A Probabilistic Wind Speed Prediction Method and System Based on Multi-Scale Information

Abstract

An embodiment of the present invention provides a probability wind speed prediction method and system based on multi-scale information, which uses a multi-layer convolutional neural network to extract sufficient multi-scale features from input samples, and uses a long short-term memory (LSTM) and attention mechanism to further extract temporal features and encode them into low-dimensional feature vectors; then, by means of the proposed non-crossing loss, the conditional quantile difference between adjacent quantiles is obtained, and the conditional quantiles of all given quantiles are obtained through cumulative addition and subtraction; it can not only extract sufficient multi-scale features from limited data, but also provide high-quality and reliable probability prediction results, while solving the crossing problem of the quantile model.

Description

Technical Field

[0001] The embodiments of the present invention relate to the technical field of renewable energy development and utilization, and particularly to a probability wind speed prediction method and system based on multi-scale information. Background Technique

[0002] With the rapid development of the world economy, the demand for energy by humans is increasing continuously, and non-renewable fossil energy is facing depletion. At the same time, the combustion of fossil energy is also accompanied by the emission of a large amount of toxic substances and greenhouse gases, seriously polluting the ecological environment. Therefore, it is urgent to find new energy sources. As a clean and renewable energy source, wind energy has received extensive attention and developed rapidly in recent years. However, the intermittency and randomness of wind bring difficulties to the large-scale grid connection of wind power. Accurate and effective wind power prediction is an effective means to address this challenge. In addition, wind power prediction is also helpful for formulating dispatch plans, calculating spinning reserve capacity, reducing system operation costs, performing peak shaving and frequency modulation work, maintaining system power balance, and arranging maintenance work.

[0003] Current wind power prediction methods can be divided into two categories: direct prediction and indirect prediction. Direct prediction takes the power prediction value as the output of the model; indirect prediction is to first predict the wind speed and then convert it into a power prediction value through the wind power curve. Currently, various wind speed prediction methods have been proposed one after another, and these models can be roughly divided into physical models, statistical models, artificial intelligence models, and hybrid models.

[0004] Physical models predict future wind speeds based on meteorological factors such as air temperature, air pressure, and humidity. Numerical weather prediction is a common physical model, which predicts wind speeds by solving equations reflecting atmospheric processes and the change of the atmosphere over time. Physical models require a large amount of computational effort and time, and are generally only used for long-term wind speed prediction. Statistical models usually take historical wind speeds as inputs, and typical statistical models include autoregressive models, autoregressive moving average models, and autoregressive integrated moving average models. This type of model can be used for short-term wind speed prediction, but has limited ability to handle non-linear data. Artificial intelligence models have better non-linear fitting ability and better prediction performance than statistical models, and are widely used in the field of wind speed prediction. Common artificial intelligence models include backpropagation neural networks, Elman neural networks, extreme learning machines, echo state networks, generalized regression neural networks, radial basis function neural networks, support vector machines, deep learning models, etc. Hybrid models combine different types of prediction models to integrate their advantages to improve prediction accuracy.

[0005] The above methods are all deterministic predictions, and their purpose is to provide predicted values close to the observed values. However, the uncertainty of wind speed will cause a deviation between the predicted value and the true value. Without providing relevant information on the deviation, scheduling and decision-making based on the deterministic predicted value with a large error will bring risks to the operation of the power system. Therefore, wind speed probability prediction has become a research hotspot in recent years. This method can generate fluctuation intervals or probability density curves of future wind speeds at different confidence levels to reflect the uncertainty level and provide comprehensive references for the scheduling and operation of the power system.

[0006] Generally, probability wind speed prediction models are divided into two categories: parametric models and non-parametric models. Typical parametric models include the delta method, MVE, and Gaussian Process Regression (GPR). However, since it is difficult to ensure that wind speed strictly follows a specific distribution, the parametric method lacks sufficient reliability. Non-parametric methods have no distribution assumptions and mainly include lower-upper bound estimation and quantile regression. Among them, quantile regression can generate conditional quantiles at multiple quantiles at one time to provide prediction intervals at multiple different confidence levels. To make up for the limitations of quantile regression in dealing with non-linear data, hybrid models based on artificial intelligence models and quantile loss have been successively proposed in recent years. Although this method can greatly improve the prediction performance of the model, the estimated values of conditional quantiles often cross, violating the property that the conditional quantiles at different quantiles increase monotonically. Compared with discrete prediction intervals, the Probability Density Function (PDF) can provide more comprehensive, intuitive, and detailed information. Kernel density estimation can convert discrete conditional quantiles into continuous PDFs without making any assumptions about the shape of the distribution.

[0007] There are three deficiencies in the current prediction models. First, the insufficiency of features will reduce the prediction performance of the model. Convolutional neural networks are widely used as feature extractors. In multi-layer convolutional neural networks, the features extracted by different layers have different levels. However, most previous studies only use the features of the last convolutional layer and ignore the multi-scale features (MSFs) of the remaining convolutional layers. Since the MSFs are not fully utilized, it may lead to information loss and thus limit the prediction performance of the model. Second, previous studies have always focused on deterministic prediction, while the uncertainty of wind speed makes the reliability of the prediction results unable to be guaranteed. In contrast, probability prediction can quantify the relevant uncertainty and provide more reliable and comprehensive information. In existing methods, using a quantile-based model is an effective strategy that can provide prediction intervals at multiple confidence levels. Third, the quantile model often suffers from the quantile crossing problem, which limits the reliability of the probability prediction results. However, this problem is always ignored in many studies. Summary of the Invention

[0008] An embodiment of the present invention provides a probability wind speed prediction method and system based on multi-scale information, which can not only extract sufficient multi-scale features from limited data, but also provide high-quality and reliable probability prediction results, and at the same time solve the cross problem of the quantile model.

