Forecasting Monthly Wind Energy Using an Alternative Machine Training Method with Curve Fitting and Temporal Error Extraction Algorithm

Abstract

The aim of this research was to forecast monthly wind energy based on wind speed measurements that have been logged over a one-year period. The curve type fitting of five similar probability distribution functions (PDF, pdf), namely Weibull, Exponential, Rayleigh, Gamma, and Lognormal, were investigated for selecting the best machine learning (ML) trained ones since it is not always possible to choose one unique distribution function for describing all wind speed regimes. An ML procedural algorithm was proposed using a monthly forecast-error extraction method, in which the annual model is tested for each month, with the temporal errors between target and measured values being extracted. The error pattern of wind speed was analyzed with different error estimation methods, such as average, moving average, trend, and trained prediction, for adjusting the intended following month’s forecast. Consequently, an energy analysis was performed with effects due to probable variations in the selected Lognormal distribution parameters, according to their joint Gaussian probability function. Error estimation of the implemented method was carried out to predict its accuracy. A comparison procedure was performed and was found to be in line with the conducted Markov series analysis.

1. Introduction

There has been a great deal of interest and research in the renewable wind energy field for the last two decades due to environmental and clean energy impacts that might require near and longtime wind speed forecasting for proper power energy operation and planning [1]. It is important to note that whereas the global total installed wind generation was around 100 GW in 2020, it has increased to nearly 600 GW in 2021. The trend has increased even more in the year 2022. However, the random nature of the wind regime is a potential challenge for accurate forecasting. The focus of forecasting analysis is mainly on deterministic and probabilistic types of forecasting in which several algorithms and procedural methodologies have been implemented [2]. Deterministic models are based on physical and statistical approaches to the logged data. Whereas physics-based models are based on the interpretation of theoretical physics on meteorological data, statistical methods are based on trends and predictions of data, as well as data analytics such as artificial intelligence (AI) and machine learning (ML). Recently, models based on the chaotic nature of wind speed have been investigated. Models based on a hybrid of both deterministic and stochastic approaches have also been attempted [3]. Abbreviations of the above main methods and algorithms are listed in the Appendix A.

1.1. Literature Review

Wind speed forecasting can generally be modeled using either physical interpretation of meteorological data and complex formulation of air flow hydrodynamics or the use of deterministic forecasting analysis, which offers single-values predictions of output results. Traditionally, deterministic models based on time series operations of wind speed data are utilized for better accuracy, such as auto regressive moving average (ARMA) [4,5] and auto regressive integrated moving average (ARIMA) [6,7]. However, many of these models rely on simple error formulations such as mean absolute error, mean squared error, and root mean squared error. Following this, data analytics and machine learning (ML) models are attempted, such as the grey method [8], artificial neural networks (ANN) [5,9], support vector machine (SVM) [10,11,12], and optimal kernel function [13]. Whereas ARMA uses stationary data, ARIMA can make use of non-stationary data, but both techniques require a lot of high-quality historical data that may be unavailable or impersistent. Grey formulation is useful in cases when there is a gap of uncertainty or insufficient data. The SVM technique maps non-linear predictions onto linear regression. Artificial neural networks employ a trial-and-error approach to improve machine learning. In all these methods, accuracy is an essential requirement, especially for short-term forecasting and cases of stochastic nature and intermittences that require good knowledge of random variables characteristics and probability distribution functions. It was common to utilize smaller wind turbine generators for the peak load demands, and hence short-term prediction is necessary, but overall, this can be undetermined and could be risky for power control and proper power operation. However, with the utilization of wind farms and larger wind turbines, it is possible to target the base demand instead, and therefore medium and long-term forecasting can be a predominant requirement.

The hybrid approach of modeling wind speed forecasting is proved to be useful in improving short-term accuracy and overcoming the impersistent nature of wind regimes. It is assumed that machine learning hybrid models tackling daily, and monthly requirements can be useful, such as the work of [14], which used wavelet frequency analysis. In the work of [15], an empirical wavelet decomposition with a long short-term memory approach was recently attempted. The work of [16] used three different neural networks for prediction: the empirical wavelet transforms decomposition (EWT), the ensemble learning method, and deep learning networks. The recent work of [17], based on hybrid time series decomposition augmented with parameters optimization, was also attempted. On the other hand, probabilistic forecasting [18], unlike deterministic forecasting, can provide further information about uncertainties of the forecasts, which can be useful in cases such as decision-making problems and optimal control of power flow and availability. Probabilistic forecasting is based on either a parametric approach or non-parametric models, such as quantile regression (QR) [19], kernel density estimation (KDE) [20], lower upper bound estimation (LUBE) [21,22], and bootstrap [23]. Novel algorithms based on fuzzy logic [24] and ML [25] technologies were conducted in many references, with a focus on data preprocessing and postprocessing of the forecasting process.

Probability density functions are the essence of any attempted probabilistic forecasting method; for example, in the work of [26], a pdf was constructed to outperform several mixture models by using two different datasets, and in the work of [27], an improved deep mixture density network (IDMDN) algorithm was applied in multiple wind farms. In contrast, the work of [28] suggested a data decomposition hybrid model for wind energy forecasting based on an improved empirical mode decomposition (IEMD) algorithm. Gaussian processes [29] were used to evaluate wind speed probability, with dataset manipulations such as moving average (MA), autoregressive part (AR), or even both, normally termed ARMA. Errors in forecasting determination were formulated in the work of [30] based on the extraction of spatial-temporal wind speed information. Figure 1 depicts a classification of major attempted methods and algorithms that were applied for wind speed forecasting.

