Wind speed interpolation and prediction method based on functional space variable coefficient model

Abstract

This invention discloses a wind speed interpolation prediction method based on a functional spatial variable coefficient model. By constructing a functional spatial variable coefficient mixed effect model, the method uses a spatially dependent Matérn distribution family spatial correlation function to characterize spatial correlation in the principal component scores, fully drawing on information from geographically adjacent locations to analyze the spatial correlation of wind speeds between wind turbines. The spatial variable coefficient model can more effectively describe this spatial variation pattern and eliminate large-scale biases caused by data aggregation, thereby improving the accuracy of wind speed prediction. In addition, this invention can also effectively perform interpolation prediction for wind speed at any spatial location within the range.

Description

Technical Field

[0001] This invention belongs to the field of wind speed prediction technology, specifically a wind speed interpolation prediction method based on a functional spatial variable coefficient model. Background Technology

[0002] Rapid economic and population growth has led to a year-on-year increase in electricity demand. With the continuous growth of global energy demand and the increasing pollution from fossil fuels, the application of renewable energy cannot be ignored worldwide. In recent years, wind energy has attracted widespread attention from scientists and governments globally due to its clean, pollution-free, vast reserves, and renewable nature. Governments worldwide have provided substantial support for the construction and research of wind power generation. However, the daily operation of wind farms heavily relies on accurate wind speed forecasting to increase the stability of wind power grid connection, reduce grid pressure, and lower economic operating costs. Predicting wind speeds at multiple spatial locations within a wind farm can provide more information for wind power generation.

[0003] With the continuous development of data acquisition technology, the density of wind speed data collection is becoming increasingly dense, resulting in functional data. This can address the problem that traditional statistical analysis methods for discrete data fail to fully extract the potential information from the data. Furthermore, functional data collected at different times or spatial locations are often naturally correlated. Recently, spatiotemporally correlated functional data has been widely applied in many fields, but its application in the renewable energy sector is still limited. For the few works on this type of data, functional principal component analysis (FPCA) is used to model spatiotemporal dependencies. Spatial correlation is typically nested within the principal component scores of the Karhunen-Loéve decomposition.

[0004] Variable coefficient models are an effective tool for studying the spatiotemporal effects of covariates and have been widely used to model longitudinal data. Furthermore, there are related studies on multilevel covariate and multilevel dependency structure models. All of these studies are based on time variations, while research on spatial variation coefficients for specific regions is relatively limited. Summary of the Invention

[0005] In view of this, the purpose of this invention is to provide a wind speed interpolation prediction method based on a functional spatial variable coefficient model, which can predict the short-term future trend of wind speed and the wind speed interpolation at unsampled locations.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] A wind speed interpolation prediction method based on a functional spatial variable coefficient model includes the following steps:

[0008] Step 1: Obtain wind speed data and covariate data at multiple spatial locations in the target wind farm. The covariates include the current ambient temperature around the turbine and the wind speed at the previous moment.

[0009] Step 2: Assuming the wind speed data is defined as a functional response process in the time domain, a functional spatial variable coefficient mixed effect model is constructed. This model includes a fixed effects component, a random effects component, and random observation errors. The fixed effects component parameterizes the spatially correlated coefficient function using tensor product B-splines. The random effects component is decomposed into a mean function, orthogonal characteristic functions, and Gaussian random field principal component scores using principal component analysis, and spatial correlation is characterized based on the Matérn distribution family spatial correlation function.

[0010] Step 3: Optimize the number of nodes in the tensor product B-spline using the BIC criterion, and estimate the coefficient function of the fixed effects part and the mean function of the random effects part;

[0011] Step 4: Estimate the covariance function of the random effects component based on the three-dimensional tensor product B-spline, and extract the principal component scores and spatial correlation function through principal component analysis;

[0012] Step 5: Interpolate and predict the wind speed at unsampled locations, calculate the best linear unbiased prediction using the spatial correlation function kriging method, and generate the predicted wind speed sequence for the target location by combining it with data from neighboring locations.

[0013] Furthermore, in step two, the constructed functional spatial variable coefficient mixed effect model is expressed as follows:

[0014] Y(s i ,t k ) = Z T (s i ,t k )β(s i )+δ(s i ,t k )+ε(s i ,t k )

[0015] Where: Y(s,t) is the functional response process; Z(s,t)β(s) is the fixed effects component; Z(s,t) is a d-dimensional covariate vector; β(s) = β(s) x ,s y Y(s) is the coefficient function corresponding to the covariate Z(s,t); δ(s,t) is the random effects component, representing the potential random process; ε(s,t) is the random observation error; ... random observation error. i ,t k ) represents any given time point t k and position s i Measurement; s i For spatial location; t k For a point in time; sx and s y These represent the horizontal and vertical coordinates of the spatial point s, respectively; i = 1, ..., n are the indices of the turbine location; k = 1, ..., K are the indices of the time points.

[0016] Furthermore, the expression for the partial decomposition of random effects using principal component analysis is as follows:

[0017]

[0018] Where: μ(t) is the mean function of the random effects component; ψ j (t) represents the orthogonal characteristic function of the random effects component; ξ(s) represents the principal component score of the Gaussian random field in the random effects component.

