated in a new position, it will be possible to assess whether this update is feasible in terms of making the system even more efficient. In this sense, the main contribution of this article is the prediction of chaotic time series, based on the Wavelet Neuro-Fuzzy algorithm (WNF), to improve the performance of active solar trackers for power forecasting. It’s also presented an active solar tracker to provide data for time series forecasting associated with power generation.

Active solar trackers can be categorized as [4]: i) microprocessor-based; ii) computer-controlled date and time based; iii) auxiliary bifacial solar cell-based; and iv) a combination of the three previously mentioned categories. The active solar tracker presented in this study is based on LDRs. The sequence of this introduction provides a literature review focusing on recent works involving the PV generation forecasting considering uncertainties in the system.

In [5] a reliable power management system and an accurate power forecasting model were used to solve the problem of geographical dispersion and output power fluctuations for the PV grid connection. Hong and Liu [6] studied how to solve stability problems caused by uncertainty and intermittency in power generation when PV and wind turbine generators were assumed. Wan et al. [7] presented a new and efficient probabilistic prediction approach to accurately quantify the variability and uncertainty of photovoltaic energy conversion. The model was based on linear programming and built using Extreme Learning Machine.

Specifically, studies were carried out focused on system optimization through generation forecasting, which is the theme of this article. Al-Dahidi et al. [8] considered an optimized method to improve the prediction of 24-hour photovoltaic solar energy conversion. The Artificial Neural Network (ANN) was optimized by expanding the number of hidden neurons, while diverse training datasets were assumed to build distinct ANN models.

Focused on the optimal operation of micro-grids, in [9] a study about the application of a heuristic algorithm to evaluate the power generation of a PV system and other renewable energy sources were presented. As shown in [10], it is possible to perform the forecast on solar trackers. However, the use of hybrid algorithms can improve efficiency in forecasting [11], as will be proposed in this article.

As noticed on the previously mentioned work and emphasized by Panait and Tudorache [12], ANN’s are frequently adopted in applications associated to solar energy systems due to their ability to extract solutions of nonlinear problems with variable parameters, which is exactly the case of solar trackers that present the integration between sensors and actuators.

Electric power generation forecast studies are also carried out on PV panels about the assessment of environmental conditions. The work presented in [13] had the objective of predicting the 24-hour power generation capacity of a microgenerator from climate information such as air temperature, wind speed, air humidity, pressure, and solar radiation. The proposed method was assumed for residential demand planning for the next day.

In the research proposed by Huang, Chen, Yang, and Kuo [14], an intelligent method to predict one-day-ahead hourly photovoltaic power generation was proposed. For the initial phase, the method based on historical data of PV energy generation considering various climates. In the training phase, five models were built in different environmental conditions. In the sequence, the Fuzzy inference was considered to select an appropriate prediction model from the trained ones.

Huang and Kuo [15] considered a high-precision deep neural network model to predict the output power of a photovoltaic system. The algorithm generated a 24-hour probabilistic and deterministic forecasting of PV power output based on meteorological information, such as temperature, solar radiation, and historical data about the PV generation. The importance of predicting power generation of PV systems using machine learning was highlighted in [16], considering that this type of system is highly sensitive to climate variation and seasonal factors. In this case, accurate forecasting of PV output was stated as necessary to ensure the reliability of the system.

In the article written by Elobaid, Abdelsalam, and Zakzouk [17], it was reported that during partial shading and rapid changes in environmental conditions, artificial intelligence-based techniques are superior to classic Maximum Power Point Tracker. Additionally, Manjili, Vega, and Jamshidi [18] showed a combination of data-analytic approaches consisting of a composition of artificial intelligence and statistical techniques for solar energy forecasting.

Chaotic time series prediction methods are gaining ground among researchers, and according to [19], some advantages in terms of accuracy in forecasting associated with renewable energies generation were reported. In [20] the Adaptive Neuro-Fuzzy Inference System is assumed for solar radiation prediction, where a case study in Seoul city with the need to correctly update the position of the dual-axis positioning PV system was presented.

Hybrid methods for time series forecasting are showing promising results, as it can be seen in the work [21], in which the Neuro-Fuzzy time series model is based on integrated nonlinear feature selection. Moreover, the use of advanced optimization techniques can further improve the model for forecasting time series [22]. In [23], both particle swarm optimization and proportions of intervals are combined with fuzzy for time series forecasting.

Specifically for forecasting related to the energy problem, in [24], the hybrid neuro-fuzzy model was applied to energy consumption prediction. Related to the electricity generation issue, works [25] and [26] presented models for wind speed forecasting. To forecast electric power load, in [27] and [28] both studies presented fuzzy time series models showing promising results in terms of accuracy.