In summary, wind forecasting has been undergoing several main stages of development. The deterministic forecasting technique is traditionally used for point forecasts with the implementation of different models depending on different datasets, as well as different data time scales and topographic conditions. This technique embraces several modeling methods with different features, one of which is the ARMA-ARIMA model, which can predict time series values through measurement regression, whereas grey analysis is useful for insufficient or missing data. The ANN model essentially relies on the evolution of input data and their weighted characteristics. The SVM model maps a dataset into high-dimensional features through non-linear to linear regression. Physical modeling mainly depends on atmospheric conditions. Most deterministic forecasting approaches are currently used for the proper control and operation of wind energy systems. Hybrid models have recently been implemented to improve accuracy with the advantages of combined deterministic methods. Such hybrid methods combine statistical and physical models in several ways to improve the accuracy with the advantages of both methodologies.

In order to increase the accuracy of uncertainty predictions, probabilistic forecasting is now used, rather than spot forecasts, to provide deeper insights into predictions. The basic classification and features of probabilistic models are developed into several methodologies, such as parametric probabilistic forecasting, which assumes a specific forecast distribution but requires a large amount of data. The KDE method also does not assume a specific forecast distribution and hence requires a large amount of data too. The QR method overcomes uncertainty forecasts in the form of predictive estimation. The LUBE method simplifies the interval prediction in one step with the benefit of a lower computational cost. The bootstrap technique involves resampling of original data and hence simplified interval prediction. The ensemble diversity technique uses numerical weather prediction models, which normally offer better results in short-term as well as medium-term forecasting. Recently, research has focused on spatio-temporal forecasting, which uses prediction information from neighboring wind farms. Due to the vast installations of wind power systems worldwide, an attempt [31] was conducted by combing multi-model systems as another approach to probabilistic forecasting.

ML models normally require machine training that is based on output error adjusting without studying or analyzing the error trends. Hence, larger data frames are essential for optimal machine learning mechanisms. It can be noticed that in almost all previous attempted machine training models, the error of predicted output compared to the measured one was employed dominantly for adjusting the model, i.e., the error is used for this purpose only, which suggests that this approach is a single-ended one-dimensional control. It can be viewed as a one-phase static scanning of the logged data with no memory enhancement. For medium and long-term forecasting, there is also a lack of knowledge between sampling periods. The need for extra control and knowledge of the forecasting process is further extrapolated for random and calm wind regimes, such as at the logging site of this research. In this work, a hybrid model based on ML was proposed that relies on temporal error extraction, in which forecasting was utilized with the prediction of the error trends. Errors in subsequent periods were extracted and analyzed. The trend of error projection was used to adjust future forecasting, which can also compensate for the requirement of longer periods of measurement. The structure of this study was to first to curve-fit the logged data into five different probability distributions for machine learning. Secondly, the parameters of each of the pdf-s were used for model abstractions to determine the best-performing distribution function. Thirdly, the forecasted wind speed and power and energy analyses were performed on the model. Lastly, an adaptive analysis was performed to check the extent of algorithm accuracy by evaluating the probabilities of variations in the implemented pdf parameters.

1.2. Motivation

Although the wind speed probability curve always follows a Weibull pdf with different scale and shape factors [32], other similar pdf-s can fit in describing the wind speed patterns since it is not always possible to choose one unique distribution function for describing all wind speed regimes. In this work, five different distribution functions were investigated, namely the Rayleigh, Gamma, Lognormal, Exponential, and Weibull. These probabilities are related since Rayleigh random variable is a special case of the Rician random variable that deals with Gaussian coordinate conversion from cartesian to polar. Similarly, an Exponential variable is a special case of the Gamma random variable, whereas Lognormal distribution is a special case of Gaussian or normal distribution. Their patterns also depend on the data sampling speed of the wind regime. It must be stressed here that although wind speed is random in nature, it is still physically bound by upper and lower limits. Moreover, it is not always appropriate to consider the logged data as time series values due to the abrupt and chaotic nature of wind speed variations at random times.

As the aim of this work was to aid in the operation and proper planning of power distribution control, it was intended to forecast the monthly extracted energy accurately, based on the analysis of machine learning [33,34], and the extracted temporal errors from the foregoing logged measurements, as it is useful to analyze the attained error patterns.

1.3. Main Contribution

In this context, an optimal machine learning model was initially performed and trained using the best-fitted curve to the wind speed distribution, as several similar pdf-s are attempted for the ML model. The Lognormal distribution of the annual wind speed at the logged Fujairah (UAE) site is attained for being best trained in the back propagation (BP) feedforward neural network. The trained model was employed to be a standard tool for validation and testing on each successive month. The predicted wind speed following each month was compared with the actual measurement; hence monthly errors were consequently extracted and analyzed. The pattern of the error extraction was used for updating and tuning the forecast. In this study, the error pattern was analyzed based on the average, moving average, trend, as well as fitted parabolic function. Other methods of analyzing the error can be performed, such as using a Gaussian distribution if the nominal and variance values were interpolated accurately. Since only 12 months of model testing was performed, the Gaussian distribution of the error pattern could not be confirmed.

Changes in the wind speed and, consequently, energy were considered due to adaptive variations in the Lognormal pdf parameters. A joint Gaussian distribution was derived in order to study the various effects of the two parameters of the selected Lognormal pdf. It was intended to estimate the overall errors as well as check the accuracy of the implemented algorithm in this study with other tools, such as Markov series analysis.

1.4. Algorithm Justification

Across a great many references in the literature survey, the ARMA method was one of the most utilized algorithms that have been implemented in several case studies. However, it bears several disadvantages, such as requiring a lot of stationary and historical data. The accuracy is reduced for smaller wind speed excursions, which is the case where the Fujairah site is located. Longer forecasting horizons are also limited. An SVM-based model, another neural network, short-term forecasting algorithm, is the most suitable for high dimensional feature space, whereas in this case, there are only two features. Relying on a neural network as an input-output black box result in the lack of further system information. In this work, these disadvantages were mitigated by analyzing the extracted error raised from applying an ML model that develops with time. Hence, accurate forecasting can be achieved even with a non-ideal NN model. Time sampling of the 12-month raw data may be varied for better results, although the focus of this study was on the methodology; hence monthly average data were used.