[0009] In a first aspect, an embodiment of the present invention provides a probability wind speed prediction method based on multi-scale information, including:

[0010] Step S1, obtain historical wind speed time series data, and divide the wind speed time series data into a training set and a test set;

[0011] Step S2, construct a neural network model, the neural network model includes a multi-scale feature extraction module and an LSTM network module; the multi-scale feature extraction module is used to extract one-dimensional feature vectors of the input vector, and parallel splice to obtain multiple groups of feature subsequences at different levels; the LSTM network module is used to extract the time series features of the feature subsequences, and perform weighted combination based on the local features of the time series features to determine the low-dimensional feature vectors of the feature subsequences, and connect the low-dimensional feature vectors of all the feature subsequences into a prediction feature vector for prediction;

[0012] Step S3, construct a quantile loss function for training the neural network model, determine the initial difference of the conditional quantiles of any two adjacent quantiles, and convert the initial difference of the conditional quantiles into a set of conditional quantile differences greater than 0 through approximate absolute difference conversion; determine the conditional quantile prediction values of the remaining quantiles based on the conditional quantiles and the conditional quantile differences of two adjacent quantiles;

[0013] Step S4, perform neural network model training and verification based on the training set and the test set, determine the wind speed probability prediction value based on the trained neural network model, convert the wind speed probability prediction value into the conditional quantile prediction values of all quantiles, and combine the conditional quantile prediction values of two different quantiles to obtain a prediction interval with a corresponding confidence level; use all the conditional quantiles as inputs, and convert the discrete conditional quantile prediction values into a continuous probability density curve through KDE.

[0014] Preferably, the step S1 specifically includes:

[0015] Collect historical wind speed time series data, and perform normalization processing on the historical wind speed time series data:

[0016]

[0017] In the above formula, x is the wind speed time series in the historical wind speed time series data, x max and x minare the maximum and minimum values in the historical wind speed time series data, respectively, x normal is the normalized data;

[0018] Divide the historical wind speed time series data into a training set and a test set according to a preset ratio. The training set is used for the training of the neural network, and the test set is used for testing the prediction performance of the neural network model.

[0019] Preferably, in step S2, the multi-scale feature extraction module includes three stacked one-dimensional convolutional layers. Each convolutional layer includes multiple convolutional kernels. The formula for the convolutional kernel operation is:

[0020]

[0021] In the above formula, where is the output of the h-th neuron in the (k - 1)-th convolutional layer, M j is the input feature vector set of the j-th neuron, and are the bias and output of the j-th neuron in the k-th convolutional layer, respectively, is the weight matrix from the h-th neuron in the (k - 1)-th layer to the j-th neuron in the k-th layer, f(·) is the activation function, and * represents the convolution operation;

[0022] Each of the convolutional kernels is used to parallelly splice the one-dimensional feature vectors extracted from the input samples into a multi-dimensional subsequence, so as to obtain three groups of feature subsequences {c i} i＝1,2,3 .

[0023] Preferably, in step S2, the LSTM network module is specifically used for:

[0024] For the feature subsequence c i , extract the time series features based on the LSTM network module which are the hidden states of all units corresponding to the i-th LSTM; L i is the number of units of the i-th LSTM network module; through a fully connected operation, h i is transformed into h i ’, and h i′ and are added to obtain h i″ :

[0025]

[0026] Among them, h i″ is the newly obtained feature vector, h i′ is the feature vector obtained through the fully connected operation, is the hidden state of the last cell of the i-th LSTM network module; based on the attention mechanism, weights are assigned to the feature vectors in h i″ :

[0027]

[0028] where and are the l-th and k-th feature vectors of h i″ respectively, is the attention probability of the feature vector ; according to the obtained attention probability, the feature vectors in h i″ are weighted to obtain a low-dimensional feature vector:

[0029]

[0030] where v i is the low-dimensional feature vector corresponding to the feature subsequence c i ; the feature vectors corresponding to all feature subsequences are concatenated into a prediction feature vector V for prediction.

[0031] Preferably, the quantile loss function is:

[0032] Q Y (τ|X) = f1(X, β(τ))

[0033] In the above formula, Q Y (τ│X) is the estimated value of the input vector X at the quantile τ, X = (x1, x2,..., x N ); the target variable Y = (y1, y2,..., y N ); f1(·) is a non-linear function, β(τ) is the regression coefficient of the model, and its estimated value can be obtained by minimizing the following loss function value:

[0034]

[0035] where is the estimated value of β(τ), N is the number of input samples, y i is the value of the target variable corresponding to the input vector x i ; is the conditional quantile of the quantile τ, is the quantile loss:

[0036]

[0037] Preferably, the output of the conditional quantile includes the conditional quantile of the median quantile Initial difference in conditional quantiles between any two adjacent quantiles

[0038] τ = {τ k} k＝1,2,…,K , 0 < τ1 < τ2 < … < τ k0 <… < τ K <1, τ k0 = 0.5; where:

[0039]

[0040]

[0041]

[0042] Convert into a set of positive differences in conditional quantiles:

[0043]

[0044] where θ is a positive number; is the positive difference in conditional quantiles between two adjacent quantiles;

[0045] Based on the obtained conditional quantiles and the difference in conditional quantiles the estimated values of conditional quantiles at the remaining quantiles can be obtained by cumulative addition and subtraction:

[0046]

[0047] where, is the estimated value of the conditional quantile at quantile τ k ;

[0048] Calculate the total quantile loss value for training the neural network model:

[0049]

[0050] In the above formula, L represents the loss value.

[0051] Preferably, the probability density curve is:

[0052]

[0053]

[0054] In the above formula, is the obtained probability density equation, B is the bandwidth, and K E (α) is the Epanechnikov kernel equation.

[0055] In a second aspect, an embodiment of the present invention provides a probability wind speed prediction system based on multi-scale information, including:

[0056] A data acquisition module, which acquires historical wind speed time series data and divides the wind speed time series data into a training set and a test set;

[0057] A model construction module, which constructs a neural network model. The neural network model includes a multi-scale feature extraction module and an LSTM network module. The multi-scale feature extraction module is used to extract one-dimensional feature vectors of the input vector and obtain multiple groups of feature subsequences at different levels by parallel splicing. The LSTM network module is used to extract the time series features of the feature subsequences, perform weighted combination based on the local features of the time series features to determine the low-dimensional feature vectors of the feature subsequences, and connect the low-dimensional feature vectors of all the feature subsequences into a prediction feature vector for prediction;

[0058] A loss function construction module, which constructs a quantile loss function for training the neural network model, determines the initial difference of the conditional quantiles of any two adjacent quantiles, and converts the initial difference of the conditional quantiles into a group of conditional quantile differences greater than 0 through an approximate absolute difference transformation. Based on the conditional quantiles and the conditional quantile differences of two adjacent quantiles, the conditional quantile prediction values of the remaining quantiles are determined;

[0059] A prediction module, which trains and validates the neural network model based on the training set and the test set, determines the wind speed probability prediction value based on the trained neural network model, converts the wind speed probability prediction value into the conditional quantile prediction values of all quantiles, combines the conditional quantile prediction values of two different quantiles, and can obtain a prediction interval with a corresponding confidence level. Taking all the conditional quantiles as inputs, the discrete conditional quantile prediction values are converted into a continuous probability density curve through KDE.