In addition to hybrid forecasting models based on Neuro-Fuzzy, ensemble models have gained ground in time series prediction research. These models are being explored because they usually require less computational effort than known models based on deep learning or other combinations of algorithms [29]. When solar systems for electrical energy generation are considered, the efficiency evaluation for energy conversion became necessary to improve the generation capacity and to assess the availability of the electrical system [30, 31].

The ensemble models are combined algorithms based on weak learners for specific applications to obtain a more effective model [32]. It is possible to organize these learners in different ways to obtain several variations of the ensemble model, such as Bagging, Random Forest, Boosting, and Stacking [33].

An ensemble Stacking model to predict energy consumption is presented by Khairalla et al. [34], the results showed that the model is an encouraging methodology for complex time series forecasting tasks. Another interesting work that combined the ensemble and fuzzy strategy for time series forecasting was proposed by Yang, Jiang, and Lu [35]. With the application of a hybrid model, results showed an improvement in the precision and stability of the system.

The proposition of ensemble models considering more complex architectures based on deep learning can be stated as a promising proposal for time series forecasting [36]. This approach was developed and presented in [37], showing that deep learning has more advantages when the forecasting horizon increases.

The contributions of this article for the photovoltaic electric power generation research are summarized in the following:

The first contribution is focused on improving solar energy capture through the use of trackers. Assuming this type of system, it is possible to maximize the generation of photovoltaic panels by updating the position of the panel concerning the sun, maintaining the system with the best inclination dynamically.

The second contribution refers to finding an algorithm that is more accurate in predicting a time series in the context of power forecasting of photovoltaic systems. For this purpose, this work uses a hybrid method combined with the wavelet transform to reduce the noise of the time series and improve the precision of the model.

The last contribution is focused on showing that it is necessary to forecast the photovoltaic generation to act in the control of the proposed dynamic system, once it is not promising to use more energy to update the position than the generation capacity improved with the proposed update in the panel position.

In this sense, the problem issue in this manuscript is related to the verification of the technical feasibility in carrying out a generation forecast for a solar tracker. The main objective of this work is to evaluate the technical feasibility in forecasting electricity generation assuming solar trackers and artificial intelligence algorithms. This article presents a research with quantitative methodology through the development and evaluation of a prototype for the proposed algorithm [38].

The specific objectives of this work are: i) To assess whether the Wavelet transform improves the predictive capacity of the algorithm; ii) To assess whether the proposed algorithm is superior to other algorithms that can be applied to the same problem; iii) To develop a prototype of an efficient solar tracker.

Based on this need, the research hypothesis for this project is: A hybrid time series forecasting method based on a fuzzy inference system can improve the efficiency of photovoltaic systems when associated with solar trackers.

The remainder of this article is organized as follows. Section 2 presents and discusses the design of the prototype, focusing on its constructive characteristics. Section 3 addresses the method applied for time series forecasting, emphasizing Wavelet transform (WT) technique that was assumed for feature extraction. Section 4 evaluates the predictive performance of the algorithm, showing the results of this study. Finally, Section 5 presents the final considerations associated with the applicability of the technique and presents some suggestions for future works.

2 Dual-axis solar tracker prototype

The mechanical structure of the dual-axis solar tracker system allows the panel angle to be adjusted, from east to west (daily follow) and from north to south (seasonality) to ensure that the PV panels always follow the elevation and position of the sun. Due to a solar tracking system, modules can provide the best performance. In the dual-axis azimuth system, the main movement is realized by rotating modules in their vertical axis of fixation. Thus, continuous movements become necessary on both axes to obtain vertical rotation and lifting motion more precisely [10].

The amount of sunlight incidence on the face of the module is directly dependent on the angle of incidence of the light source. The largest area available for capture occurs whenever solar rays are parallel to the normal of the module face. Keeping this angle as close as possible to 0°, a maximization of the module output power will occur, because in this way, there is a maximization of the incidence of solar radiation. To keep the PV cells perpendicular to the sunbeam, the position provided by the solar tracker must consider the compensation of changes in the altitude angle of the sun (throughout the day), the latitudinal offset of the sun (seasonality), and changes in azimuth angle.

Solar tracking can be performed assuming a control algorithm that guides the positioning of modules optimally, where the input is latitude and longitude, or by sensors on a feedback control loop. Active trackers are solar trackers based on the micro-processing module, in which panels position is calculated through a control algorithm. This type of system has electrically operated positioning drivers that require controlled motors, gears, and couplings [39].

The method assumed in this article to track the sun is based on a closed-loop system with photosensors used to determine the position of the sun based on the comparison of luminous incidence. The information provided by the photosensors is transmitted to the controller, which modifies the signal and commands the actuators. The PV system with an electronic follower is effective if the following condition is met: $ɛ = (P_{T} - P_{F}) - P_{C} ⪢ 0,$ (1) where P_T is the energy produced by the module with the solar tracker, P_F is the energy produced by the same module but disregarding the solar tracker and P_C is the energy consumption to perform solar tracking, which is directly proportional to the tracking accuracy.