2. Procedural Methodology

Due to the sudden and rapid variations on site, it is not always possible to rely on previous time series for future analysis since the minimum period of sampling to achieve this condition cannot be resolved ideally. On the other hand, data sampling at very small-time intervals is inadequate for very large data, yet the variation between maximum and minimum levels is the same, although being random in nature. Hence, wind speed data are random from one month to the next, with little correlation. Figure 2 displays a summary of the procedural steps conducted in this study.

Five different methods of curve fitting were performed on the wind speed data, namely, Weibull, Rayleigh, Gamma, Lognormal and Exponential pdf-s, since they have similar wind speed probability distribution functions. The monthly parameters of the five fitted curves were used for the machine learning and machine training procedure as inputs, whereas the following months’ data are the targets. The procedure was repeated for each month and checked for minimum error. The best-trained curve fitting function was selected as the system model. The next stage was to use the model for each month to test the model, and consequently, errors were extracted. The error probability function was predicted to rectify it for the forecast of future values.

3. Wind Speed Curve Fitting

Wind speed data were measured over a period of 18 months at the Fujairah site, and the attributes are shown in

Table 1. Physical parameters of Fujairah site [35].

Parameter	Value
Latitude (deg N)	25007′ N
Longitude	56018′ E
Mean wind speed at 10 m	4.072664 mph
Mean wind direction	182.46510
Average temperature	28 °C
Mean pressure	900–1100 m bar
Relative humidity	50–100%
Air density	1.188 kg/m3
Terrain	flat land
Obstacles	Hills
Surface roughness class	0.5 Sa

Note: The site is surrounded by low–high hills that might impose wind speed obstructions and restrictions. The wind regime is generally calm at site, with occasional abrupt and chaotic increases in wind speed due to seasonal temperature changes.

. The site is a coastal region located NE of the UAE, surrounded mainly by volcanic hills. Due to the nature of this site, the wind regime is random and chaotic, with occasional sporadic wind spells that vary in speed and direction. The terrain is generally an arid flat area surrounded by hills. Weather data can vary from year to year, according to previous observations. Logger measurements are registered every 10 s and averaged over one minute at a location 12 m above ground level (AGL). Wind speed and direction, as well as various meteorological data such as temperature, humidity, pressure, light, and UV index, were registered and programmed on a LabVIEW^® platform.

Statistical variations in the raw measurements of major meteorological data are depicted in Figure 3 over a period of 18 months starting from 22 February 2020 as Day 1. The wind speed is generally calm except during spring and fall, whereas wind direction varies continuously from NW to SE. It can be noted that whereas the temperature does not change much, humidity varies throughout the seasons.

An annual 30-bin histogram of the logged wind speed data is shown in Figure 4, which was used to fit the curves of five different probability distributions that are similar in appearance. Accordingly, the parameters and factors of these pdf-s are evaluated to be used for training the implemented NN model later. More bins can be used, yet the focus in this study is on methodology and not accuracy.

One deficiency of the attempted work of this research is the limited 18-month period of measurements, but this has been readily compensated for with the use of the extracted error pattern algorithm used in this study. It should be noted that under current climate change conditions and environmental changes, the stretch of the machine training period cannot be located, and there is no guarantee that forecasting could be met either, yet this handicap element can be managed with the proposed error extraction algorithm. Furthermore, it should be noted that the input abstractions employed in the algorithm are parameters of five different distribution functions that may vary with different periods of machine training. This accounts for any misinterpreted forecasting due to different lengths of logging periods. Another deficiency is single logging site measurements, yet this is limited due to the generally calm and random nature of the wind regime at the site. It should be made clear that Fujairah is a limited area of the Emirates and has a geography that is different from the rest of the land; hence this work constitutes an experimental pilot study that is focused on the proposed methodology. Further studies are needed for other sites of interest.

Figure 5 depicts the five different distributions used to fit the logged wind speed data. It should be noted that all the considered pdf-s have their patterns nearly fitting the logged wind speed data except for the Exponential distribution. At the mean x-y values of the graph, the Lognormal distribution has a maximum probability, occurring at speeds lower than the mean value, whereas the Rayleigh distribution has the minimum probability; this may reflect the generally calm nature of the wind at the site except for some abrupt variations. Curve fitting of the wind speed depends on the method of measured data sampling, such as daily, weekly, or monthly sampling of the considered logged data since different distributions can be obtained. In this study, we considered monthly analysis since it reflects a better curve-fitting analysis with the wind speed being of a random nature instead of a time series nature.

Since the distributions of these five pdf-s are similar, they were all considered in the machine learning analysis to select one that can be used to train the neural network with minimum error. The relationships that relate to the above different pdf-s were considered in this study, and their parameters or constant factors are as follows [36]:

1.

Rayleigh PDF, where σ is the scale parameter.

$$f_X\left(x\right)=f\left(x;\sigma \right)=\frac{x}{{\sigma }^2}\exp \left(-\frac{x^2{}^{}}{2{\sigma }^2}\right)$$

(1)

2.

Weibull PDF, where a is the inverse of the scale parameter, and b is the shape parameter.

$$f_X\left(x\right)=f\left(x;a,b\right)=ab{x}^{b-1}\exp \left(-a{x}^b\right)$$

(2)

3.

Lognormal PDF, where μ and σ are related to the mean (or average) and standard deviation parameters.

$$f_X\left(x\right)=f\left(x;\mu ,\sigma \right)=\frac{1}{x\sqrt{2\pi {\sigma }^2}}\exp (-\frac{{(\ln \left(x\right)-\mu )}^2}{2{\sigma }^2})$$

(3)

4.