[0019] Furthermore, the covariance function of the random effects component is:

[0020]

[0021] Where: Y(s1,t1) and Y(s2,t2) are the wind speeds at positions s1, s2 and t1, t2, respectively; C j (s1,s2) is the covariance function;

[0022] When s1 = s2, the covariance function of the random effects component simplifies to:

[0023]

[0024] Where: ω j and ψ j (·) is R T Eigenvalues ​​and eigenfunctions of (·,·);

[0025] By allowing principal component scores of different orders to have different spatial covariances, the covariance function is a common-regional model, which simplifies to a separable structure:

[0026] C j (·)=ω j ρ j (·)

[0027] Where: ρ j (·) represents the spatial correlation function of component j.

[0028] Furthermore, in step three, the parameter function is obtained by approximating the coefficient function β(s) through a linear combination of the tensor product B-spline basis functions. The mean function μ(t) of the random effects component is approximated using B-splines;

[0029]

[0030] Where: B(s) x ,s y () is a vector of cubic B-spline basis functions of the tensor product; Let represent the parameter vector of the tensor product.

[0031] Using the mean function μ(t) of the random effects component approximated by B-splines, we obtain:

[0032]

[0033] in: It is an approximation of the mean function μ(t);

[0034] Parameter function and parameters It is obtained by minimizing the first objective function, and the first objective function is:

[0035]

[0036] in: Represent the parameter space. Given a spline order k=3, optimize the number of nodes in the tensor product B-spline using the BIC criterion:

[0037]

[0038] Furthermore, in step four, the method for estimating the covariance function of the random effects component based on the three-dimensional tensor product B-spline is as follows:

[0039] The covariance function R(u,t1,t2) of the random effects component is treated as a function in the three-dimensional domain and estimated using three-dimensional tensor product B-splines:

[0040]

[0041] in: R(u,t1,t2) is the estimator of the covariance function; B(u), B(t1), and B(t2) are three sets of B-spline basis functions; u is the distance between two spatial locations, within a predetermined spatial distance Δ. For parameter vector estimation;

[0042] Parameter vector It is obtained by minimizing the second objective function, and the second objective function is:

[0043]

[0044] Where: v is the vector of parameters to be estimated;

[0045] After obtaining the three-dimensional covariance function, estimate the covariance function:

[0046]

[0047] in: The weight function is a non-negative bounded function, and when u∈[0,Δ], Otherwise, it is 0.

[0048] Furthermore, in step five, let s0 be a new position without any observations, then:

[0049] If s0∈{s1,K,s n}∈ D Then, the interpolation prediction curve trajectory of the spatial location s0 in a specific region is:

[0050]

[0051] like The interpolation prediction curve trajectory of the spatial location s0 in a specific region is:

[0052]

[0053] Where: Z(s0,t) k ) is the covariate of the interpolation position s0; Z(s0,t) k The tensor product B-spline approximation function; (s0) is the parameter function obtained by approximating the coefficient function β(s0) through a linear combination of tensor product B-spline basis functions; This is an approximation of the mean function for the random effects portion; The principal component score ξ of the Gaussian random field j The best linear unbiased prediction of (s0); (t k ) is the estimate of orthogonal characteristic functions.

[0054] The beneficial effects of this invention are as follows:

[0055] This invention presents a wind speed interpolation prediction method based on a functional spatial variable coefficient model. By constructing a functional spatial variable coefficient mixed effect model, it uses a spatially dependent Matérn distribution family spatial correlation function to characterize spatial correlation in the principal component scores, fully leveraging information from geographically proximate locations to analyze the spatial correlation of wind speeds between wind turbines. The spatial variable coefficient model can more effectively describe this spatial variation pattern and eliminate large-scale biases caused by data aggregation, thereby improving the accuracy of wind speed prediction. In addition, this invention can also effectively perform interpolation prediction for wind speeds at any spatial location within the range.

[0056] The present invention also has the following technical effects:

[0057] This invention considers the spatial variability of covariates and the spatiotemporal correlation of wind speed data, predicting the short-term future trend of wind speed and interpolating wind speed at unobserved locations. It introduces spatial correlation with random biases through the Matérn distribution across specific spatial locations to account for inter-regional dependencies. The dependency complexity of the functional spatial variable coefficient mixed-effects model presents computational challenges, especially with large sample sizes. This invention proposes a corresponding estimation method with the following technical advantages:

[0058] (1) The proposed functional spatial variable coefficient mixed effect model can be used for spatial interpolation of wind speed at unsampled locations. Covariates at different spatial locations are quantified by nonparametric spatial functions and parameterized using cubic tensor product B-splines. For the functional estimation of spatially varying principal component scores, the optimal linear unbiased estimate (BLUP) of principal component scores is found using the spatial correlation function kriging method, which can flexibly predict wind speed sequences at unsampled locations.

[0059] (2) Spatial and temporal correlations between wind speed sequences are considered. Most existing studies are based on ignoring spatial correlations, considering predictions of a single time series or wind speed sequences from multiple spatial locations. Ignoring spatial effects can lead to specific errors. This invention connects the wind speed correlations at different spatial locations using the Matérn distribution family of spatial correlation functions. Attached Figure Description

[0060] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the following figures are provided for illustration:

[0061] Figure 1 This refers to the relative spatial distribution of wind turbines in a wind farm.