For analysis of this experiment, the instantaneous power is used, measured in Watts. To facilitate the understanding of this research, it presents the power expressed in mW. To justify automatic solar tracking, both economically and energetically, the efficiency parameter ɛ should be as high as possible.

The proposed dual-axis system can be considered a hybrid strategy because of the controller, after receiving data from the sensors, it calculates the position of the sun based on the time and date of the year. Using four sensors, this design tends to minimize directional errors, since the process is adaptive concerning the luminosity information of each sensor. In this way, the system can provide a considerably higher yield in unfavorable climatic conditions, such as rainy and passing clouds situations [10].

Based on previously mentioned benefits, a dual-axis solar tracker was designed and built. To build the prototype, medium density fiberboard was cut through a laser cut machine and all components were integrated (see Figure 1). The active tracking system was designed based on Light Dependent Resistors (LDR) in a polar tracker structure. In this prototype, 5 mm LDR’s were used. These components vary their resistance according to the light intensity that they are exposed to.

Fig. 1

Dual-axis solar tracker prototype.

A microprocessor controller Arduino Uno R3 type was considered for both reading and interpreting data obtained by the LDR’s. Data was recorded through a specific shield in an Arduino for data recording in ’.txt’ format a SD (Security Digital) card was used [40]. In this step, the Java language was used for serial communication. To move the structure, two 9g (SG 90) micro-servos were adopted, these providing a range up to 180°. In this way, it is possible to determine how many steps the motor shaft should rotate, and consequently how many degrees. Figure 2 (in blue) shows the installed micro-servo for daily tracking.

Fig. 2

Micro-servo 9g assumed to change the PV angles.

The micro-servos have a torque of 1.2 kg . cm (around 0.1177 N . m) and operate at 4.8 V via Pulse Width Modulation (PWM). Both motors are powered by an external source, so there is no influence on the PVs generation, this was assumed due to testing conditions. In this study, polycrystalline silicon photovoltaic cells of 1,000 mW with 6 Vcc were assumed, those producing a current up to 200 mA.

To simulate an ideal linear load, sets of 25 Ω resistors of 1,000 mW were used. Besides, resistors were assumed to make the associations with the LDR’s to reach 10 kΩ. This association was defined to obtain adequate signal values for the controllers. For comparison purposes, an equivalent PV was used to measure without tracking. This fixed system was positioned with an ideal angulation for the local latitude of 23°, oriented to the North Geographic.

3 Chaotic time series forecasting

In this section, it is presented the technique assumed to predict chaotic time series associated with solar energy conversion is presented. In particular, the WNF will be evaluated based on data from the dual-axis solar tracker presented in the last section. To improve the algorithm, the WT will be used for feature extraction.

Both the settings and the evaluation criterion assumed for WNF will be discussed in this section. Also, it will be presented the Nonlinear Autoregressive with eXogenous inputs (NARX) model, which will be assumed for comparison will be presented. Observations were performed using a sampling time of Δt = 1s. A time series is considered stationary if there are no large variations in the variance of values observed over time. In this way, the sequence of values should be stable and provides regular behavior.

In time series forecasting, already known values of the system output, until time t, are assumed to predict the value at some point P in the future, t + P [41]. This procedure creates a mapping from sampled data points n, acquired every Δt units in time. The dataset assumed for the prediction can be defined as x = [x (t - (n - 1) Δt) , …, x (t - Δt) , x (t)], where a future predicted value can be defined as: $y (t + P) = f (x),$ (2) where y (t + P) represents the predicted value for the time series, and f (x) a mapping function considering n regressors.

Among the chaotic time series forecasting techniques, it was highlighted in the introduction of this article the WNF method, which was considered in this work and is presented below.

3.1 Neuro-fuzzy system

Neuro-Fuzzy systems are a combination of ANNs with Fuzzy systems. This type of system has the advantage of allowing an easy translation of the final system into a set of if-then rules, and the Fuzzy system can be viewed as a neural network structure with knowledge distributed throughout connection strengths [42].

Applications of neural and Fuzzy hybrid systems are useful in fields such as the applicability of existing algorithms for ANNs, and direct adaptation of knowledge articulated as a set of Fuzzy linguistic rules, some examples of applications can be found in [43].

An adaptive network is a structure that consists of nodes and directional links. The input-output behavior can be determined by the values of a set of modifiable parameters through which the nodes are connected [44]. The adaptive Neuro-Fuzzy uses a hybrid learning algorithm to identify parameters of Sugeno-type Fuzzy inference systems. Usually, this system applies a combination between the least-squares and the back-propagation gradient descent methods for training the Fuzzy inference system membership function parameters to emulate a given training dataset [45].

