Wind speed simulation method based on time-varying autoregression model and maximum entropy method

Abstract

The invention provides a wind speed simulation method based on a time-varying autoregression model and a maximum entropy method. The wind speed simulation method based on the time-varying autoregression model and the maximum entropy method comprises the following steps: S1, determining a non-stationary characteristic of a target wind speed through a specified evolution power spectral density, namely an EPSD function, calculating a pulse matrix of a time-varying autoregression, namely a TVAR model, and considering a single-variable non-Gaussian non-stationary wind speed process x (t); the mean value and the standard deviation of the time change are respectively expressed as mu x (t) and sigma x (t), and t represents time. By adopting the wind speed simulation method combining the time-varying autoregression model and the maximum entropy method, the coexistence problem of non-Gaussian property and non-stationarity of special wind speeds such as typhoon and thunderstorm is effectively solved, and the non-stationarity characteristic in the wind speed process is captured through the time-varying autoregression model, so that the simulation is more accurate.

Description

Technical Field

[0001] The invention relates to the technical field of random process simulation, and in particular to a wind speed simulation method based on a time-varying autoregressive model and a maximum entropy method. Background Art

[0002] The wind-resistant design of some nonlinear structures often requires wind speed time histories, which can be obtained through wind tunnel tests, simulation methods based on computational fluid dynamics, and random simulation methods. The rapid development of computers has made random simulation methods a widely used wind speed simulation method. For simplicity, it is generally assumed that the wind speed process is stable and its probability distribution obeys Gaussian distribution. These assumptions are valid for large-scale steady strong winds on open and flat ground, but they are invalid for some "special winds" such as typhoons and thunderstorms, which often show the coexistence of non-Gaussianity and non-stationarity. Traditional simulation methods can simulate the non-stationary characteristics of wind speed well, but it is difficult to accurately capture the strong non-Gaussianity of wind speed. In addition, there are regional restrictions on the simulation of non-Gaussian characteristics, and its application is still limited.

[0003] Therefore, an efficient and accurate simulation method is needed, which can accurately summarize the non-stationary and non-Gaussian characteristics of wind speed and significantly improve the scope of applicability. Summary of the invention

[0004] 1. Technical issues to be resolved

[0005] In view of the shortcomings of the prior art, the present invention provides a wind speed simulation method based on a time-varying autoregressive model and a maximum entropy method, which solves the problem that some "special winds", such as typhoons and thunderstorms, often exhibit coexistence of non-Gaussianity and non-stationarity, resulting in invalid wind speed simulation.

[0006] (II) Technical solution

[0007] To achieve the above objectives, the present invention is implemented through the following technical solutions: a wind speed simulation method based on a time-varying autoregressive model and a maximum entropy method:

[0008] S1. First, the non-stationary characteristics of the target wind speed are determined by specifying the evolution power spectral density, i.e., EPSD function, and the impulse matrix of the time-varying autoregressive, i.e., TVAR model is calculated;

[0009] Consider a univariate non-Gaussian non-stationary wind speed process x(t), whose mean and standard deviation over time are represented by μ x (t) and σ x (t), where t represents time, the normalized stationary non-Gaussian process vector y(t) with zero mean and STD can be easily obtained by the following formula:

[0010] y(t)=[x(t)-μx (t)] / σ x (t) (1)

[0011] The evolving power spectral density (EPSD) of y(t) can be obtained from the EPSD S of x(t). x (ω,t) determines:

[0012]

[0013] Where ω represents the circular frequency. Without loss of generality, the simulation target in this patent is y(t);

[0014] The time-varying autoregressive model, as a parametric model, can be expressed as

[0015]

[0016] Where n represents the nth moment, i.e., t = nΔt; N represents the total number of time points; p is the model order; a i (n) is the model coefficient, a i (0) = 1; w(n) is the model input, which is a zero-mean white noise; y(n) is the model output, a zero-mean discrete non-stationary process;

[0017] Furthermore, the TVAR model can be expressed in vector matrix form as the output Y N×1 Between the input N×1 Relationship, that is

[0018] Y N×1 =H N×N W N×1 (4)

[0019] Among them, H N×N is the pulse matrix of the TVAR model, and the non-stationarity of the target wind speed is reflected here. The specific solution is as follows:

[0020] After determining the model order p, multiply both sides of equation (3) by y(nj), j = 1, 2, ..., p and take the expectation to obtain

[0021]

[0022] Where R yy is the correlation function of the non-stationary wind speed y(t), which can be expressed as follows

[0023]

[0024] where E[·] refers to the mathematical expectation; S(ω,t) is the evolving power spectrum of y(t), ω represents the circular frequency; e represents the exponential function; Represents an imaginary unit.