[0060] In a third aspect, an embodiment of the present invention provides an electronic device, including a memory, a processor, and a computer program stored on the memory and executable on the processor. When the processor executes the program, the steps of the probability wind speed prediction method based on multi-scale information as described in the first aspect embodiment of the present invention are implemented.

[0061] In a fourth aspect, an embodiment of the present invention provides a non-transitory computer-readable storage medium, on which a computer program is stored. When the computer program is executed by a processor, the steps of the probability wind speed prediction method based on multi-scale information as described in the first aspect embodiment of the present invention are implemented.

[0062] A probability wind speed prediction method and system based on multi-scale information provided by an embodiment of the present invention uses a multi-layer convolutional neural network to extract sufficient multi-scale features from input samples, and further extracts temporal features using a long short-term memory (LSTM) and an attention mechanism and encodes them into low-dimensional feature vectors; then, by means of the proposed non-crossing loss, the conditional quantile difference between adjacent quantiles is obtained, and the conditional quantiles of all given quantiles are obtained through cumulative addition and subtraction; it can not only extract sufficient multi-scale features from limited data, but also provide high-quality and reliable probability prediction results, and at the same time solve the crossing problem of the quantile model. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings.

[0064] Figure 1 FIG. is a flowchart of a probability wind speed prediction method based on multi-scale information according to an embodiment of the present invention;

[0065] Figure 2 FIG. is a schematic diagram of the principle of approximate absolute value conversion according to an embodiment of the present invention;

[0066] Figure 3 FIG. is a schematic diagram of the interval prediction results of the model MSF-DNQR on Dataset A and Dataset B according to an embodiment of the present invention;

[0067] Figure 4 FIG. is a schematic diagram of the entity structure according to an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0068] In order to make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the drawings in the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts fall within the scope of protection of the present invention.

[0069] In the embodiments of the present application, the term "and / or" merely describes the associated relationship of associated objects, indicating that there can be three relationships. For example, A and / or B can represent three situations: A exists alone, A and B exist simultaneously, and B exists alone.

[0070] The terms "first" and "second" in the embodiments of the present application are only for descriptive purposes, and cannot be construed as indicating or implying relative importance or implicitly specifying the quantity of the indicated technical features. Thus, features defined with "first" and "second" may explicitly or implicitly include at least one such feature. In the description of the present application, the terms "comprising" and "having" and any variations thereof are intended to cover non-exclusive inclusion. For example, a system, product, or device that includes a series of components or units is not limited to the listed components or units, but may optionally further include components or units not listed, or may optionally further include other components or units inherent to these products or devices. In the description of the present application, the meaning of "a plurality" is at least two, such as two, three, etc., unless otherwise specifically defined.

[0071] Reference to "embodiments" herein means that a particular feature, structure, or characteristic described in connection with the embodiments can be included in at least one embodiment of the present application. The phrase appears in various places in the specification and does not necessarily refer to the same embodiment, nor is it an independent or alternative embodiment mutually exclusive with other embodiments. Those skilled in the art will explicitly and implicitly understand that the embodiments described herein can be combined with other embodiments.

[0072] There are three deficiencies in the current probabilistic wind speed prediction model. First, the inadequacy of features will reduce the prediction performance of the model. Convolutional neural networks are widely used as feature extractors. In a multi-layer convolutional neural network, the features extracted by different layers have different levels. However, most previous studies only use the features of the last convolutional layer and ignore the multi-scale features (MSFs) of the remaining convolutional layers. Since the MSFs are not fully utilized, information loss may occur, which in turn limits the prediction performance of the model. Second, previous studies have always focused on deterministic prediction, while the uncertainty of wind speed makes the reliability of the prediction results unable to be guaranteed. In contrast, probabilistic prediction can quantify the relevant uncertainty and provide more reliable and comprehensive information. In existing methods, using a quantile-based model is an effective strategy that can provide prediction intervals at multiple confidence levels simultaneously. Third, the quantile model often suffers from the quantile crossing problem, which limits the reliability of the probabilistic prediction results. However, this problem has always been ignored in many studies.

[0073] Therefore, the embodiments of the present invention provide a probability wind speed prediction method and system based on multi-scale information, which uses a multi-layer convolutional neural network to extract sufficient multi-scale features from input samples, and uses a long short-term memory (LSTM) and attention mechanism to further extract temporal features and encode them into low-dimensional feature vectors. Then, the conditional quantile difference between adjacent quantiles is obtained by means of the proposed non-crossing loss, and the conditional quantiles of all given quantiles are obtained by cumulative addition and subtraction. It can not only extract sufficient multi-scale features from limited data, but also provide high-quality and reliable probability prediction results, while solving the crossing problem of the quantile model. The following will be elaborated and introduced through multiple embodiments.

[0074] Figure 1 The embodiments of the present invention provide a probability wind speed prediction method based on multi-scale information, including:

[0075] Step S1: Obtain historical wind speed time series data, and divide the wind speed time series data into a training set and a test set;

[0076] Collect historical wind speed time series data, and perform normalization processing on the historical wind speed time series data:

[0077]

[0078] In the above formula, x is the wind speed time series in the historical wind speed time series data, x max and x min are the maximum value and the minimum value in the historical wind speed time series data respectively, and x normal is the normalized data;

[0079] Divide the historical wind speed time series data into a training set and a test set according to a preset ratio. The training set is used for the training of the neural network, and the test set is used for testing the prediction performance of the neural network model.