The main idea is that the network learns in two main phases: the forward phase and the backward phase. In the first one, the consequent parameters identify the least squares estimate, while in the second one, the derivatives of the squared error concerning each node output, propagate backward from the output layer to the input layer. In this backward phase, the premise parameters are updated by an optimization method [46].

Considering a time series forecasting application, the Neuro-Fuzzy algorithm can manipulate incomplete, imprecise, complex, and non-linear data. Besides, models created using these strategies could adapt and group common data into clusters [41].

Additionally, in this article, we assumed Fuzzy C-Means (FCM) to improve training. The FCM method allows one piece of data to belong to two or more clusters. The data is partitioned into Fuzzy sets by minimizing the sum of square error for groups. Moreover, the FCM automatically chooses the number of clusters and randomly assigns the coefficients to each data point until a convergence criterion is met.

The process of FCM starts by calculating the centroid c_j of each cluster j for each point, which can be expressed by: $c_{j} = \frac{\sum_{i = 1}^{n} μ_{i, j}^{m} x_{i}}{\sum_{i = 1}^{n} μ_{i, j}^{m}},$ (3) where any point of x_i has a set of coefficients concerning the degree μ_i of the cluster, and m is the Fuzzy partition matrix exponent for controlling the degree of Fuzzy overlap. The centroid of a cluster is the average of all points, weighted by their degree of belonging.

The FCM system attempts to separate the dataset in a finite collection of x elements, those called of clusters, based on a given criterion [47]. Thus, the objective function to be minimized J_m, with N clusters, can be expressed by $J_{m} = \sum_{i = 1}^{n} \sum_{j = 1}^{N} μ_{i, j}^{m} {∥ x_{i} - c_{j} ∥}^{2},$ (4) wherein, $μ_{i, j} = \frac{1}{\sum_{k = 1}^{N} {(\frac{∥ x_{i} - c_{j} ∥}{∥ x_{i} - c_{k} ∥})}^{\frac{2}{m - 1}}} .$ (5)

For a complete evaluation, the application of the WT for normalization and extraction of signal characteristics was evaluated in comparison to raw signal.

3.2 Wavelet transform

The use of the Wavelet transform is a promising technique for the evaluation of time series, considering that it can reduce signal noise, maintaining its fundamental characteristics. WT uses signal energy, variance, and reconstructed component to generate the filtered signal based on Shannon entropy. Band filters based on frequency rejection may disregard important characteristics at high frequencies that may have relevant information in this analysis [48].

The WT uses sliding windows over time intervals to decompose a signal through different levels of resolution and scales [49]. One of the greatest advantages of using WT is the ability in manipulating data and extracting features or compressed characteristics of the function. Considering g (t) a time domain function to be decomposed, the WT φ (a, b) can be defined as: $φ (a, b) = \int_{- \infty}^{\infty} g (t) \frac{1}{\sqrt{| a |}} ψ (t) * (\frac{t - b}{a}) dt,$ (6) where a and b are real parameters of the function: $ψ_{a, b} (t) = \frac{1}{\sqrt{| a |}} ψ (t) * (\frac{t - b}{a}) .$ (7)

This transform can be rewritten as the internal product of the functions g (t) and ψ_a,b (t) as: $φ (a, b) = h (g (t), ψ_{a, b} (t)) = \int_{- \infty}^{\infty} g (t) ψ_{a, b} (t) dt .$ (8)

In equation (7), ψ (t) is the main function and ψ_a,b (t) represents a secondary function. The b parameter indicates that the function has been translated t over a distance of b. The parameter a generates a scale change, which can be increased when a > 1, or decreased when a < 1 [50].

3.3 Algorithm configuration

To standardize the analyzes, the maximum number of iterations was set to 1,000 with an adaptive learning rate. The algorithm was used with the hybrid training method, with an initial step equal to 0.01, a decrease rate equal to 0.9, and the rate of increase equal to 1.1. The hybrid neural network optimization method uses the combination of least-squares estimation and back-propagation for training.

For network training, the adjustment of the fuzzy inference system was of the Sugeno type. This model generates a single-output and tunes the system parameters using the specified input and output for training data. In the Adaptive Neuro-Fuzzy Inference System, 5 layers were used.

To optimize the network during training, two methods were considered: the backpropagation (B), and a hybrid (H) method. Backpropagation uses gradient descent to calculate the parameters. The hybrid method assumes a combination of backpropagation, which was assumed to calculate input participation function parameters, and least-squares estimation, which was considered to calculate the output parameters [51].