[0062] Figure 2 For all wind turbines;

[0063] Figure 3 The SCCF between the wind speed sequence and its lag sequence;

[0064] Figure 4 (a) SCCF contour map; Figure 4 (b) Global Moran Index;

[0065] Figure 5 For local Moran scatter plots;

[0066] Figure 6 This is the wind speed fitting curve;

[0067] Figure 7 This is a wind speed residual plot;

[0068] Figure 8 This is the coefficient function for the fixed effects portion;

[0069] Figure 9 Let be the mean function and characteristic function of wind speed;

[0070] Figure 10 Forecast curves for some wind turbines;

[0071] Figure 11 A scatter plot of wind speed in spatial forecasting;

[0072] Figure 12 This is the wind speed interpolation curve;

[0073] Figure 13 Interpolation curves and box plots for the new location. Detailed Implementation

[0074] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0075] 1. Methods and Steps

[0076] This embodiment presents a wind speed interpolation prediction method based on a functional spatial variable coefficient model, which includes the following steps.

[0077] Step 1: Obtain wind speed data and covariate data at multiple spatial locations in the target wind farm. The covariates include the current ambient temperature around the turbine and the wind speed at the previous moment.

[0078] (1) Study area and dataset

[0079] The data in this example comes from the Spatial Dynamic Wind Power Prediction Dataset (SDWPF) released on the official website of the Baidu KDD Cup (https: / / aistudio.baidu.com / aistudio / Competition / detail / 152 / 0 / datasets). This dataset contains 245 days of wind speed data for 134 wind turbines in a large wind farm in China, including the relative positions of the turbines, the ambient temperature outside the turbines, and the internal temperature of the turbine nacelles. This dataset was provided by Longyuan Power Group, the largest wind power producer in China and Asia, and collected by a SCADA system. Figure 1 The relative spatial distribution of 134 wind turbines is shown, with horizontal and vertical coordinates in meters.

[0080] In this dataset, generators 24, 25, 31, 38, 40, 53, 61, 67, 68, 84, 102, 121, and 122 had missing data for extended periods and were therefore excluded from the study. For the 121 generators tested, missing data was less frequent, and standard imputation methods were used. In the analysis, data from 11 consecutive days was selected, yielding 31,944 data points. The data sampling interval was 1 hour. The first 9 days of data were used as the training set, and the last 2 days were used as the test set.

[0081] (2) Spatiotemporal characteristics of the data

[0082] In theory, the spatiotemporal correlation of wind speed is an external manifestation of atmospheric motion. On the one hand, the wind speed at the same spatial point has a specific correlation with its historical state, i.e., temporal correlation. On the other hand, there is also a specific correlation between wind speeds at different locations within the same time frame, i.e., spatial correlation. Therefore, wind speed exhibits spatiotemporal correlation, and understanding the spatiotemporal patterns of wind speed is crucial for wind power generation. It can be used to improve forecasting, make wind farm operations more efficient, and rationalize resource utilization.

[0083] like Figure 2 As shown, the time series graph of wind speed from 121 wind turbines reveals that wind speed varies within the range of 0–20 m / s. Wind speed varies slightly at different spatial locations because it is influenced by various practical factors, including topography and meteorological conditions, and is accompanied by some random variations.

[0084] To reveal the correlation between wind speed time series, the sampled cross-correlation function (SCCF) is used to measure the correlation between any two time series. Let a t and b t Given two time series with a time lag of h, the SCCF is calculated as shown in formulas (1) and (2). g is the length of the time series; a M and b M They are a i and b i The mean of C. aa (0) and C bb (0) represents the correlation coefficient C when h = 0, b = a, or a = b, respectively. ab The value of (h). r ab The larger the value of (n), the higher the correlation between the two time series. Each wind speed time series is used as a reference, and the relationship between that series and its lagged series is calculated. For illustration, three generators are selected, such as... Figure 3As shown, the temporal correlation between sequences decreases with increasing time lag h, indicating that the wind speed at the same location is related to its historical state. In other words, the wind speed sequence has a strong time dependence, so temporal correlation needs to be considered when modeling wind speed.

[0085]

[0086] Similarly, the SCCF between each wind turbine and wind speed sequences at other locations was calculated to reveal spatial correlations. Figure 4 (a) The contour map illustrates an example of generator No. 1. This map reveals the fundamental characteristics of the spatial correlation of wind speeds in a wind farm. On the one hand, the spatial correlation between wind speeds is absolute within a specific range; there is a correlation between the wind speed of generator No. 1 and that of every other generator. On the other hand, the degree of spatial correlation is relative. It is characterized by a higher correlation between geographically adjacent generators and a relatively lower correlation between generators that are spatially far apart.

[0087] Next, we will further discuss spatial correlation using Moran's I index. The global Moran's I index aims to determine whether there is spatial correlation in regional wind speeds from a holistic perspective, and its value ranges from [-1, 1]. If it is greater than 0, the observed samples are spatially positively correlated. If it is close to 0, the sequences are spatially uncorrelated. The local Moran's I index refines the correlation between adjacent regional units from a microscopic perspective, revealing whether there is a clustering effect in local space.