[0080] Step S2: Construct a neural network model, which includes a multi-scale feature extraction module and an LSTM network module. The multi-scale feature extraction module is used to extract one-dimensional feature vectors of input vectors and obtain multiple groups of feature subsequences at different levels by parallel splicing. The LSTM network module is used to extract the temporal features of the feature subsequences, perform weighted combination based on the local features of the temporal features to determine the low-dimensional feature vectors of the feature subsequences, and connect the low-dimensional feature vectors of all the feature subsequences into a prediction feature vector for prediction;

[0081] The model constructed in the embodiments of the present invention is named a deep non-crossing quantile regression model based on multi-scale features, that is, MSF-DNQR. The specific model implementation steps are as follows:

[0082] In a multi-layer convolutional neural network, different layers have different levels of features. Most previous studies only use the features of the last convolutional layer while ignoring the MSFs of the remaining convolutional layers, which may lead to information loss and limit the prediction performance of the model. An embodiment of the present invention designs a multi-scale feature extraction module to extract sufficient information from limited data and compress it into a low-dimensional feature vector to reduce the computational cost. The processing steps of the module can be divided into the following two steps:

[0083] I. Multi-scale feature extraction; construct a neural network model stacked by three one-dimensional convolutions. Each convolutional layer contains multiple convolutional kernels, and the formula for the convolutional kernel operation is:

[0084]

[0085] In the above formula, where is the output of the h-th neuron in the (k-1)-th convolutional layer, M j is the input feature vector set of the j-th neuron, and are the bias and output of the j-th neuron in the k-th convolutional layer respectively, is the weight matrix from the h-th neuron in the (k-1)-th layer to the j-th neuron in the k-th layer, f(·) is the activation function, and * represents the convolution operation;

[0086] Each of the convolutional kernels is used to parallelly splice the one-dimensional feature vectors extracted from the input samples into a multi-dimensional subsequence to obtain three sets of feature subsequences {c i} i＝1,2,3 .

[0087] II. Temporal feature extraction and encoding; for each feature subsequence c i , extract the temporal feature which is the hidden state of all units corresponding to the i-th LSTM; L i is the number of units of the i-th LSTM network module; through a fully connected operation, h i is transformed into h i ', and h i′ and are added to obtain h i″ :

[0088]

[0089] Among them, h i″ is the newly obtained feature vector, h i′ is the feature vector obtained through the fully connected operation, is the hidden state of the last unit of the i-th LSTM network module; based on the attention mechanism for hi″ Assign weights to the eigenvectors in

[0090]

[0091] where and are the l-th and k-th eigenvectors of h i″ respectively, is the attention probability of the eigenvector ; weight the eigenvectors in h i″ according to the obtained attention probability to obtain a low-dimensional eigenvector:

[0092]

[0093] where v i is the low-dimensional eigenvector corresponding to the eigen-subsequence c i ; connect the eigenvectors corresponding to all eigen-subsequences into a prediction eigenvector V for prediction.

[0094] Step S3, construct a quantile loss function for training the neural network model, determine the initial difference of conditional quantiles between any two adjacent quantiles, and convert the initial difference of conditional quantiles into a set of conditional quantile differences greater than 0 through an approximate absolute difference transformation; determine the conditional quantile prediction values of the remaining quantiles based on the conditional quantiles and the conditional quantile differences between two adjacent quantiles;

[0095] The embodiment of the present invention constructs a non-crossing quantile loss function; given a set of input vectors X=(x1, x2,..., x N ) and target variables Y=(y1, y2,..., y N ), quantile regression is used to establish a mapping relationship between the input and the conditional quantiles of the target variables. The combination of an artificial intelligence model and a quantile loss can also handle non-linear problems:

[0096] Q Y (τ|X)=f1(X,β(τ)) (6)

[0097] In the above formula, Q Y (τ│X) is the estimated value of the input vector X at the quantile τ, X=(x1, x2,..., x N ); the target variable Y=(y1, y2,..., y N ); f1(·) is a non-linear function, and β(τ) is the regression coefficient of the model, and its estimated value can be obtained by minimizing the following loss function value:

[0098]

[0099] where is the estimated value of β(τ), N is the number of input samples, and y i is the input sample x i corresponding to the value of the target variable, and Q yi (τ|x i ) is the conditional quantile of the quantile τ, is the quantile loss:

[0100]

[0101] To construct multiple prediction intervals and the probability density function of the target variable, the conditional quantiles of multiple quantiles need to be estimated. However, the obtained estimated values always have a crossing problem, that is, for any two given quantiles τ1, τ2, 0 < τ1 < τ2 < 1, the estimated value of the conditional quantile of τ1 is greater than or equal to the estimated value of the conditional quantile of τ2. This problem violates the monotonicity of the conditional quantile and makes the prediction results unreliable. Therefore, the embodiments of the present invention propose a non-crossing quantile loss to solve this problem, which can be specifically divided into the following steps:

[0102] Assume there are K quantiles τ = {τ k} k＝1,2,…,K , 0 < τ1 < τ2 < … < τ k0 <… < τ K < 1, τ k0 = 0.5; The conditional quantiles The output includes the conditional quantile of the middle quantile τ k0 and the initial difference of the conditional quantiles between any two adjacent quantiles

[0103] Figure 2 As shown in Figure 2 , through the approximate absolute value conversion, is converted into a set of positive conditional quantile differences:

[0104]

[0105] where θ is a positive number; is the positive difference of the conditional quantiles between two adjacent quantiles;

[0106] Based on the obtained conditional quantiles and the conditional quantile difference the estimated values of the conditional quantiles of the remaining quantiles can be obtained by cumulative addition and subtraction:

[0107]

[0108] where, is the quantile τk Conditional quantile estimates;

[0109] Calculate the total quantile loss value for training the neural network model:

[0110]

[0111] In the above formula, L represents the loss value. Since the difference in conditional quantiles between adjacent quantiles is positive, the conditional quantiles at different quantiles are monotonically increasing, and the crossing problem is eliminated.