To perform a complete analysis of the learning capabilities of the algorithm, several configurations were considered. Statistical analysis was assumed as the signal was segmented into 4 windows of 5,400 samples each (1 hour and 30 minutes recording each set). A comparison was made between the amount of data used for both training and testing, where 25%, 50%, and 75%. of the entire dataset was assumed for training in different cases. The preliminary analysis was performed based on day 1, this day’s characteristics will be presented in subsection 3.5.

The algorithm response was evaluated according to the variation of the number of clusters in the FCM structure considering an interval from 2 to 10 clusters. This interval was defined based on previous tests, those showed that the use of more than 10 clusters did not significantly improve the results in terms of performance, enhancing the computational effort.

3.4 Performance evaluation

The performance evaluation criterion adopted in this study was based on the regression of the network (presented as accuracy in %) and in time for convergence. Considering the network output given by y (t) = a₀ + a₁ x + e, where a₀ and a₁ are the linear and angular coefficients, e is the residual between the model and observations. The regression error (Reg . error) of the network is obtained through the least square method, given by: $Reg . error = \sum_{i = 1}^{n} {e_{i}}^{2} = \sum_{i = 1}^{n} {{(y}_{i} - a_{0} - a_{1} x)}^{2} .$ (9)

So, there are two equations and two unknowns. These equations are called normal equations, these minimizing the sum of the squared differences. They can be simultaneously obtained by: $a_{1} = \frac{n \sum x_{i} y_{i} - \sum x_{i} \sum y_{i}}{n \sum x_{i}^{2} - {(\sum x_{i})}^{2}}, a_{0} = \bar{y} - a_{1} \bar{x} .$ (10)

To calculate the error, the Root Mean Square Error (RMSE) was assumed: $RMSE = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} {(y_{i} - x_{i})}^{2}} .$ (11)

3.5 Statistical analysis

For the statistical analysis, average value, standard deviation, variance, and covariance were considered. The variance V_i measures statistical dispersion, indicating "how far" in general their values are from the expected value, and is presented as: $V_{i} = \frac{1}{n - 1} \sum_{p = 1}^{n} {(y_{i, p} - {\bar{y}}_{i})}^{2} .$ (12)

In equation (12), y_i,p is the value of the predicted output i in object p, ${\bar{y}}_{i}$ is the average of the variable i. In (13) y_j,p is the value of the variable j in object p, ${\bar{y}}_{j}$ is the average of the value of the variable j.

Standard deviation is a measure of dispersion around the population mean of a random variable and is defined as the square root of the variance. Covariance C_i,j is the linear correlation between two random variables, according to the equation: $C_{i, j} = \frac{1}{n - 1} \sum_{p = 1}^{n} (y_{i, p} - {\bar{y}}_{i}) (y_{j, p} - \bar{y_{j}}) .$ (13)

Based on both eigenvalues and eigenvectors, the main components can be obtained considering the accumulated percentage variability. Factors with the highest eigenvalues are selected and the indicators of each factor are then calculated. The influential characteristics are chosen based on the evaluation of indicators considering the most significant factors.

For statistical evaluation, data were recorded for 6 hours (from 12h to 18h) on 6 separate days. During day 1 there were clear skies on a post-frost day, with 0 mm of rain. On day 2, there was a clear sky most of the day, with cloudiness for approximately 1h, near the end of the afternoon, and there was no rain either.

On days 3 and 4 there was 5 mm of rain in the evening (not occurring in the measurement period) and the sky was cloudy. Day 5 was cloudy in the morning and clear in the afternoon, during the measurement period, with no rain on the day. Finally, on the 6 the sky was cloudy and there was no rain during the day.

Measurements were performed in Curitibanos city, in the state of Santa Catarina in the southern region of Brazil. The city has an altitude of 987 m above sea level and is located at latitude: 27° 16′ 58″ S, and longitude: 50° 35′ 04″ W. Measurements were carried out during the winter season.

3.6 Additional models for comparisons

In time series forecasting, parametric models can be assumed to reduce the error between real data and the estimated output of the model. The nonlinear input-output (NIO) model can be defined according to equation (14) for the past values of series x (t). $f (x) = f (x (t - 1), \dots, x (t - d)) + ɛ_{t} .$ (14)

Another way to predict the time series is through a Nonlinear AutoRegressive (NAR) model, wherein y (t) given d past values of y (t). The NAR model can be presented as: $f (x) = f (y (t - 1), \dots, y (t - d)) + ɛ_{t} .$ (15)

The Nonlinear Autoregressive with eXogenous inputs (NARX) was also assumed [52]. The NARX model relates the current value of a time series to past values of the same series and the current and past values of exogenous series, and can be stated as: $\begin{matrix} f (x) = f (x (t - 1), \dots, x (t - d), y (t - 1), \dots, \\ y (t - d)) + ɛ_{t} . \end{matrix}$ (16)

In this case, the predicted output of the model (time series estimated output) y (t) given d past values of y (t) and another series x (t). The summary flowchart of the method adopted for time series forecasting and comparisons that were performed in this article is presented in Figure 3.