[0025] Combining equations (5) and (6), the model coefficient a i (n) can be solved and further synthesized into A N×N In matrix form:

[0026]

[0027] The impulse matrix H of the final TVAR model is N×N Equivalent to H N×N =A -1 N×N ;

[0028] S2, based on the first k-order time-varying statistical moments E[y k (n)] to solve the first k-order time-varying statistical moments E[w k (n)], and then based on the obtained time-varying PDF reconstructed by the maximum entropy method, i.e., MEM, when H N×N When the matrix is ​​determined, it is only necessary to simulate the corresponding non-Gaussian non-stationary white noise process W N×1 Input into the TVAR model, the target non-Gaussian non-stationary random process can be simulated. First, the simulation of the non-Gaussian non-stationary white noise process must be completed. The input process W needs to be determined. N×1 That is, the statistical moment of non-Gaussian non-stationary white noise. Due to the linear transformation of the TVAR model, the output process Y N×1 The specified target statistical moment is related to the input process W N×1 The statistical moments of the input process and the output process of the TVAR model are not the same, so it is necessary to determine the mapping relationship between the statistical moments of the input process and the output process. N×1 The time-varying statistical moment of , the corresponding time-varying PDF of the non-Gaussian white noise process can be easily constructed;

[0029] Based on the target power spectrum, time-varying variance The standard deviation σ can be calculated by formula (6) y (t), then, the higher-order statistical moments of y(n) can be calculated as follows:

[0030]

[0031] Where k represents the order of the statistical moment; γ k Represents the target skewness value, kurtosis value, excess skewness value, and excess kurtosis value;

[0032] The first four order moments are sufficient to represent common probability distributions. For some non-Gaussian processes, the first four order moments may not be sufficient to characterize their probability distributions, so higher order statistical moments are needed to summarize their distribution characteristics. In order to better simulate non-Gaussian non-stationary wind speeds, after comprehensively considering accuracy and efficiency, it is necessary to further derive the relationship between the higher order moments of the input process and the output process.

[0033] Assuming that the input process and the output process have zero mean and are unrelated at any two different moments, the relationship between their second to sixth order marginal moments can be expressed as

[0034]

[0035]

[0036] Once the input process W is obtained N×1 The time-varying statistical moment of , the corresponding time-varying PDF of the non-Gaussian white noise process can be constructed using a moment-based model. The maximum entropy method is used to construct the PDF to ensure that the estimated PDF has high-order moment information and can better reflect the probability distribution of high-order moments. According to the time-varying statistical moment of , the corresponding time-varying PDF can be obtained by the maximum entropy method through the following optimization problem:

[0037]

[0038] Where f W (w,n) is the time-varying PDF of w(n); H(n) is the time-varying information entropy of w(n); E[w k (n)] is the k-order origin moment of w(n), and K represents the total number of statistical moments used to construct the PDF;

[0039] Introducing the Euler-Lagrange equation in Eq(14), the PDF of w(n) is approximately:

[0040]

[0041] Where λ 0 ,λ 1 ,…λ k is the Lagrange multiplier, which can be determined by the following unconstrained optimization problem:

[0042]

[0043] Where K represents the number of moments used to determine MEM. The larger the K value, the more accurate the MEM estimation is. As K increases, the estimation efficiency of MEM decreases. In order to strike a balance between the accuracy and efficiency of the estimation, the following convergence mechanism is used to reasonably determine the appropriate value of K.