Gamma PDF, where a is the shape parameter and b is the inverse of the scale parameter.

$$f_X\left(x\right)=f\left(x;a,b\right)=\frac{b^a{x}^{a-1}{e}^{-bx}}{\Gamma \left(a\right)}$$

(4)

5.

Exponential PDF, where b is the rate parameter.

$$f_X\left(x\right)=f\left(x;b\right)=b{e}^{-bx}$$

(5)

4. ML Forecasting Model

Machine training was performed using a feedforward, multilayer, back propagation neural network to select the parameters of one out of the five curve fitting pdf-s that can best train the NN with minimum error, as demonstrated in Figure 6. This is performed to explore a selected pdf that can be selected as the output target for the NN when a 5-layer perceptron with different neurons is used as the input abstracts. It was found that this network arrangement can improve the performance of forecasting analysis.

It was intended to use all parameters of the five similar pdf-s as possible abstractions, with the parameters of each individual pdf to be targets at a time. As seen from Section 3 above, there are 8 different curve pdf parameters that may be used as inputs for the forecasting model, two of which are for the Rayleigh, Gamma, and Lognormal curves, whereas Weibull and Exponential curves have one parameter each. In the training process of this neural network, pdf parameters for each curve in a sequence were assigned as targets, while the other parameters were assigned as input abstracts. This is to avoid network training saturation. Similar neural network configurations are kept for all five simultaneous cases of the studied models.

Table 2. Performances of NN case studies.

Case	Input Abstracts	Output Target	All R
1	b of Exponential pdf a and b of Weibull pdf a and b of Gamma pdf μ and σ of Lognormal pdf	σ of Rayleigh pdf	0.659
2	σ of Rayleigh pdf a and b of Weibull pdf a and b of Gamma pdf μ and σ of Lognormal pdf	b of Exponential pdf	0.785
3	σ of Rayleigh pdf b of Exponential pdf a and b of Gamma pdf μ and σ of Lognormal pdf	a and b of Weibull pdf	0.652
4	σ of Rayleigh pdf b of Exponential pdf a and b of Weibull pdf μ and σ of Lognormal pdf	a and b of Gamma pdf	0.358
5	σ of Rayleigh pdf b of Exponential pdf a and b of Weibull pdf a and b of Gamma pdf	μ and σ of Lognormal pdf	0.981

Note: All R is regression index for training + validating + training processes. Only 12 months of monthly data were used.

shows NN training performance based on the overall regression index for the five studied cases.

The wind speed raw data, which were logged over a period of 18 months, were fitted onto five different probability distribution functions pdf-s, which can normally describe the wind variations over the entire period. The parameters of these distribution functions are used for training a neural network. During the first stage of training, the neural network is trained in rounds where each pdf’s parameters are used individually as targets of the neural network, and parameters of all pdf-s are used as input abstractions. The parameters of the pdf that can train the neural network with minimum error compared to others were selected as a benchwork for tuning the network in the second stage of the implemented algorithm. In this context, the error is between the forecasted and logged values of the parameters of the pdf-s. During this second stage, the trained network is checked in each period, say monthly, to predict output pdf parameters, with an extracted error when compared with the logged pdf’s parameters during curve fitting. The pattern of the extracted errors was used to tune the forecasting for the subsequent period. Further information can also be drawn from this error pattern. It is expected that this error is alternating around the zero level and will be smooth when the network is fully trained.

Figure 7 demonstrates a flowchart of the major analysis that was carried out on the proposed model of this study. The first step was to fit the logged wind speed data onto five major probability distribution functions (PDF-s) that normally describe the wind. This considers any distributions that may vary with time, location, and periods of measurement, due to climate change and meteorological variations that are happening all the time. Parameters of the expressed wind speed distributions are used in training a neural network. Different abstractions are used to select the best distribution for the machine training. In each round, different output parameters of any distribution among the five distributions are used. At a later stage, the entire network is trained for the entire logging period with the best-selected arrangement of distribution functions. In this work, a method of error extraction was used to train for accurate forecasting. The trained network is used for each logging period, monthly in this case, to compare forecasted wind speeds with the measured ones, and the error is extracted. The registered extracted errors were used to update the forecasting and to check unprecedented values of the selected pdf parameters that might arise. The forecasted parameters of the Lognormal pdf were used to evaluate wind speed, power density, and energy, as seen in Section 5 of Results and Analysis. This methodology can be repeated for different sampling scenarios, yet this was not implemented, considering that it is out of this study’s scope and that further research is needed.

It can be seen above that network performance is best for Lognormal pdf, with Figure 8 depicting its regression index being best trained compared with other cases. It can be emphasized that although the best-trained pdf was used as an output for the model, all five curves were employed in the ML training. This algorithm model was then used to test for individual monthly target parameters in sequence, and consequently, the attained errors were extracted.

Theoretically, the regression coefficient B in the trained multivariate model was evaluated from the matrix equation [28], $Y=XB+\epsilon$,$\left(X=x_1, x_2,\dots ,x_m\right).$ We would want to best fit the predicted value by minimizing $\epsilon$, which is related to the sum of squared residuals (SSR) as $SSR={\epsilon }^2=|Y-XB{|}^2={(Y-XB)}^T\left(Y-XB\right)$, that can be reduced to,

$$SSR=Y^{T}Y-B^T{X}^{T}Y-Y^{T}XB+B^T{X}^{T}XB$$

(6)

by taking the derivative of SSR with respect to each B_i and equate to zero, i.e.,

$$\frac{\partial SSR}{\partial B_i}=-X^{T}Y+\left(X^{T}X\right)B=0$$

(7)

hence,

$$B={(X^{T}X)}^{-1}{X}^{T}Y$$

(8)