Fig. 3

Method flowchart.

4 Analysis and discussion

All measurements were made with the PV with a dual-axis solar tracker compared to a fixed solar panel. There was an improvement in power generation especially on days with higher cloudiness, as can be seen in Figure 4. Assuming the solar tracker, it is possible to capture higher brightness by reflection. The presented measurements were performed on a cloudy winter day in the late afternoon (dataset # 04).

Fig. 4

Power generation comparison between a fixed panel and the prototype with solar trackers.

According to Figure 4, there was an improvement of 30.91% when the solar tracker was assumed. From the results using the solar tracker, WNF was assumed for time series forecasting purposes.

The algorithm responses were evaluated in terms of accuracy by changing both the network configuration parameters and the training method. Table 1 presents the results considering optimization based on backpropagation (B) and the hybrid method (H) for training. In both, variations in the amount of data for training and testing were considered and evaluated.

Table 1

Accuracy variation for the validation phase based on the configuration of both optimization method and Wavelet parameters according to the percentage of data assumed for training and testing procedures

Opt. / Train	Dataset	Dataset	Dataset	Dataset
/ Test	# 01 (%)	# 02 (%)	# 03 (%)	# 04 (%)
B / 25 / 75	98.890	98.337	98.738	99.504
B / 50 / 50	98.889	98.315	98.751	99.511
B / 75 / 25	98.902	98.352	98.754	99.511
H / 25 / 75	98.921	98.400	98.759	98.508
H / 50 / 50	98.922	98.429	98.767	99.516
H / 75 / 25	98.936	98.433	98.771	99.516

In this section, the best results from each dataset are underlined and the best overall results are in bold. In terms of accuracy, the best results are those with values closer to 100% and in terms of time, with shorter analysis time. Accuracy values are based on the equation (9), to validate the results based on the values obtained through network training.

The results on dataset # 04 were superior in terms of accuracy, mainly because there was a lower number of clouds in the evening providing a more linear behavior, facilitating learning procedure. The best relative result was obtained using 75% for training and 25% for testing, so this setting was also assumed for the other evaluations that will be discussed in the sequence.

The variation in the number of levels of Wavelet packet decomposition yielded equivalent results from the fifth level, and the use of more than one node in this analysis caused the signal to lose its fundamental characteristic. Thus, 5 levels of decomposition with 1 node were assumed. Moreover, the application of the WT was evaluated using different cluster numbers compared with the raw data, as shown in Table 2.

Table 2

Proposed method accuracy evaluation

Clusters	Dataset	Dataset	Dataset	Dataset
/ Data	# 01 (%)	# 02 (%)	# 03 (%)	# 04 (%)
02 / Raw	98.936	98.433	98.771	99.516
04 / Raw	98.934	98.496	98.793	99.520
06 / Raw	98.975	98.598	98.962	99.532
08 / Raw	98.985	98.645	98.833	99.536
10 / Raw	98.993	98.737	98.842	99.543
02 / WT	99.607	99.373	99.485	99.782
04 / WT	99.609	99.389	99.494	99.784
06 / WT	99.617	99.418	99.504	99.789
08 / WT	99.621	99.412	99.510	99.792
10 / WT	99.623	99.443	99.516	99.796

The application of the wavelet transform when compared to the raw signal can be seen in Figure 5 (dataset # 04). As can be verified, the WT application reduces signal noise, while maintaining its characteristics. In terms of accuracy, the increase in cluster numbers has generally yielded superior results. Using 10 clusters in most analyzes yielded a higher accuracy result. However, the time required for the algorithm to converge was considerably longer, as presented in Table 3.

Fig. 5

Wavelet transform (WT) compared to raw signal.

Table 3

Proposed method time evaluation

Clusters	Dataset	Dataset	Dataset	Dataset
/ Data	# 01 (s)	# 02 (s)	# 03 (s)	# 04 (s)
02_Raw	3.377	3.350	3.577	6.322
04_Raw	6.176	6.003	6.035	8.336
06_Raw	9.549	9.313	9.253	11.906
08_Raw	13.410	13.258	13.378	15.951
10_Raw	18.136	18.345	17.984	23.350
02_WT	3.327	3.386	3.628	5.797
04_WT	6.341	6.426	6.357	8.504
06_WT	9.399	9.802	9.321	12.079
08_WT	13.626	13.602	13.673	17.916
10_WT	18.731	19.138	18.858	20.781

WT application for feature extraction provided superior accuracy results with equivalent convergence time. The application of WT for feature extraction yielded accurate results with equivalent convergence times. It took relatively more time to converge based on dataset # 4 as there was less data variability during generation logging.