[0088] Figure 4 (b) The global Moran's index over the study timeframe is presented. It is readily apparent from the figure that, with a few exceptions, most time points show positive values, fluctuating primarily between 0 and 0.6, indicating positive spatial autocorrelation in the study area within the example. Building upon this, we further analyzed local spatial correlation. Figure 5 The local Moran scatter plot shows that the points are distributed in all four quadrants, with a greater concentration in the first and third quadrants, indicating that the wind speed of the wind farm has a spatial clustering effect.

[0089] Step 2: Assuming the wind speed data is defined as a functional response process in the time domain, a functional spatial variable coefficient mixed effect model is constructed. The functional spatial variable coefficient mixed effect model includes a fixed effect component, a random effect component, and random observation errors. The fixed effect component is parameterized using tensor product B-splines to represent the coefficient function related to spatial location. The random effect component is decomposed into a mean function, an orthogonal characteristic function, and Gaussian random field principal component scores through principal component analysis, and spatial correlation is characterized based on the spatial correlation function of the Matérn distribution family.

[0090] Based on the spatiotemporal analysis of wind speed in the previous section, a functional spatially variable coefficient mixed effect model (FSVC-ME) is proposed. This model can explore the spatiotemporal patterns of wind speed more efficiently and accurately.

[0091] Suppose that the function response process Y(s,t) defined in the time domain t∈T is derived from the function response process in the spatial domain. For upsampling, the constructed functional spatially variable coefficient mixed-effects model is expressed as:

[0092] Y(s i ,t k ) = Z T (s i ,t k )β(s i )+δ(s i ,t k )+ε(s i ,t k (3)

[0093] Where: Y(s,t) is the functional response process; Z(s,t)β(s) is the fixed effects component; Z(s,t) is a d-dimensional covariate vector; β(s) = β(s) x ,s y Y(s) is the coefficient function corresponding to the covariate Z(s,t); δ(s,t) is the random effects component, representing the potential random process; ε(s,t) is the random observation error; ... random observation error. i ,t k ) represents any given time point t k and position s i Measurement; s i For spatial location; t k For a point in time; s x and s y These represent the horizontal and vertical coordinates of the spatial point s, respectively; i = 1, ..., n are the indices of the turbine location; k = 1, ..., K are the indices of the time points. It is the measurement error, which satisfies ε ik ~ i.i.d. N(0,σ 2 ).

[0094] The dependencies between function responses at different spatial locations are reflected by a latent stochastic process δ(s,t), which connects the correlations between function responses at different spatial locations. This embodiment decomposes this process using the standard Karhunen-Loeve function. Specifically, the expression for the random effects portion decomposed using principal component analysis is as follows:

[0095]

[0096] In the standard function principal component analysis (FPCA) framework: μ(t) = E{δ(s,t)} is the mean function of the random effects component; ψ j (t) is the orthogonal characteristic function of the random effects part. If j = j′, then ∫ψ j (t)ψ j′ (t) = 1, otherwise 0; ξ j (s) represents the principal component score of the Gaussian random field in the random effects component, and the principal component score ξ j (s) is a function with zero mean and variance ω. j Gaussian random fields.

[0097] In practice, the Karhunen-Loeve expansion in equation (4) is truncated by p, and the covariance function is:

[0098] C j (s1,s2)=cov{ξ j (s1),ξ j (s2)},j=1,…,p

[0099] Where p is usually chosen by interpreting the variance score (FVE).

[0100] In addition, ε ik Principal component scores ξ of the feature components j (s) Independent. Then, the covariance function for the random effects portion can be written as:

[0101]

[0102] Where: Y(s1,t1) and Y(s2,t2) are the wind speeds at positions s1, s2 and t1, t2, respectively; C j (s1,s2) is the covariance function.

[0103] When s1 = s2, the covariance function of the random effects component simplifies to:

[0104]

[0105] Where: ω j and ψ j (·) is R T Eigenvalues ​​and eigenfunctions of (·,·).

[0106] By allowing principal component scores of different orders to have different spatial covariances, the covariance function (6) is a common-region model. It simplifies to a separable structure:

[0107] C j (·)=ω j ρ j (·)

[0108] Where: ρ j (·) represents the spatial correlation function of component j.

[0109] Step 3: Optimize the number of nodes in the tensor product B-spline using the BIC criterion, and estimate the coefficient function of the fixed effects part and the mean function of the random effects part.

[0110] Because B-splines possess ideal numerical stability and practical computational efficiency, this embodiment uses the tensor product of B-splines to approximate the coefficient function β(s) of the fixed effect. x ,s y ).set up Two sets of B-spline basis functions, with L1 and L2 nodes respectively. θ v ={θ1 T ,…,θ T d} T Represents the cubic B-spline basis function of the tensor product The vector is approximated by a bivariate coefficient function through a linear combination of tensor product B-spline basis functions:

[0111]

[0112] in: This represents the tensor product regression coefficient vector estimate of the coefficient function.

[0113] In this embodiment, B-splines are also used to approximate the mean function μ(t) of the random effects component, resulting in:

[0114]

[0115] in: It is an approximation of the mean function μ(t).

[0116] Parameter function and parameters It is obtained by minimizing the first objective function, and the first objective function is:

[0117]

[0118] in: Represents the parameter space.