Furthermore, it can be seen from the following model:

$$y=F\left(x\right)+\epsilon$$

(9)

that the expected prediction error when estimating F(x) is $E\left[\left(y-F\left(x\right){)}^2\right]=E\right[y^2-2F\left(x\right)y+F^2\left(x\right)]$, reduced to

$$error=E\left[y^2\right]-2E\left[y\right]E\left[F\left(x\right)\right]+E\left[F^2\left(x\right)\right]$$

(10)

and for a random variable X with probability $P_X \left(x\right)$ and expectation $E\left[X\right],$ the expectation of squared error (SE) is $E\left[{\left(X-E\left[X\right]\right)}^2\right]=E\left[X^2\right]-2E\left[X\right]E\left[X\right]+E^2\left[X\right]$, i.e.,

$$\mathrm{SE}=E[X^2]-2E^2\left[X\right]+E^2\left[X\right]=E[X^2]-E^2\left[X\right]$$

(11)

hence,

$$E\left[X^2\right]=E[\left(X-E\left[X\right]{)}^2\right]+E^2\left[X\right]$$

(12)

and substituting X by $y-F\left(x\right)$, Equation (12) leads to the total error (TE) equal to $E[\left(y-F\left(x\right){)}^2\right]$, i.e.,

$$\begin{gathered} TE=E\left[\left(y-E\left[y\right]{)}^2\right]+E^2\left[y\right]-2E\left[y\right]E\left[F\left(x\right)\right]+E^2\left[y\right]-2E\left[y\right]E\left[F\left(x\right)\right] E\right[(F\left(x\right)-E\left[F\left(x\right)]{)}^2\right] E[(F\left(x\right) \\ -E\left[F\left(x\right)]{)}^2\right]+E^2\left[F\left(x\right)\right] \end{gathered}$$

(13)

That is, the expectation of squared error is a composite of three types of errors, namely variance error of $E[(F\left(x\right)-E\left[F\left(x\right)]{)}^2\right]$, a bias error of ${(E\left[y\right]-E\left[F\left(x\right)\right])}^2$, and a noise error of $E[\left(y-E\left[y\right]{)}^2\right]$. Variance error reflects the sensitivity to small variations in the train set, whereas a bias error depicts the error between the true and expected estimation. These errors must be removed to avoid generalization. High variance results in overfitting, whereas high bias results in underfitting, and hence the tradeoff is to balance them for the specified model. It must be noted here that the curve fitting process also contributes to the error, but this error is reduced since five similar distributions are combined in the NN model training.

5. Analysis and Results

Achieved results of this work can be summarized as follows:

(a)

New concept of neural network modeling:

The performed NN modeling is based on a temporal error extraction algorithm. The error pattern is used in adjusting and tuning the forecast.

(b)

Probability distribution functions’ parameters are used as input NN abstracts:

The input abstracts of the used neural network are not raw wind speed data but parameters of wind speed pdf, in which five combined pdf-s are used. The NN targets are factors of a Lognormal pdf that shows the best-trained output for the NN performance, hence selected in a forecast analysis.

(c)

Annual wind energy forecast is performed:

Monthly based data of the daily average logged wind measurements are used for energy forecasting analysis, which implies better management and planning of the utility power. Daily data sampling does not necessarily secure accurate forecasting due to the random nature of wind. The wind regime is normally different at night than at daytime; hence wind speed pdf varies during the 24 h in a day, leading to inaccurate analysis.

(d)

Adaptive analysis of extracted energy:

As the energy analysis is heavily reliant on parameters of the wind speed pdf, accurate prediction of the forecast is performed with variations in parameters based on the Normal distribution. A joint Gaussian distribution function of variations in the pdf parameters is derived and applied to the performed energy analysis.

(e)

Accuracy of the adopted method:

The accuracy of the wind speed algorithm is analyzed using a MAP error estimation method. It is found to be in line and comparable with the performed Markov series analysis. The details of the analyses and achieved results of this study are discussed in the following sections.

5.1. Lognormal Pdf Prediction

Figure 9 shows the measured and predicted values of parameter u of the Lognormal pdf, with its trend value of 1.1, and Figure 10 shows the same analysis for the second Lognormal pdf parameter σ with a value of 0.58. It can be seen that the predicted parameters of the Lognormal pdf are comparable with actual measured values.

Table 3. Measured and predicted values of the Lognormal PDF parameters.

Month	Measured		Predicted
	μ	σ	μ	σ
1	1.4508	0.5174
2	1.484	0.4347
3	1.5061	0.3442	1.52	0.35
4	1.3654	0.243	1.52	0.24
5	1.377	0.2113	1.21	0.13
6	1.3457	0.184	1.22	0.15
7	1.2603	0.2582	1.25	0.17
8	1.1027	0.1092	1.19	0.28
9	1.1267	0.2478	1	0.17
10	1.3717	0.2874	0.99	0.26
11	1.4854	0.2938	1.21	0.35
12	1.1135	0.5519	1.43	0.4

displays measured and predicted values of the Lognormal pdf parameters μ and σ over the 12 months period.

From the depicted table, an initial trend of the two pdf parameters can be determined for a subsequent month by using Excel functions as $\mu =1.27$ and $\sigma =0.6$.

Table 4. Error prediction of the Lognormal pdf.

Month	Error of Parameter μ		Error of Parameter σ
	μ Error	% μ	σ Error	% σ
1
2
3	0.0139	0.92	0.0058	1.69
4	0.1546	11.32	0.003	1.23
5	0.167	12.13	0.0813	38.48
6	0.1257	9.34	0.034	18.48
7	0.0103	0.82	0.0882	34.16
8	0.0873	7.92	0.1708	156.41
9	0.1267	11.25	0.0778	31.4
10	0.3817	27.83	0.0274	9.53
11	0.2754	18.54	0.0562	19.13
12	0.3165	28.42	0.1519	27.52

displays the total accumulated error estimations of the two parameters of the Lognormal distribution over the 12 logged months as $\mu =1.6591$ ($12.849\%$) and $\sigma =0.6964$ ($33.803\%$). It should also be noted that although Lognormal PDF curve fitting is comparable with other distributions, it is found to be the best-performing distribution with the trained ML model.