[0112] Step S4: Based on the training set and the test set, train and validate the neural network model, determine the wind speed probability prediction value based on the trained neural network model, convert the wind speed probability prediction value into conditional quantile prediction values at all quantiles, and combine the conditional quantile prediction values at two different quantiles to obtain a prediction interval corresponding to the confidence level; using all the conditional quantiles as inputs, convert the discrete conditional quantile prediction values into a continuous probability density curve through KDE.

[0113] Input the test sample into the trained model and denormalize the output result of the model:

[0114]

[0115] where f is the denormalized predicted value. Convert the predicted value into conditional quantile estimates at all quantiles through formula (9) and formula (10). By combining the conditional quantiles at two different quantiles, a prediction interval corresponding to the confidence level can be obtained. Using all the conditional quantiles as inputs, the discrete conditional quantile prediction values can be converted into a continuous probability density curve through KDE, thereby intuitively reflecting the relevant uncertainty: The probability density curve is:

[0116]

[0117]

[0118] In the above formula, is the obtained probability density equation, B is the bandwidth, and K E (α) is the Epanechnikov kernel equation.

[0119] Through the above operations, multi-scale features can be extracted and fully utilized from limited data, multiple prediction intervals at different confidence levels and a continuous probability density curve can be obtained, and at the same time, the quantile crossing problem can be solved.

[0120] To verify the reliability of the solution of the embodiments of the present invention, two wind speed datasets in South Dakota are used in the embodiments of the present invention to evaluate the effectiveness of the proposed model. The time scale of the datasets is 1 hour, and the time span is the whole year of 2012. Among them, the data of the first eight months are used to generate the training sample set, and the data of the remaining four months are used to generate the test sample set. For the convenience of description, these two datasets are named Dataset A and Dataset B respectively. To measure the probability prediction performance of different models, 7 evaluation metrics are used, namely Prediction Interval Coverage Probability (PICP), Average Coverage Error (ACE), Prediction Interval Normalized Average Width (PINAW), Mean Prediction Interval Width (MPIW), Coverage Width - based Criteria (CWC), Scoring Function (SC), and Pinball Loss (PL).

[0121] To evaluate the probability prediction performance of the proposed model, multiple probability prediction comparison models are introduced in the embodiments of the present invention, which can be divided into traditional probability prediction models and artificial - intelligence - based probability prediction models. Traditional probability prediction models include Gaussian Process Regression (GPR) and Linear Quantile Regression (LQR). Artificial - intelligence models include Lower and Upper Bound Estimation (LUBE) and quantile - loss - based models. LUBE can generate both the upper and lower bounds of the prediction interval for a given confidence level. The artificial - intelligence - based quantile - loss model generates the conditional quantiles of the target variable by combining artificial - intelligence techniques and quantile loss, including Quantile Regression - Back Propagation Neural Network (QR - BP), Quantile Regression - Convolutional Neural Network (QR - CNN), Quantile Regression - Long Short - Term Memory (QR - LSTM), and Quantile Regression - CNN and LSTM Hybrid Model (QR - CNN - LSTM). When verifying the performance of the proposed MSF module, the loss functions of the quantile - loss - based models should be kept consistent to ensure the fairness of the comparison. Therefore, the MSF module and the quantile loss are combined to introduce a new comparison model MSF - DQR.

[0122] The parameter settings of the comparison models follow the relevant literature, and the parameter values of the deep - learning models are obtained by training with the Adam optimizer. In the embodiments of the present invention, 199 quantiles are selected at equal intervals from 0.05 to 0.995, the number of input nodes and the lag time length of the input samples are both set to 12. In addition, in the proposed loss function, the value of θ is set to 1×10 -8 . To illustrate the performance of the probabilistic wind speed prediction, the embodiments of the present invention compare the prediction intervals at 85%, 90%, and 95% confidence levels, and the results are shown in Table 1 and Table 2.

[0123] Table 1 Probability prediction results of different models on Dataset A

[0124]

[0125] Table 2 Probability prediction results of different models on Dataset B

[0126]

[0127]

[0128] First, by comparing the probability prediction metric results of GPR, LUBE, LQR, QR-NN, QR-CNN, QR-LSTM, QR-CNN-LSTM, and MSF-DQR at different confidence levels in different datasets, it can be seen that the performance of the proposed model MSF-DQR is better than that of the other seven comparison models. Deep learning models other than LUBE are better than the traditional probability prediction models GPR and LQR. Among the deep learning models, the models based on quantile loss are better than LUBE. Among the deep learning comparison models, the model QR-CNN based on convolutional neural network has the best performance, but it is still inferior to the deep learning model MSF-DQR based on multi-scale feature extraction technology. Since QR-NN, QR-CNN, QR-LSTM, QR-CNN-LSTM, and MSF-DQR have the same loss function, the reason may be that the sufficient extraction and utilization of sample features by the multi-scale feature extraction module can improve the prediction performance of the model.

[0129] Secondly, by comparing the prediction results of MSF-DQR and MSF-DNQR, it can be found that they have similar prediction performance. Since the model structures are the same, MSF-DNQR only improves the loss function compared with MSF-DQR, indicating that the introduction of the non-crossing loss function has no negative impact on the prediction performance of the model.

[0130] Since MSF-DNQR is better than other probability prediction comparison models in terms of reliability (PICP, ACE), interval width (MPIW, PINAW), comprehensive index (CWC, SC), and accuracy (PL) except MSF-DQR, it can be considered that the proposed model can generate probability prediction results with higher quality and accuracy.

[0131] Figure 3 The prediction intervals of MSF-DNQR at 85%, 90%, and 95% confidence levels are shown. It can be observed that the upper and lower boundaries of the prediction intervals fluctuate with the observed values, and the intervals can maintain a relatively narrow width while covering the vast majority of the observed values, which further indicates that MSF-DNQR can provide high-quality and accurate probability prediction values.

[0132] To convert the output of the neural network into the conditional quantile difference between two adjacent quantiles, in addition to the approximate absolute value conversion shown in formula (9), the embodiments of the present invention also propose three conversion strategies based on the softmax, softplus, and ReLU activation functions as shown in formula (15), formula (16), and formula (17):

[0133]

[0134]

[0135]

[0136] Where θ is a positive number to ensure that the obtained difference is greater than 0. Keeping the structure of the neural network unchanged and replacing the loss function of MSF-DNQR with the above three equations, three different models can be obtained, denoted as MSF-DNQR-SM, MSF-DNQR-SP, and MSF-DNQR-ReLU. Table 3 to Table 4 show the probability prediction results of the above three models and MSF-DNQR on different datasets.