[0119] Given a spline order κ = 3, optimize the number of nodes in the tensor product B-spline using the BIC criterion:

[0120]

[0121] The estimated mean function is used for ensemble observation data:

[0122] Step 4: Estimate the covariance function of the random effects component based on the three-dimensional tensor product B-spline, and extract the principal component scores and spatial correlation function through principal component analysis.

[0123] Next, the covariance function R(u,t1,t2) is estimated, where u represents the distance between two spatial sites. This distance must be limited to a predetermined spatial distance Δ, because spatial dependence typically decays to zero after a certain distance. Specifically, in this embodiment, the method for estimating the covariance function of the random effects component based on three-dimensional tensor product B-splines is as follows:

[0124] The covariance function R(u,t1,t2) of the random effects component is considered as a function over the three-dimensional domain H:=[0,Δ]×T×T, and estimated using three-dimensional tensor product B-splines:

[0125]

[0126] in: R(u,t1,t2) is the estimator of the covariance function; B(u), B(t1), and B(t2) are three sets of B-spline basis functions, with L4, L5, and L6 nodes respectively; u is the distance between two spatial locations, within a predetermined spatial distance Δ. Let be the parameter vector. And:

[0127]

[0128] After the spline approximation of the nonparametric function, the parameter vector is estimated. That's enough. Specifically, the parameter vector. It is obtained by minimizing the second objective function, and the second objective function is:

[0129]

[0130] Where: Θ is the vector of parameters to be estimated.

[0131] After obtaining the three-dimensional covariance function, estimate the covariance function:

[0132]

[0133] in: The weight function is a non-negative bounded function, when u∈[0,α], Otherwise, it is 0.

[0134] Then, the characteristic functions can be obtained through functional principal component analysis (FPCA). The estimation and eigenvalues ​​{ω} jThe estimation of the eigenvalues ​​of the truncation expansion equations is performed. To preserve sufficient information in the initial steps of the algorithm, FVE ≥ 99% is set as the criterion for calculating the number of eigencomponents retained in the truncation expansion equations.

[0135] In addition, through The obtained spatial covariance function and spatial correlation function are derived from... The corresponding variance function is given by Γ=Var{Y(s,t)}=R(0,t,t)+σ. 2 The variance parameter of random error is given by Provided.

[0136] Step 5: Interpolate and predict the wind speed at unsampled locations, calculate the best linear unbiased prediction using the spatial correlation function kriging method, and generate the predicted wind speed sequence for the target location by combining it with data from neighboring locations.

[0137] For a new location s, the spatially correlated function kriging method is used to predict the random field. Let s0∈D n Let Y(s0,t) be a new location with no observations. The goal here is to interpolate Y(s0,t) using information from neighboring locations. It is ξ j The best linear unbiased prediction (BLUP) for (s0). Let N(s0,Δ) be the set of sampling locations within the range Δ from s0. is the observation vector from a neighboring location. It is the mean vector obtained by interpolation from the mean function. Assume... The covariance matrix is ​​obtained by interpolating the spatiotemporal covariance function R(·,·,·), and the covariance matrix of the observed data in the neighborhood is... So ξ j The best linear unbiased prediction (BLUP) of (s0) is given by the following equation:

[0138]

[0139] in:

[0140] At this point, all parameters of the FSVC-ME model have been obtained. In the final step, the spatial interpolation performance of the proposed FSVC-ME model is demonstrated using the leave-one-out method. This is achieved using a spatial location s0∈{s1,K,s...} n}∈D n Using the data as test data, the remaining data and the fitted model are used to predict the curve for this spatial location, and the above steps are repeated for all spatial locations. The interpolated predicted curve trajectory for the spatial location s0 in a specific region is as follows:

[0141]

[0142] If the interpolation position The covariate Z(s,t) is modeled as a function approximated by the tensor product B-spline. in coefficient It can be obtained using the least squares method, for positions {s1,…,s n}∈D n Z(s,t). Thus, the covariance at the interpolation position s0 can be obtained. Conversely, the interpolation prediction curve trajectory of the spatial location s0 in a specific region is given by formula (12), where

[0143]

[0144] Where: Z(s0,t) k ) is the covariate of the interpolation position s0; Z(s0,t) k The tensor product B-spline approximation function; The parameter function is obtained by approximating the coefficient function β(s0) through a linear combination of tensor product B-spline basis functions; Approximation of the mean function for the random effects portion; The principal component score ξ of the Gaussian random field j The best linear unbiased prediction of (s0); This is for estimating orthogonal characteristic functions.

[0145] 2. Results and Analysis

[0146] 2.1 Fitting Results and Fixed Effects Analysis

[0147] Based on the proposed FSVC-ME model, this embodiment analyzes wind speed data from wind farms. A strong correlation exists between wind turbine locations, ranging from 1.8 km. Therefore, Δ = 1.8 km is chosen as the spatial lag distance for the model, and the current ambient temperature of the wind turbines and the previous wind speed are applied as covariates to the model, yielding a series of fitting results.