It can also be seen from Table 4 that the Lognormal parameter μ varies in the range of 1.1–1.5 with a nominal value of 1.33, whereas σ varies in the range of 0.11–0.55, with a nominal value of 0.3. Figure 11 depicts variations in parameters μ and σ on the probability of wind speed distribution. Parameter μ has a more dominant effect on the probability distribution than parameter σ.

5.2. Error Extracting Algorithm

In order to tune the trained model used above in Section 4, the monthly errors of the Lognormal pdfs’ two parameters {μ and σ} were used as targets for the performed model using six inputs of the other pdf’s parameters, which are listed in Table 2 as: {σ of Rayleigh pdf, b of Exponential pdf, a and b of Weibull pdf, a and b of Gamma pdf}. The error was then compared with the measured values. This process is depicted in Figure 12, in which the trained network for the entire 12-month period of measurements was used as a checking tool for each consecutive month to extract the error by comparing the monthly target F_i(x) with measured values y_i, producing error e₁. The parameters of the selected Lognormal pdf were designated as inputs {x}. The pattern of errors {e₁, e₂,..,e_n} was used to update and tune the following months’ forecasted values. The error pattern constitutes an added information and forecasting check. It is expected that this error is alternating between positive and negative values as the extraction process proceeds from one month to the next, but the trend is useful for indicating the progress of the machine training. Contrary to the studied 12 months analysis, it is expected that for larger logging periods, the error of the alternations is smaller, and the pattern is smoother.

Table 5. Error estimation of the Lognormal PDF parameters.

Trained μ	Trained σ	Actual μ	Actual σ	Error μ	Error σ
1.4238	0.3469	1.484	0.4347	−0.0602	−0.0878
1.4052	0.3424	1.5061	0.3442	−0.1009	−0.0018
1.3502	0.2351	1.3654	0.243	−0.0152	−0.0079
1.3609	0.1886	1.377	0.2113	−0.0161	−0.0227
1.3504	0.1934	1.3457	0.184	0.0047	0.0094
1.2733	0.2469	1.2603	0.2582	0.013	−0.0113
1.124	0.1798	1.1027	0.1092	0.0213	0.0706
1.1266	0.1796	1.1267	0.2478	1.0E−4	−0.0682
1.4028	0.3408	1.3717	0.2874	0.0311	0.0534
1.3307	0.2602	1.4854	0.2938	−0.1547	−0.0336
1.1742	0.4185	1.1135	0.5519	0.0607	−0.1334

displays machine training errors that are extracted by comparing trained and actual measured parameters μ and σ of the Lognormal pdf for each successive month. It is noted that the trained parameters are matched with the measured values. This is depicted in Figure 13, which shows the monthly evaluated errors of the forecasting Lognormal pdf’s two parameters using an ML method. The error is alternating around zero value as machine training is in progress. It should be noted that without augmenting the average, moving average, trend, and parabolic trend functions, the error increases with progressing months. This might be due to the short period of logging measurement; nevertheless, the trend of error is important here.

Updating the values of Lognormal pdf parameters, performed from ML training, with extracted error augmentation reduces the forecast error and improves forecasting.

Table 6. Comparison of implemented methods.

Method	μ	σ
Measured	1.1135	0.5519
Predicted	1.43	0.4
Trained without error extraction	1.1742	0.4185
Trained with moving average of extracted error	1.1242	0.3385
Trained with trended extraction error	1.1242	0.2685
Trained with average extracted error	1.1545	0.4620

depicts a comparison of the forecasting methods of Lognormal pdf parameters μ and σ, based on the error extraction algorithm, when compared with the actual measured values. It can be seen that predicting with tools such as Excel gives a maximum error, whereas a more reliable forecast is accomplished when employing the method of NN trained with average error extraction of preceding months. Further error extraction patterns, such as the Gaussian distribution, can also be investigated. This was not conducted in this study. It is expected that with a greater time period than the 12 months considered, a more accurate interpolation of the extracted error can be obtained.

5.3. Forecasting Wind Energy

Contrary to the fact that wind speed distributions are normally expressed as a Weibull pdf, we used the Lognormal pdf since it offers the best performance when trained with the ML model, as shown in previous sections. It can be deduced from the previous analysis that the Lognormal pdf maps the measured data with minimum error. Parameter μ is varied in the range of 1.1–1.5 with a nominal value of 1.3, whereas parameter σ varies in the range of 0.15–0.55 with a nominal value of 0.35. These figures are rounded up due to unexpected noisy errors in measurement.

The mean monthly wind speed of the mapped Lognormal pdf can be determined from Equation (3) with the following expected relationship

$$V_{mean}={\int }_0^{\infty }v{f}_V\left(v\right)={\int }_0^{\infty }\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp (-\frac{{(\ln \left(v\right)-\mu )}^2}{2{\sigma }^2})$$

(14)

By integrating (14) over a wind speed range of 0 to 15 mph (6.7 m/s), $V_{mean}$ was evaluated according to

Table 7. Error estimation of the Lognormal PDF.

Method	${\mathit{V}}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}}$	Error w.r.t Measurement
Measurement	3.3526
Predicted	4.3573	+300%
Trained	3.4915	+4%
Trained with moving error	3.2565	−2.8%
Trained with trend error	3.1907	−4.8%
Trained with average error	3.4537	+3%

for the different methods implemented in this study. It can be noted that trained values are highly comparable, and they constitute much lower errors compared with the conventional trend value evaluated by Excel.