[0137] Table 3 Probability prediction results based on different non-crossing loss functions on dataset Dataset A

[0138]

[0139] Table 4 Probability prediction results based on different non-crossing loss functions on dataset Dataset B

[0140]

[0141] First, MSF-DNQR has the best performance among all models. The PICP of this model not only meets the given confidence level, but also the values of ACE, MPIW, PINAW, CWC, PL and the absolute value of SC are the smallest in most cases. Secondly, the performance of MSF-DNQR-ReLU is similar to that of MSF-DNQR, but it lacks sufficient reliability because its PICP is sometimes lower than the specified confidence level. MSF-DNQR-SM and MSF-DNQR-SP perform the worst. It can be found from Tables 3 and 4 that although the PICP of these two models can meet all the specified confidence levels compared with MSF-DNQR-ReLU, the values of PINAW and MPIW are relatively large. Therefore, the guarantee of their reliability is at the cost of sacrificing the width of the interval. At the same time, it can also be found that the values of CWC, SC and PL of these two models are relatively poor, indicating that the comprehensive quality of their intervals and the accuracy of the prediction results are relatively low. In summary, compared with the three non-crossing strategies based on softmax, softplus and ReLU activation functions, the MSF-DNQR proposed in the embodiments of the present invention has the best probability prediction performance.

[0142] The embodiments of the present invention further provide a probability wind speed prediction system based on multi-scale information, which is based on the probability wind speed prediction method based on multi-scale information in the above embodiments, and includes:

[0143] A data acquisition module, which acquires historical wind speed time series data and divides the wind speed time series data into a training set and a test set;

[0144] A model construction module, which constructs a neural network model. The neural network model includes a multi-scale feature extraction module and an LSTM network module; the multi-scale feature extraction module is used to extract one-dimensional feature vectors of input vectors and obtain multiple groups of feature subsequences at different levels by parallel splicing; the LSTM network module is used to extract the time series features of the feature subsequences, perform weighted combination based on the local features of the time series features to determine the low-dimensional feature vectors of the feature subsequences, and connect the low-dimensional feature vectors of all the feature subsequences into a prediction feature vector for prediction;

[0145] A loss function construction module, which constructs a quantile loss function for training the neural network model, determines the initial difference of conditional quantiles between any two adjacent quantiles, and converts the initial difference of conditional quantiles into a group of conditional quantile differences greater than 0 through an approximate absolute difference conversion; determines the conditional quantile prediction values of the remaining quantiles based on the conditional quantiles and the conditional quantile differences between two adjacent quantiles;

[0146] A prediction module trains and validates a neural network model based on the training set and the test set, determines a wind speed probability prediction value based on the trained neural network model, converts the wind speed probability prediction value into conditional quantile prediction values at all quantiles, and combines the conditional quantile prediction values at two different quantiles to obtain a prediction interval at a corresponding confidence level; taking all the conditional quantiles as inputs, the discrete conditional quantile prediction values are converted into a continuous probability density curve through KDE.

[0147] Based on the same concept, an embodiment of the present invention further provides a schematic diagram of an entity structure, as Figure 4 shown. The server may include: a processor 810, a communication interface 820, a memory 830, and a communication bus 840. Among them, the processor 810, the communication interface 820, and the memory 830 communicate with each other through the communication bus 840. The processor 810 can call the logical instructions in the memory 830 to execute the steps of the probability wind speed prediction method based on multi-scale information as described in the above embodiments. For example, it includes:

[0148] Step S1, obtain historical wind speed time series data, and divide the wind speed time series data into a training set and a test set;

[0149] Step S2, construct a neural network model, and the neural network model includes a multi-scale feature extraction module and an LSTM network module; the multi-scale feature extraction module is used to extract one-dimensional feature vectors of the input vector, and parallelly splice them to obtain multiple groups of feature subsequences at different levels; the LSTM network module is used to extract the time series features of the feature subsequences, perform weighted combination based on the local features of the time series features to determine the low-dimensional feature vectors of the feature subsequences, and connect the low-dimensional feature vectors of all the feature subsequences into a prediction feature vector for prediction;

[0150] Step S3, construct a quantile loss function for training the neural network model, determine the initial difference between the conditional quantiles of any two adjacent quantiles, and convert the initial difference between the conditional quantiles into a set of conditional quantile differences greater than 0 through approximate absolute difference conversion; determine the conditional quantile prediction values of the remaining quantiles based on the conditional quantiles and the conditional quantile differences between two adjacent quantiles;

[0151] Step S4: Based on the training set and the test set, train and validate the neural network model. Determine the wind speed probability prediction value based on the trained neural network model, convert the wind speed probability prediction value into conditional quantile prediction values at all quantiles, and combine the conditional quantile prediction values at two different quantiles to obtain a prediction interval corresponding to a confidence level. Use all the conditional quantiles as inputs, and convert the discrete conditional quantile prediction values into a continuous probability density curve through KDE.

[0152] In addition, when the logical instructions in the above-mentioned memory 830 are implemented in the form of software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, in essence, or the part that contributes to the prior art, or a part of this technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions for causing a computer device (which may be a personal computer, a server, or a network device, etc.) to execute all or part of the steps of the methods described in various embodiments of the present invention. The aforementioned storage medium includes: various media such as USB flash drives, mobile hard disks, read-only memories (ROM, Read-Only Memory), random access memories (RAM, Random Access Memory), magnetic disks, or optical discs that can store program codes.