[0148] The FSVC-ME model was used to fit the wind speed sequence of all 121 wind turbines in the wind farm. Figure 6 The figure shows the fitting results for three randomly selected wind turbine units. As can be seen from the figure, the fitted curve is consistent with the actual curve, and the fitting result is good. Figure 7The composite residual analysis plot for all wind turbines in the wind farm is shown. The residual plot displays random fluctuations centered at zero, with the amplitude of variation remaining stable within a frequency band. The residual values ​​fall between these two values, validating the model's accuracy. The density histogram shows that the residuals follow a normal distribution, indicating that the model is applicable to wind speed data.

[0149] Figure 8 This shows the coefficient function β1(s) of the fixed effects part. x ,s y ),β2(s x ,s y (Image of ). Figure 8 (a) shows that the correlation between wind speed and ambient temperature is higher in the upper right part of the wind turbine location and lower in the lower left part. Numerically, the correlation between wind speed and ambient temperature at the current spatial location is very small. Similarly, from Figure 8 (b) It can be seen that the correlation between wind speed and the wind speed at the previous moment is higher at local wind turbine locations in the lower left part of the coordinate system, and lower at other locations. Overall, the correlation between the current instantaneous wind speed and the previous instantaneous wind speed is not significantly different at each wind turbine location within this region.

[0150] 2.2 Random Effects Analysis

[0151] Mean function of wind speed and characteristic function The estimation results are as follows Figure 9 As shown in (a), it can be observed that the mean function of the random effects is close to zero. First characteristic function. It explains 95.36% of the variance and is the dominant variance pattern in the random effects component. It exhibits a non-linear fluctuation trend. It extracts low-peak information from the early morning of the second day, noon of the eighth day, and evening of the fourth day. This is based on the first principal component score of the wind turbine. As the absolute value of increases, the aforementioned peak and trough information will become more pronounced. Second characteristic function Only 2.35% of the variation component was explained, showing a regular periodic fluctuation trend. The wind speed variation pattern from one date to the next can be represented by the significant changes in the approximately 3-day time period and the intervals at the boundaries of each 3-day period.

[0152] Figure 9 (b) shows a heatmap of the first and second principal component scores for 121 wind turbines. It can be seen that the first principal component scores of the turbines located in the upper right section are more concentrated. In contrast, the scores of the turbines on the left side of the map are less concentrated. Because... When the value is negative, the wind speed in the blue area of ​​the heatmap shows a more pronounced trend of the first characteristic function. This can be analyzed as a hotspot, and the underlying reasons can be further explored. The second characteristic function can be used as a reference, but it only explains a small portion of the changes; the lower left part of the second principal component score presents a small hotspot area.

[0153] 2.3 Analysis of Prediction Results

[0154] 2.3.1 Comparison of Overall Prediction Results

[0155] To verify the performance of the proposed model, comparative experiments were conducted with existing time series prediction models. A range of wind speed prediction models were selected for comparison, including the statistical model ARIMA, the single-input time series LSTM model, the multi-feature input LSTM model, and the CNN-LSTM model with a spatial neighborhood matrix as input. In the analysis, the mean absolute error ε was chosen as the primary factor. MAE (MAE), mean squared error ε MSE (MSE), root mean square error ε RMSE (RMSE) and mean relative error ε MAPE (MAPE) is defined as an evaluation indicator as follows:

[0156]

[0157]

[0158] The prediction error on the test set was calculated based on the metrics mentioned in Table 1. The input set varies depending on the specific requirements of each model architecture. The input set for the ARIMA and LSTM-S models consists only of the current wind speed sequence. In contrast, the input set for the LSTM-M model comprises a multi-feature input sequence, including the sequence from the previous wind turbine and the sequence of ambient temperature, in addition to the current wind speed sequence. The CNN-LSTM model aims to capture spatial dependencies using a graph convolutional neural network (CNN) and then learn the weights of historical wind speed data using a long short-term memory network (LSTM) for current prediction. As shown in Table 1, the MAE and RMSE of FSVC-ME are 0.1853 and 0.2404, respectively. These two metrics are 0.6657 and 0.8384 lower than the CNN-LSTM model, 0.7912 and 0.9531 lower than LSTM-M, 0.8503 and 1.0517 lower than LSTM-S, and 2.1641 and 2.4986 lower than ARIMA. The FSVC-ME model has significant advantages, with all prediction and evaluation metrics showing smaller errors than other comparative models, and consistently maintaining the highest prediction accuracy.

[0159] It's important to note that LSTM-S, LSTM-M, and ARIMA models do not consider the spatial correlation between wind turbines, resulting in lower prediction accuracy. The CNN-LSTM model, however, considers all input data, including the spatial matrix. This model requires expanding the spatial wind speed matrix, and some spatial information needs to be recovered, leading to a decrease in prediction accuracy later on. Furthermore, the FSVC-ME model, to some extent, learns and extracts the spatial correlation features between units and accurately distinguishes the wind speed distribution information of different units at different times.