[0044] ‖H e -H e-2‖ / ‖H e ‖≤ε H (17)

[0045] H e It can be determined by the estimated probability density function in equation (14), ε H is the relative error, and it is recommended to take 10 -5 ;

[0046] S3. Establish the piecewise translation function of the moment and determine the parameter μ of the piecewise translation function according to the time-varying PDF of w(n) - / + (n), σ - / + (n), and the moment-based HPM model Finally, the standard Gaussian wind speed process z(n) is input into the piecewise translation function to obtain w(n);

[0047] According to the translation process theory, the underlying standard Gaussian random process Z N×1 =[z(1),z(2),…,z(N)] T Perform nonlinear memoryless transformation to obtain the input non-Gaussian white noise process W N×1 :

[0048] w(n)=g[z(n),n]=F W -1 {Φ G [z(n)],n}(18)

[0049] Where g[·]=F W -1 {Φ G [·]} is the transfer function, where Φ G [·] and F W [·] are the cumulative distribution functions (CDFs) of the standard Gaussian white noise process and the non-Gaussian white noise process, respectively. Note that F W and F W -1 They are inverse functions of each other and can be obtained by the following formula:

[0050]

[0051] In order to simulate non-Gaussian processes more intuitively and conveniently, moment-based models such as HPM and JTM can usually be used as translation functions. Through these models, non-Gaussian processes can be explicitly expressed as functions of the underlying standard Gaussian process. However, the parameters of these models are only determined by the first four moments of the target non-Gaussian process, and these models have limitations in the feasible region. When simulating processes with obvious non-Gaussian characteristics, the first four moments cannot well reflect the probability characteristics of the non-Gaussian process, and are affected by the monotonic limit, resulting in unsatisfactory performance of these moment-based models. This study uses a moment-based piecewise translation function and expands it to a time-varying form to adapt to the white noise input generated in non-stationary conditions.

[0052]

[0053] Where μ - (n), σ - (n) and The new process The time-varying mean, standard deviation, and kurtosis of PDF Symmetric around the median of w(n), about w ≤ w mid (n) when it satisfies Similarly + (n), σ + (n) and The new process The time-varying mean, standard deviation, and kurtosis of PDF Symmetric around the median of w(n), with respect to w ≥ w mid (n) when it satisfies and yes Yuwa The moment-based HPM model determined in this study has the first three moments of 0, 1, and 0, respectively, namely:

[0054]

[0055] Where k(n) and l(n) are time-varying polynomial coefficients, which can be calculated as follows:

[0056]

[0057] Next, the newly defined moment μ in equation (20) is given - / + (n), σ - / + (n) and First, the symmetric point w of the PDF of the input process is calculated mid (n):

[0058]

[0059] Then μ - (n), σ - (n) and It can be calculated by the following formula:

[0060] μ - =w mid

[0061]

[0062] Similarly + (n), σ + (n) and It can be calculated by the following formula:

[0063] μ + =w mid

[0064]

[0065] S4. Based on w(n), we further obtain W N×1 =[w(1),w(2),…,w(N)] T , and sum it with the pulse matrix H N×N Input the time-varying autoregressive model to obtain the target non-Gaussian non-stationary wind speed process Y N×1 .

[0066] (III) Beneficial effects

[0067] The present invention provides a wind speed simulation method based on a time-varying autoregressive model and a maximum entropy method. It has the following beneficial effects:

[0068] By adopting a wind speed simulation method that combines a time-varying autoregressive model with the maximum entropy method, the problem of coexistence of non-Gaussianity and non-stationarity in dealing with special wind speeds such as typhoons and thunderstorms is effectively solved. The time-varying autoregressive model is used to capture the non-stationary characteristics of the wind speed process, making the simulation more accurate. The maximum entropy method can utilize high-order moment information when constructing the probability density function of non-Gaussian white noise, thereby better reflecting the complex statistical characteristics of the wind speed process. It also avoids the limitations of traditional models in simulating obvious non-Gaussian wind speed processes by combining the piecewise translation function of the moment with the high-order moment model. It can not only maintain simulation accuracy in a wider range of scenarios, but also improve computational efficiency through a reasonable convergence mechanism, ensuring the accuracy and practicality of wind speed simulation under complex meteorological conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] Figure 1 Sample target wind speeds simulated for both methods;

[0070] Figure 2 The figure is a comparison between the method of the present invention and the traditional TVAR-based method. DETAILED DESCRIPTION

[0071] The technical solutions in the embodiments of the present invention are described clearly and completely. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0072] Embodiment 1:

[0073] The embodiment of the present invention provides a wind speed simulation method based on a time-varying autoregressive model and a maximum entropy method, including:

[0074] S1. First, the non-stationary characteristics of the target wind speed are determined by specifying the evolution power spectral density, i.e., EPSD function, and the impulse matrix of the time-varying autoregressive, i.e., TVAR model is calculated;