[0153] Based on the same concept, an embodiment of the present invention further provides a non-transitory computer-readable storage medium. The computer-readable storage medium stores a computer program, and the computer program includes at least one segment of code. The at least one segment of code can be executed by the main control device to control the main control device to implement the steps of the probability wind speed prediction method based on multi-scale information as described in the above embodiments. For example, it includes:

[0154] Step S1: Obtain historical wind speed time series data, and divide the wind speed time series data into a training set and a test set;

[0155] Step S2: Construct a neural network model. The neural network model includes a multi-scale feature extraction module and an LSTM network module. The multi-scale feature extraction module is used to extract one-dimensional feature vectors of the input vector, and parallelly splice them to obtain multiple groups of feature subsequences at different levels. The LSTM network module is used to extract the time series features of the feature subsequences, perform weighted combination based on the local features of the time series features to determine the low-dimensional feature vectors of the feature subsequences, and connect the low-dimensional feature vectors of all the feature subsequences into a prediction feature vector for prediction;

[0156] Step S3: Construct a quantile loss function for training the neural network model, determine the initial difference in conditional quantiles between any two adjacent quantiles, and convert the initial difference in conditional quantiles into a set of conditional quantile differences greater than 0 through an approximate absolute difference transformation; determine the predicted values of conditional quantiles at the remaining quantiles based on the conditional quantiles and the conditional quantile differences between two adjacent quantiles.

[0157] Step S4: Based on the training set and the test set, train and validate the neural network model, determine the predicted value of wind speed probability based on the trained neural network model, convert the predicted value of wind speed probability into the predicted values of conditional quantiles at all quantiles, and combine the predicted values of conditional quantiles at two different quantiles to obtain a prediction interval corresponding to the confidence level; using all the conditional quantiles as inputs, convert the discrete predicted values of conditional quantiles into a continuous probability density curve through KDE.

[0158] Based on the same technical concept, an embodiment of the present application further provides a computer program, which, when executed by a main control device, is used to implement the above method embodiment.

[0159] The program can be stored in whole or in part on a storage medium packaged together with the processor, or can be stored in whole or in part on a memory not packaged together with the processor.

[0160] Based on the same technical concept, an embodiment of the present application further provides a processor, which is used to implement the above method embodiment. The above processor can be a chip.

[0161] In summary, a probability wind speed prediction method and system based on multi-scale information provided by an embodiment of the present invention uses a multi-layer convolutional neural network to extract sufficient multi-scale features from input samples, uses a long short-term memory (LSTM) and an attention mechanism to further extract temporal features and encode them into low-dimensional feature vectors; then obtains the conditional quantile differences between adjacent quantiles by means of the proposed non-crossing loss, and obtains the conditional quantiles at all given quantiles through cumulative addition and subtraction; it can not only extract sufficient multi-scale features from limited data, but also provide high-quality and reliable probability prediction results, and at the same time solve the crossing problem of the quantile model.

[0162] The various embodiments of the present invention can be combined arbitrarily to achieve different technical effects.

[0163] In the above embodiments, it can be implemented in whole or in part by software, hardware, firmware, or any combination thereof. When implemented using software, it can be implemented in whole or in part in the form of a computer program product. The computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, the processes or functions described in this application are generated in whole or in part. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable devices. The computer instructions can be stored in a computer-readable storage medium or transmitted from one computer-readable storage medium to another. For example, the computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center by wire (such as coaxial cable, fiber optic, digital subscriber line) or wirelessly (such as infrared, wireless, microwave, etc.). The computer-readable storage medium can be any available medium that the computer can access or a data storage device such as a server or data center that includes one or more available media integrated. The available medium can be a magnetic medium (such as a floppy disk, hard disk, magnetic tape), an optical medium (such as a DVD), or a semiconductor medium (such as a solid-state disk).

[0164] Those of ordinary skill in the art can understand that all or part of the processes in the above embodiments of the method can be completed by computer programs instructing relevant hardware. The program can be stored in a computer-readable storage medium. When the program is executed, it can include the processes of the above method embodiments. The foregoing storage medium includes: various media such as ROM or random access memory RAM, magnetic disk, or optical disc that can store program codes.

[0165] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions described in the foregoing embodiments or perform equivalent replacements for some of the technical features. These modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A probability wind speed prediction method based on multi-scale information, characterized in that, Including: Step S1: Obtain historical wind speed time series data and divide the wind speed time series data into a training set and a test set; Step S2: Construct a neural network model, the neural network model includes a multi-scale feature extraction module and an LSTM network module; the multi-scale feature extraction module is used to extract the one-dimensional feature vector of the input vector, and parallelly splice to obtain multiple groups of feature subsequences at different levels; the LSTM network module is used to extract the time series features of the feature subsequences, and perform weighted combination based on the local features of the time series features to determine the low-dimensional feature vectors of the feature subsequences, and connect the low-dimensional feature vectors of all the feature subsequences into a prediction feature vector for prediction; Step S3: Construct a quantile loss function for training the neural network model, determine the initial difference of the conditional quantiles of any two adjacent quantiles, and convert the initial difference of the conditional quantiles into a group of conditional quantile differences greater than 0 through approximate absolute difference conversion; Determine the predicted values of the conditional quantiles of the remaining quantiles based on the conditional quantiles and the conditional quantile differences between two adjacent quantiles; Step S4: Based on the training set and the test set, train and verify the neural network model, determine the wind speed probability prediction value based on the trained neural network model, convert the wind speed probability prediction value into the predicted values of the conditional quantiles of all quantiles, and combine the predicted values of the conditional quantiles of two different quantiles to obtain a prediction interval corresponding to the confidence level; take all the conditional quantiles as inputs, and convert the discrete predicted values of the conditional quantiles into a continuous probability density curve through KDE; The quantile loss function is: Q Y (τ|X) = f1(X, β(τ)) In the above formula, Q Y (τ│X) is the estimated value of the input vector X at the quantile τ, and X = (x1, x2, …, x N ); the target variable Y = (y1, y2, …, y N ); f1(·) is a non-linear function, and β(τ) is the regression coefficient of the model. Its estimated value can be obtained by minimizing the following loss function value: Among them, is the estimated value of β(τ), N is the number of input samples, and y i is the input vector x i corresponding to the value of the target variable, is the conditional quantile of the quantile τ, is the quantile loss: The conditional quantile outputs the conditional quantile of the median and the initial difference of the conditional quantiles of any two adjacent quantiles τ = {τ k} k＝1,2,…,K , 0 < τ1 < τ2 < … < τ k0 < … < τ K < 1, τ k0 = 0.5; where: Convert into a set of conditional quantile differences greater than 0: where θ is a positive number; is the positive difference in conditional quantiles between two adjacent quantiles; Based on the obtained conditional quantiles And the difference in conditional quantiles The estimated values of conditional quantiles at other quantile points can be obtained through cumulative addition and subtraction: Among them, is the conditional quantile estimate of the quantile τ k ; Calculate the total quantile loss value for training the neural network model: In the above formula, L represents the value of the loss.