[0160] Table 1 Comparison Results of Evaluation Indicators

[0161]

[0162] 2.3.2 Comparison of Prediction Results for Selected Wind Turbine Generators

[0163] Of the 121 wind turbine locations, three were selected for illustration, located on the right, center, and upper left of the relative turbine location map, namely turbines 12, 13, and 125. The predicted curves for each model are shown below. Figure 10 As shown. Figure 10 While both models predicted actual wind speeds to some extent, the FSVC-ME model's prediction curve is closer to the actual value. The ARIMA model can only predict the approximate range of actual wind speeds to a certain extent, but it needs to better capture wind speed trends. Other comparative models, although able to predict actual wind speed trends, suffer from time lag because they use observations from the previous time point to predict the next. Specifically, when wind speed changes are stable, the prediction errors of other models are relatively stable, but their prediction errors increase when wind speed fluctuates. On the other hand, the FSVC-ME model's prediction curve more closely matches the actual curve and outperforms the predictions of other models.

[0164] exist Figure 10 The 40th time point marks a turning point in the trend, and the proposed model accurately captures this change. Other comparative models also captured this change, but with larger errors. These three locations exhibit similar trends within the intervals of increasing, stable, and decreasing wind speeds, further validating the spatial correlation of wind speeds at different locations. The next subsection will focus on the ability to predict overall wind speeds at precise moments.

[0165] 2.3.3 Spatial Prediction Results

[0166] The model proposed in this embodiment demonstrates good performance in time series forecasting and is highly capable of providing spatial information about wind farms, which is of significant reference value for grid dispatching. This embodiment uses a wind speed scatter heatmap to visualize the distribution of wind speed in the wind farm. Three time points from the test set of predicted and actual values ​​are selected for illustration using the wind speed scatter heatmap, such as... Figure 11 As shown, these represent the peaks at hours 6 and 42, and the trough at hour 21, respectively. The actual values ​​on the left and the predicted values ​​on the right match very well across the 121 wind turbines, demonstrating accurate spatial prediction. At hour 6, wind speeds are higher at the top of the relative location map of the wind farm and lower in the lower left corner, gradually decreasing from top to bottom. At hour 21, wind speeds are generally lower across the wind farm, with higher speeds in the center. At hour 42, the wind speed distribution exhibits a pattern of high speeds in the center and lower speeds on the left and right sides.

[0167] 2.4 Wind speed interpolation without observation locations

[0168] This embodiment illustrates the effectiveness of the proposed FSVC-ME method in interpolating wind speeds at new locations. In the dataset, one location is considered the test data, and the remaining data are used to interpolate the wind speed at that location. The MAE, MSE, RMSE, and MAPE of the predicted and actual values ​​are calculated, and this experiment is repeated for all locations. For comparison, we also performed the same experiment on Kriging interpolation of the FPCA function, and the comparison results are shown in Table 2. It is easy to see from the table that all evaluation metrics of the FSVC-ME method are much smaller than those of the Kriging interpolation method, which confirms that the prediction error of the proposed FSVC-ME method for the FPCA function is much smaller than that of the Kriging method. In addition, we randomly selected three wind speed interpolation curves for illustration, such as... Figure 12 As shown in the figure, the proposed FSVC-ME method is very effective in interpolating new spatial locations.

[0169] Table 2. Spatial Interpolation Comparison Results

[0170]

[0171] In addition, we added three new spatial locations to further test the effectiveness of the proposed model. The coordinates of these three new locations, γ1, γ2, and γ3, are (3826.30, 2733.56), (518.77, 11101.00), and (4850.88, 11044.45), respectively. We interpolated the wind speeds at these three new locations using the FSVC-ME model and cubic B-splines. To achieve more accurate interpolation, we plotted the wind speeds using the observed data. Figure 1The distribution map of wind turbines shows that they are relatively densely distributed in the horizontal direction, but relatively sparse in the vertical direction. Based on these characteristics, we will... x The number of nodes in B(s) is set to 10. y The number of nodes was set to 5. Finally, the wind speed profile at the new location was obtained and compared with the wind speed profiles at neighboring locations within a distance Δ from the new location.

[0172] The results are as follows Figure 13 As shown in the figure, the interpolation curves for the new location and its neighboring locations are within a reasonable range, based on the time series plots and box plots. The wind speed curves show a similar trend to those of neighboring locations, and the interpolation curves are smoother. This is because the function terms in the model are approximated using B-splines. Overall, the proposed model is able to interpolate wind speed profiles for new spatial locations, which helps wind farms predict wind speed sequences for more spatial locations.

[0173] 3. Conclusion

[0174] This paper proposes a functional spatially variable coefficient mixed-effects model (FSVC-ME) for spatiotemporal data. The proposed model can more effectively capture the correlation between wind turbine wind speeds and accurately predict wind speeds. Furthermore, the model can also be used for wind speed interpolation at new locations where no observation data is available. Based on the analysis of wind speed data from a wind farm in China, the following conclusions are drawn.

[0175] (1) Through time series correlation analysis of wind speed, it can be seen that wind speed has a strong time correlation. Secondly, through the analysis of SCCF and Moran's I, the visualized data shows obvious spatiotemporal correlation.

[0176] (2) In this embodiment, the proposed FSVC-ME model is applied to the data. A spatial correlation function from the Matérn distribution family with spatial dependence is used for the principal component scores. This model fully leverages information from geographically proximate locations and analyzes the spatial correlation of wind speeds between wind turbines. The spatially variable coefficient model can more effectively describe this spatial variation and eliminate large-scale biases caused by data aggregation. The proposed model can effectively interpolate wind speed predictions for any spatial location within the range.