Hence, the expected wind speed obtained from averaging the four machine training algorithm methods is 3.3481 m/s. Since wind power is proportional to the cube of wind speed, the average power density that is available for extraction per unit swept area is

$$Power density={\int }_0^{\infty }(0.5\rho f\left(v\right) v^{3}dv)$$

(15)

Hence, the speed of interest in wind energy can be termed as $V_{rmc},$ the root-mean-cube speed, defined as

$$V_{rmc}=\sqrt[3]{{\int }_0^{\infty }\frac{v^2}{\sqrt{2\pi {\sigma }^2}}\exp \left(-\frac{{(\ln \left(v\right)-\mu )}^2}{2{\sigma }^2}\right)dv}$$

(16)

and the average annual power density available is $0.5\rho V_{rmc}{}^3$. The total accumulated energy that can be extracted per year is the integration of this available power over all possible velocities and is calculated by summing the power density $0.5\rho V_{rmc}{}^3$ for the total measured wind speeds over the year, where the air density ρ is taken to be 1.188 kg/m³ at the Fujairah site, i.e.,

$$Energy=\left(0.5\rho v^3\right)\left(8760\right)$$

(17)

Table 8. Power density and energy for different velocities.

Velocity (m/s)	Power Density Watt/m2	Energy kWh/m2/yr.
$V_{mean}=3.34$	22.293	195,292
$V_{rmc}=3.71$	31.636	277,800

depicts the power density and energy for both mean and $rmc$ velocities. It should be noted that the probability is similar for all speed definitions, around 16.3%.

It must be mentioned that the height of the wind speed logger was 12 m above ground level AGL), yet this speed can be adjusted according to $v_2=v_1{(\frac{H_2}{H_1})}^{\alpha }$, with α being the shear exponent, which depends on the site landscape nature. An approximate value of $\alpha =0.5$ is used for the Fujairah site, which is a coastal town with a landscape composed of rough and uneven terrain, hedgerows, and hills.

5.4. Algorithm Adaptive Analysis

The Lognormal pdf’s nominal values for the parameters μ and σ that were deduced from the previous analysis cannot be constant but may be expressed as each having a nominal value plus a noise error component that can be expressed as a Gaussian random variable, with mean m and variance c

$$f_X\left(x\right)=\frac{1}{\sqrt{2\pi c^2}}\exp (-\frac{{(x-m)}^2}{2c^2})$$

(18)

and the joint probability of this noise error is the multiplication of both probabilities since the two parameters are independent. Hence, the two pdf parameters are also varying as independent Gaussian random variables. Substituting mean values of $\mu =1.33$ and $\sigma =0.35$ with assumed variances of, say, 0.2, based on their variation ranges, yields

$$f_{U,\Sigma }\left(u,\sigma \right)=\frac{e^{-\left(\frac{{(\mu -1.33)}^2}{2{\left(0.2\right)}^2}+\frac{{(\sigma -0.35)}^2}{2{\left(0.2\right)}^2}\right)}}{2\pi {\left(0.2\right)}^2}$$

(19)

We evaluated the variations in wind speed probability v due to variation in μ, i.e.,

$$f_V\left(v\right)={\int }_0^{\infty }{f}_{V|U}\left(v|u\right) f_U\left(u\right)$$

(20)

which can be reduced to

$$f_V\left(v\right)=\frac{e^{-\left(\frac{{(\ln \left(v\right)-1.33)}^2}{2{\left(0.35\right)}^2}+\frac{{(\mu -1.33)}^2}{2{\left(0.2\right)}^2}\right)}}{2\pi v{\left(0.35\right)}^2}$$

(21)

Similarly, for probability of speed variation with respect to σ,

$$f_V\left(v\right)=\frac{e^{-\left(\frac{(\ln \left(v\right)-1.33){}^2}{2{\left(0.35\right)}^2}+\frac{{(\sigma -0.35)}^2}{2{\left(0.2\right)}^2}\right)}}{2\pi v{\left(0.35\right)}^2}$$

(22)

Figure 14 depicts wind speed probabilities for different values of μ and σ when they are varied by ±10% around their Gaussian pdf nominal values.

5.5. Estimating Algorithm Error

It can be noted from the previous sections that the main elements of the model are the parameters of the implemented function distribution, which implies that the accuracy of forecasting depends on eliminating errors within these parameters. Different types of errors exist, namely curve fitting errors, machine learning errors, and noise errors arising from variations in Lognormal pdf parameters themselves. Curve fitting errors arise from selecting Lognormal pdf among the five other similar distributions, whereas ML errors constitute a bias, a variance, and a noise error.

In order to predict overall forecasting error, the maximum likelihood estimation and Maximum-A-Posteriori (MAP) was used here to estimate these errors by choosing the most probable value for a certain random wind speed observation. In general, this estimation is a value that maximizes the conditional pdf of V that is given observation X. That is, for a given number of observations Xi among n trials; each modeled as a true value plus an error, which can be expressed as an independent Gaussian random variable with variance ${\sigma }_e^2$, i.e.,

$$f_{V|X}\left(y,x\right)=\frac{1}{{(2\pi c_e^2)}^{\frac{n}{2}}}\exp \left\{-\frac{1}{2c_e^2}{\sum }_{i=1}^n{(x_i-v)}^2\right\}$$

(23)

and using the Bayes theorem yields

$$f_{V|X}\left(v,x\right)=\frac{f_{X|V}\left(x|v\right)f_V\left(v\right)}{f_X\left(x\right)}$$

(24)

Hence, by evaluating $f_{X|V}\left(x|v\right)f_V\left(v\right)$ from above [37] and differentiating it with respect to v and setting the result equal to zero for maximum value, the value of the MAP estimator yields as:

$$MAP=\frac{\frac{1}{n}{\sum }_{i=1}^n{X}_i+\frac{{\mu }_v{\sigma }_e^2}{n{\sigma }_v^2}}{1+\frac{{\sigma }_e^2}{n{\sigma }_v^2}}$$

(25)

It should be noted that the average of all observations is $\frac{1}{n}\sum _{i=1}^n{X}_i$, that is, the MAP estimator is the average of observations, assumed 100%, skewed by prior knowledge of wind speed pdf. As an example, assuming n = 10 readings at different measurement occasions, ${\sigma }_v=4.4mph$, ${\mu }_v=2mph$, and ${\sigma }_e=10\%$ of average wind speed yields a MAP estimator value of 4.398, whereas this estimator is equal to 4.344, with ${\sigma }_e=50\%$.