2. The probability wind speed prediction method based on multi-scale information according to claim 1, characterized in that, The specific content of step S1 includes: Collect historical wind speed time series data and perform normalization processing on the historical wind speed time series data: In the above formula, x is the wind speed time series in the historical wind speed time series data, x max and x min are the maximum value and the minimum value in the historical wind speed time series data respectively, and x normal is the data after normalization; Divide the historical wind speed time series data into a training set and a test set according to a preset ratio. The training set is used for the training of the neural network, and the test set is used for testing the prediction performance of the neural network model.

3. The probability wind speed prediction method based on multi-scale information according to claim 1, characterized in that, In step S2, the multi-scale feature extraction module includes three stacked one-dimensional convolutional layers, each convolutional layer includes multiple convolutional kernels, and the formula for the convolutional kernel operation is: In the above formula, where is the output of the h-th neuron in the (k - 1)-th convolutional layer, and M j is the input feature vector set of the j-th neuron, and are the bias and output of the j-th neuron in the k-th convolutional layer respectively, is the weight matrix from the h-th neuron in the (k - 1)-th layer to the j-th neuron in the k-th layer, f(·) is the activation function, and * represents the convolution operation; Each of the convolutional kernels is used to parallelly splice one-dimensional feature vectors extracted from the input samples into a multi-dimensional subsequence, so as to obtain three sets of feature subsequences at different levels according to the input vectors {c i} i＝1,2,3 .

4. The probability wind speed prediction method based on multi-scale information according to claim 3, characterized in that, In step S2, the LSTM network module is specifically used for: For the feature subsequence c i , extract the temporal features based on the LSTM network module which corresponds to the hidden states of all units of the i-th LSTM; L i is the number of units of the i-th LSTM network module; transform h i into h i ', and obtain h i′ by adding h to get h i″ : Among them, h i″ is the newly obtained feature vector, and h i′ is the feature vector obtained through the fully connected operation, is the hidden state of the last unit of the i-th LSTM network module; weights are assigned to the feature vectors in h i″ based on the attention mechanism: Among them, and are the l-th and k-th eigenvectors of h i″ respectively, and is the attention probability of the eigenvector ; the eigenvectors in h i″ are weighted according to the obtained attention probability to obtain a low-dimensional eigenvector: where v i is the low-dimensional feature vector corresponding to the feature subsequence c i The feature vectors corresponding to all feature subsequences are concatenated into a prediction feature vector V for prediction.

5. The probability wind speed prediction method based on multi-scale information according to claim 4, wherein, The probability density curve is: In the above formula, is the obtained probability density equation, B is the bandwidth, and K E (α) is the Epanechnikov kernel equation.

6. A probability wind speed prediction system based on multi-scale information, wherein, Including: A data acquisition module, which obtains historical wind speed time series data and divides the wind speed time series data into a training set and a test set; A model construction module, which constructs a neural network model, the neural network model includes a multi-scale feature extraction module and an LSTM network module; the multi-scale feature extraction module is used to extract the one-dimensional feature vector of the input vector, and parallelly splice to obtain multiple groups of feature subsequences at different levels; the LSTM network module is used to extract the time series features of the feature subsequences, and perform weighted combination based on the local features of the time series features to determine the low-dimensional feature vectors of the feature subsequences, and connect the low-dimensional feature vectors of all the feature subsequences into a prediction feature vector for prediction; A loss function construction module constructs a quantile loss function for training the neural network model, determines the initial difference in conditional quantiles between any two adjacent quantiles, and converts the initial difference in conditional quantiles into a set of conditional quantile differences greater than 0 through an approximate absolute difference transformation; Determine the conditional quantile prediction values of the remaining quantiles based on the conditional quantiles and the conditional quantile differences between two adjacent quantiles; A prediction module trains and validates the neural network model based on the training set and the test set, determines the wind speed probability prediction value based on the trained neural network model, converts the wind speed probability prediction value into the conditional quantile prediction values of all quantiles, and combines the conditional quantile prediction values of two different quantiles to obtain a prediction interval corresponding to the confidence level; taking all the conditional quantiles as inputs, convert the discrete conditional quantile prediction values into a continuous probability density curve through KDE; The quantile loss function is: Q Y (τ|X) = f1(X, β(τ)) In the above formula, Q Y (τ│X) is the estimated value of the input vector X at the quantile τ, and X = (x1, x2, …, x N ); the target variable Y = (y1, y2, …, y N ); f1(·) is a non-linear function, and β(τ) is the regression coefficient of the model. Its estimated value can be obtained by minimizing the following loss function value: where, is the estimated value of β(τ), N is the number of input samples, and y i is the input vector x i corresponding to the value of the target variable, is the conditional quantile of the quantile τ, is the quantile loss: The conditional quantile output includes the conditional quantile of the median and the initial difference between the conditional quantiles of any two adjacent quantiles τ = {τ k} k＝1,2,…,K , 0 < τ1 < τ2 < … < τ k0 < … < τ K < 1, τ k0 = 0.5; where: Convert into a set of conditional quantile differences greater than 0: where θ is a positive number; is the positive difference in conditional quantiles between two adjacent quantiles; Based on the obtained conditional quantiles and the difference in conditional quantiles The estimated values of conditional quantiles at other quantile points can be obtained by cumulative addition and subtraction: Among them, is the conditional quantile estimate of the quantile τ k ; Calculate the total quantile loss value for training the neural network model: In the above formula, L represents the value of the loss.

7. An electronic device, comprising a memory, a processor, and a computer program stored on the memory and executable on the processor, wherein,When the processor executes the program, it implements the steps of the multi-scale information-based probabilistic wind speed prediction method according to any one of claims 1 to 5.

8. A non-transitory computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the steps of the multi-scale information-based probabilistic wind speed prediction method according to any one of claims 1 to 5.

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