[0177] (3) Through comparative analysis, the advantages and practical applicability of the proposed method compared with commonly used methods can be seen. The proposed model can simultaneously predict wind speeds at multiple locations in a large wind farm. The predicted evaluation index is smaller than that of the comparative model. For wind speeds at new locations without observation data, the proposed model performs better than the commonly used Kriging interpolation method.

[0178] In summary, the proposed FSVC-ME model has significant application value in the field of wind energy.

[0179] The above-described embodiments are merely preferred embodiments provided to fully illustrate the present invention, and the scope of protection of the present invention is not limited thereto. Equivalent substitutions or modifications made by those skilled in the art based on the present invention are all within the scope of protection of the present invention. The scope of protection of the present invention is defined by the claims.

Claims

1. A wind speed interpolation prediction method based on a functional spatial variable coefficient model, characterized in that: Includes the following steps: Step 1: Obtain wind speed data and covariate data at multiple spatial locations in the target wind farm. The covariates include the current ambient temperature around the turbine and the wind speed at the previous moment. Step 2: Assuming the wind speed data is defined as a functional response process in the time domain, a functional spatial variable coefficient mixed effect model is constructed. This model includes a fixed effects component, a random effects component, and random observation errors. The fixed effects component parameterizes the spatially correlated coefficient function using tensor product B-splines. The random effects component is decomposed into a mean function, orthogonal characteristic functions, and Gaussian random field principal component scores using principal component analysis, and spatial correlation is characterized based on the Matérn distribution family spatial correlation function. Step 3: Optimize the number of nodes in the tensor product B-spline using the BIC criterion, and estimate the coefficient function of the fixed effects part and the mean function of the random effects part; Step 4: Estimate the covariance function of the random effects component based on the three-dimensional tensor product B-spline, and extract the principal component scores and spatial correlation function through principal component analysis; Step 5: Interpolate and predict the wind speed at unsampled locations, calculate the best linear unbiased prediction using the spatial correlation function kriging method, and generate the predicted wind speed sequence for the target location by combining it with data from neighboring locations. In step two, the constructed functional spatial variable coefficient mixed effect model is expressed as follows: in: For the function response process; This is the fixed effects portion; for A dimensional vector of covariates; For corresponding covariates The coefficient function; The random effects portion represents the potential random process; This is due to random observation error; For any given point in time and location Measurement; Spatial location; For a point in time; and These represent the horizontal and vertical coordinates of the spatial point s, respectively. It is an index of the turbine's location; It is an index of a specific time point; In step three, tensor bases are used. Linear combination approximation coefficient function of spline basis functions To obtain the parameter function : Use B-splines to approximate the mean function of the random effects component ; in: A vector of cubic B-spline basis functions of the tensor basis; This represents the tensor product regression coefficient vector estimate of the coefficient function; Using B-splines to approximate the mean function of the random effects part yields: wherein: is an approximation of the mean function ​ Parameter function And parameter is obtained by minimizing a first objective function, and the first objective function is: where denotes the parameter space; the number of nodes of the tensor product B-spline is optimized by the BIC criterion under the condition that the given degree of the spline is given. In step four, the method for estimating the covariance function of the random effects component based on the three-dimensional tensor product B-spline is as follows: The covariance function of the random effects part Considered as a function on a three-dimensional domain, the estimation is performed using three-dimensional tensor-product B-splines: in: Covariance function The estimate; , and Three sets of B-spline basis functions; The distance between two spatial locations, within a predetermined spatial distance. Inside; For parameter vectors; Parameter vector is obtained by minimizing a second objective function, and the second objective function is: wherein: is the vector of parameters to be estimated; After obtaining the three-dimensional covariance function, estimate the covariance function: where: is a non-negative bounded weight function, when , , otherwise 0. 2.The wind speed interpolation and prediction method based on the functional space-varying coefficient model according to claim 1, characterized in that: The expression for the partial decomposition of random effects using principal component analysis is as follows: where: is the mean function of the random effects part; is the standard orthogonal eigenfunction of the random effects part; is the Gaussian random field principal component score of the random effects part. 3.The wind speed interpolation and prediction method based on the functional space-varying coefficient model according to claim 2, characterized in that: The covariance function for the random effects component is: where: and are the wind speed at time and are the wind speed at time is the covariance function; when When the random effects component's covariance function is simplified to: wherein: and are eigenvalues and eigenfunctions of respectively. By allowing principal component scores of different orders to have different spatial covariances, the covariance function is a common-regional model, which simplifies to a separable structure: wherein: represents a spatial correlation function of the components .

4. The wind speed interpolation prediction method based on a functional spatial variable coefficient model according to claim 1, characterized in that: In step five, set For a new location with no observations, then: If , the interpolation prediction curve trajectory of the specific region spatial position is: like Then the spatial location of a specific area The interpolation prediction curve trajectory is as follows: in: Interpolation position covariates; for The tensor product B-spline approximation function; For coefficient functions through tensor product The parametric function is obtained by a linear combination of spline basis functions. This is an approximation of the mean function for the random effects portion; Score the principal component of the Gaussian random field The best linear unbiased prediction; This is for estimating orthogonal characteristic functions.

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