For a complete statistical analysis, data were recorded on 6 separate days. The hybrid algorithm optimization was used, with 75% of training data, considering 2-cluster in FCM and with WT in the process. Statistical analysis was based on differences among all the datasets for mean training time, mean precision, standard deviation, and variance of precision. To calculate the covariance, the relationship among all days was considered as can be seen in Table 4.

Table 4

Statistical analysis

Day	Mean Time (s)	Mean Accur. (%)	Std. Deviation	Variance	Covariance
1	4.035	99.562	1.75×10^-3	3.07×10^-6	2.37×10^-6
2	3.792	99.239	1.08×10^-2	1.17×10^-4
3	7.531	99.783	2.07×10^-3	4.29×10^-6
4	3.800	99.984	2.67×10^-4	7.13×10^-8
5	5.066	99.425	6.67×10^-3	4.45×10^-5
6	4.509	99.630	3.47×10^-3	1.20×10^-5

According to the statistical analysis, it was noticed that there was little variability in the results regarding the accuracy, considering that the analysis of variance was based on different datasets, in which there was a difference in measurement time and consequently in solar intensity.

This assessment can be supported based on the global covariance values, which were obtained from the analysis of 6 different days. It was noticed that in cloud days the signal became more nonlinear and the algorithm needed a longer time to converge.

4.1 Nonlinear methods comparisons

The benchmarking of time series forecasting capacity of this article is based on methods NIO, NAR, and NARX discussed in the previous section. For these, the optimization strategies of Levenberg-Marquardt (LM), Bayesian Regularization (BR), and Scaled Conjugate Gradient (SCG) were applied [53]. The ways on how networks were configured were also evaluated, as the number of hidden neurons (NHN) and the number of delayed inputs (ND), the latter representing the regressors. All these evaluations are presented in Table 5.

Table 5
Time series forecasting evaluation through parametric models

NHN / ND Train. Method NIO (%) NAR (%) NARX (%)

05 / 05 LM 11.418 98.937 98.953

05 / 10 16.749 98.963 98.942

10 / 05 35.438 98.988 98.974

10 / 10 40.341 99.009 98.925

05 / 05 BR 12.187 99.014 99.017

05 / 10 15.379 99.034 99.056

10 / 05 12.193 99.049 99.049

10 / 10 16.171 99.092 99.113

05 / 05 SCG 8.761 98.657 98.849

05 / 10 10.147 98.677 98.769

10 / 05 11.183 98.627 98.843

10 / 10 11.301 98.670 98.844

NHN / ND	Train. Method	NIO (%)	NAR (%)	NARX (%)
05 / 05	LM	11.418	98.937	98.953
05 / 10		16.749	98.963	98.942
10 / 05		35.438	98.988	98.974
10 / 10		40.341	99.009	98.925
05 / 05	BR	12.187	99.014	99.017
05 / 10		15.379	99.034	99.056
10 / 05		12.193	99.049	99.049
10 / 10		16.171	99.092	99.113
05 / 05	SCG	8.761	98.657	98.849
05 / 10		10.147	98.677	98.769
10 / 05		11.183	98.627	98.843
10 / 10		11.301	98.670	98.844

In this analysis, the NIO method provided an insufficient result in all analyzes and thus, showing that it is not indicated for this application. Both NAR and NARX models provided equivalent results but remained inferior to the results provided by the WNF model. For this comparative analysis, configurations equivalent to the WNF system were used.

4.2 Long short-term memory comparison

The Long Short-Term Memory (LSTM) model has been widely used as an alternative for time series forecasting [54]. For comparative purposes, the LSTM model results were presented in Table 6. The optimizers used were Adaptive Moment Estimation (ADAM), Stochastic Gradient Descent with Momentum (SGDM), and Root Mean Square Propagation (RMSProp).

Table 6
Time series forecasting through LSTM

Hidden Units Train Method Time (s) Accuracy (%)

05 ADAM 60.862 91.097

10 61.817 92.992

15 60.956 94.021

20 64.051 94.066

05 SGDM 63.205 93.955

10 61.690 92.674

15 62.987 92.323

20 64.337 92.233

05 RMSProp 60.908 93.498

10 63.377 93.163

15 62.321 92.561

20 63.966 93.492

Hidden Units	Train Method	Time (s)	Accuracy (%)
05	ADAM	60.862	91.097
10		61.817	92.992
15		60.956	94.021
20		64.051	94.066
05	SGDM	63.205	93.955
10		61.690	92.674
15		62.987	92.323
20		64.337	92.233
05	RMSProp	60.908	93.498
10		63.377	93.163
15		62.321	92.561
20		63.966	93.492

As it can be observed in Table 6, the best result for accuracy was 94.07, which is lower than the WNF model proposed in this article. The time required for convergence of the LSTM algorithm is also higher, showing that the combination with Wavelet proposed in this article, is promising for the evaluation in question. Comparatively, the ADAM optimizer generated the best result in the LSTM model, considering the SGDM and RMSProp comparison. All optimizers showed in Table 6 had similar results for LSTM, providing inferior performance when compared to the WNF approach.