In order to check the accuracy of the implemented algorithm in this work, a comparison analysis was conducted to check the errors of forecasted values of the μ and σ parameters of the Lognormal pdf, in which a method based on Markov series [38] was conducted. This is based on state transition probabilities and is, in general, defined as

$$P_{i,j}=\mathrm{Pr}(X\left[k+1\right]=j |X\left[k\right]=i)$$

(26)

which are the transitioning probabilities of chain X[k] with states {x1, x2, …} changing from state i to state j in one time instant. In every month, the deviation of the parameters μ and σ are evaluated, and their probabilities of transitioning from one state time instant to another are determined. As a result, the predicted values of these parameters were evaluated and compared with the logged measured values. Maximum absolute errors are implemented for the two methods.

Table 9. Monthly forecasting error of u parameter.

Month	μ	Pu	μ1	μ2	Eu1	Eu2
1	1.4508
2	1.484	0.4	1.4238	1.6173
3	1.5061	0.4	1.4052	1.6394	0.214	0.506
4	1.3654	0.3	1.3502	1.2654	0.644	0.287
5	1.377	0.4	1.3609	1.5103	0.077	0.386
6	1.3457	0.4	1.3504	1.2123	1.104	0.792
7	1.2603	0.2	1.2733	1.1936	2.207	2.627
8	1.1027	0.2	1.124	1.0360	3.338	3.836
9	1.1267	0.4	1.1266	1.2600	5.451	3.589
10	1.3717	0.05	1.4028	1.3883	1.052	0.697
11	1.4854	0.2	1.3307	1.4187	1.199	0.448
12	1.1135	0.05	1.1742	1.0968	2.294	0.986

depicts error evaluations of the μ parameter, using the two methods with P_u as the transitioning probability, with μ₁ as the forecasted value of μ with Markov series methods, and μ₂ as the forecasted value using the implemented method. Eu₁ and Eu₂ were the estimated errors of Markov and the implemented method, respectively. It can be seen from this table that the evaluated error in the implemented method of this work is comparable and in line with the Markov series analysis. It should be noted that the implementation of Markov analysis was approximated.

Figure 15 depicts the error of Lognormal pdf parameter μ by comparing the implemented method of this work with the Markov method since it depends on the probability of random variables, which in this case, the parameters of the probability distribution function, and the two methods offer almost equal results.

Similarly,

Table 10. Monthly forecasting error of σ parameter.

Month	σ	Pσ	σ1	σ2	Eσ1	Eσ2
1	0.517
2	0.434	−0.6	0.346	0.3013	−0.1333	−0.0878
3	0.344	−0.2	0.342	0.2997	−0.0444	0.0027
4	0.243	−0.2	0.235	0.1985	−0.0444	0.1039
5	0.211	−0.6	0.188	0.0779	−0.1333	0.1356
6	0.184	−0.6	0.193	0.0506	−0.1333	0.1629
7	0.258	0.6	0.246	0.3915	0.13333	0.0887
8	0.109	−0.2	0.179	0.0647	−0.0444	0.2377
9	0.247	0.2	0.179	0.2922	0.0444	0.0991
10	0.287	0.6	0.340	0.4207	0.1333	0.0595
11	0.293	0.6	0.260	0.4271	0.1333	0.0531
12	0.551	0.1	0.418	0.5741	0.0222	−0.205

shows the monthly forecasted Lognormal pdf parameter σ and its error compared with Markov series analysis.

Figure 16 displays a comparison of error estimation of the Lognormal pdf parameter σ, where it can be noted that although discrepancies exist, their patterns are comparable, and the implemented method in this study can be competitive with the Markov series method. Whereas the Markov method relies on extensive probability analysis, this method depends on statistics. Other comparison methods from the literature can be useful to check the extent of the accuracy.

Based on the aforementioned analysis, it is imperative to validate this work with different wind patterns to check the extent of the accuracy for short-term forecasting. The shortcoming of the proposed algorithm’s accuracy is mainly due to the lack of extensive logging data for calm, chaotic, and randomly fluctuated wind situations. As was the situation at the experimented site, this algorithm can be useful to validate forecasting and to track network training with the temporal logging periods. Since this work depends on the probability distribution functions of the wind, the study of meteorological characteristics is useful [37]. Several other types of pdf-s can be incorporated into the method to reduce discrepancies in weather fluctuations and climate changes.

6. Conclusions

Wind energy forecasting was performed using a proposed method to predict wind speed effectively with the best-trained curve fitting pdf using ML and temporal error extraction algorithm. A forecast-error extract procedure was performed on the logged data over a one-year period. The pattern of the monthly extracted error was analyzed to update the forecasting with an average, moving average, trend, and predicted function. The sensitivity of the method was analyzed with variations in the parameters of the Lognormal pdf that was selected, with the best being the NN-trained distribution. A joint Gaussian probability function was implemented to describe variations in the two pdf parameters and to check their effects on the wind speed distribution. In order to enhance energy calculation incentives, a MAP error estimating analysis was conducted to check the method’s accuracy. It was found that such an error is in line with the Markov series analysis that was conducted. It is crucial to apply this algorithm to data with a different scale, such as weeks, for a better interpretation.