4.3 State-of-the-art approaches and comparisons

In the work presented in [5], results were addressed concerning the mean absolute percentage error (MAPE) and RMSE. The evaluation showed that a deep power forecasting model based on convolutional neural networks has lower MAPE and RMSE results than the deep belief network and ANN.

Whereupon the overall error on clear days is the smallest, with about 90% of MAPE being lower than 0.012. Also based on deep learning, in [15] it was presented interesting results for photovoltaic system output power forecasting. Even providing high accuracy results, deep learning techniques require more computational effort, making training time longer.

In [16] a complete review considering a wide range of PV generation forecasting models was presented. Authors showed that hybrid techniques have better forecasting accuracy when compared to individual machine learning methods. Using genetic algorithms in a hybrid architecture, the observed error was less than 5.64%.

Applying an Optimized Artificial Neural Networks, in the study presented in [8], results showed an RMSE of 12.61 (kW). Authors considered a different approach to evaluating algorithm performance for a large PV system, as presented in [6] considering a variation of a Fuzzy logic approach associated with the generation control to reduce voltage oscillations caused by large disturbances.

Using FCM, the work presented in [14] showed a complete evaluation in terms of power forecasting and also mean relative error. In the comparisons, the authors showed that the proposed algorithm provided superior results when compared to ANN and Radial Basis Function Neural Networks. Also applying a Neuro-Fuzzy scheme, in [20] it was obtained an RMSE of 0.22%, which makes it clear that the technique can be used to predict generation in PV systems.

To finish this analysis, in [7] a combination of ELM and quantile regression showed to be much faster than bootstrap based on neural network and nonparametric granule computing, providing an empirical coverage rate of up to 90.28%, with a coverage rate deviation of 0.28%, for an equivalent prediction problem.

5 Conclusion

Solar trackers improve the efficiency of PV panels by capturing maximum solar radiation. However, the use of this type of system can result in excessive power consumption to update the position of the panel. In this way, the prediction of power capacity generation can be assumed and is the point evaluated in this study.

Here, it was assumed a Neuro-Fuzzy-based approach considering Wavelet transform was assumed, promising results for predicting chaotic time series associated with PV power capacity generation, to increase the efficiency of solar tracking systems, were obtained.

The robustness of the analysis was supported by statistical evaluation, considering that little variability was obtained even changing datasets, these representing different times of the day. Additionally, low covariance was reported. In this work, the application of WNF presented better results than nonlinear classic parametric methods as NIO, NAR, and NARX.

Comparatively, the WNF model proposed in this article was superior to the LSTM model in terms of accuracy and performance, providing the lowest computational cost, this represented by higher convergence speed. Previous analysis considering different configuration for the hidden layers, varying the number of neurons and assuming distinct optimizers were conducted, and the results were still surpassed by the WNF method.

Based on the results presented using the algorithm proposed in this work, it is possible to apply forecasting algorithms to assess the technical feasibility of those who forecast the generation of photovoltaic panels through solar trackers.

Through the accuracy results presented in this article, the proposed model achieved the desired result for evaluating the application of solar forecasting. The Wavelet transform proved to be a promising technique when combined with the Neuro-Fuzzy algorithm, and the proposed method proved to be superior to well-consolidated algorithms such as LSTM and Nonlinear AutoRegressive algorithms. To conclude, the prototype proved to be functional showing better performance than a fixed panel.

In general, the results of this work show that it is possible to perform the generation forecast with high accuracy in PVs. Based on these results, it will be possible to assess whether the use of electric motors to update the position of the PVs are convenient in terms of system efficiency. Thus, the main contribution of this work is given by the ability to improve the electric energy generation from the use of solar trackers.

In terms of chaotic time series prediction, deep learning techniques have gained notoriety for generating efficient results. The comparison between Neuro-Fuzzy and deep learning is suggested for future works associated with PV generation prediction.

Footnotes

Acknowledgments

We appreciate the Ph.D. scholarship, provided by Coordination for the Improvement of Higher Education Personnel (CAPES). Thank you for the National Council for Scientific and Technological Development (CNPq) under grant 304783/2017-0.

We would like to thank the scientific initiation scholarship, provided by Support Fund for the Maintenance and Development of Higher Education (FUMDES) sponsored by the Government of the State of Santa Catarina through the State Secretariat of Education (SED), Article 171.

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