[0075] Consider a univariate non-Gaussian non-stationary wind speed process x(t), whose mean and standard deviation over time are represented by μ x (t) and σ x (t), where t represents time, the normalized stationary non-Gaussian process vector y(t) with zero mean and unit STD can be easily obtained by the following formula:

[0076] y(t)=[x(t)-μ x (t)] / σ x (t) (26)

[0077] The evolving power spectral density (EPSD) of y(t) can be obtained from the EPSD S of x(t). x (ω,t) determines:

[0078]

[0079] Where ω represents the circular frequency. Without loss of generality, in this patent, the simulation target is y(t);

[0080] The time-varying autoregressive model, as a parametric model, can be expressed as

[0081]

[0082] Where n represents the nth moment, i.e., t = nΔt; N represents the total number of time points; p is the model order; a i (n) is the model coefficient, a i(0) = 1; w(n) is the model input, which is a zero-mean white noise; y(n) is the model output, a zero-mean discrete non-stationary process;

[0083] Furthermore, the TVAR model can be expressed in vector matrix form as the output Y N×1 Between the input N×1 Relationship, that is

[0084] Y N×1 =H N×N W N×1 (29)

[0085] Among them, H N×N is the pulse matrix of the TVAR model, and the non-stationarity of the target wind speed is reflected here. The specific solution is as follows:

[0086] After determining the model order p, multiply both sides of equation (3) by y(nj), j = 1, 2, ..., p and take the expectation to obtain

[0087]

[0088] Where R yy is the correlation function of the non-stationary wind speed y(t), which can be expressed as follows

[0089]

[0090] where E[·] refers to the mathematical expectation; S(ω,t) is the evolving power spectrum of y(t), ω represents the circular frequency; e represents the exponential function; Represents an imaginary unit.

[0091] Combining equations (5) and (6), the model coefficient a i (n) can be solved and further combined into A N×N In matrix form,

[0092]

[0093] The impulse matrix H of the final TVAR model is N×N Equivalent to H N×N =A -1 N×N ;

[0094] S2, based on the first k-order time-varying statistical moments E[y k (n)] to solve the first k-order time-varying statistical moments E[w k (n)], and then based on the obtained time-varying PDF reconstructed by the maximum entropy method, i.e., MEM, when H N×N When the matrix is ​​determined, it is only necessary to simulate the corresponding non-Gaussian non-stationary white noise process WN×1 Input into the TVAR model, the target non-Gaussian non-stationary random process can be simulated. First, the simulation of the non-Gaussian non-stationary white noise process must be completed. The input process W needs to be determined. N×1 That is, the statistical moment of non-Gaussian non-stationary white noise. Due to the linear transformation of the TVAR model, the output process Y N×1 The specified target statistical moment is related to the input process W N×1 The statistical moments of the input process and the output process of the TVAR model are not the same, so it is necessary to determine the mapping relationship between the statistical moments of the input process and the output process. N×1 The time-varying statistical moment of , the corresponding time-varying PDF of the non-Gaussian white noise process can be easily constructed;

[0095] Based on the target power spectrum, time-varying variance The standard deviation σ can be calculated by formula (6) y (t), then, the higher-order statistical moments of y(n) can be calculated as follows:

[0096]

[0097] Where k represents the order of the statistical moment; γ k Represents the target skewness value, kurtosis value, excess skewness value, and excess kurtosis value;

[0098] The first four order moments are sufficient to represent common probability distributions. For some non-Gaussian processes, the first four order moments may not be sufficient to characterize their probability distributions, so higher order statistical moments are needed to summarize their distribution characteristics. In order to better simulate non-Gaussian non-stationary wind speeds, after comprehensively considering accuracy and efficiency, it is necessary to further derive the relationship between the higher order moments of the input process and the output process.

[0099] Assuming that the input process and the output process have zero mean and are unrelated at any two different moments, the relationship between their second to sixth order marginal moments can be expressed as

[0100]

[0101]

[0102] Once the input process W is obtained N×1 The time-varying statistical moment of , the corresponding time-varying PDF of the non-Gaussian white noise process can be constructed using a moment-based model. The maximum entropy method is used to construct the PDF to ensure that the estimated PDF has high-order moment information and can better reflect the probability distribution of high-order moments. According to the time-varying statistical moment of , the corresponding time-varying PDF can be obtained by the maximum entropy method through the following optimization problem:

[0103]

[0104] Where f W (w,n) is the time-varying PDF of w(n); H(n) is the time-varying information entropy of w(n); E[w k (n)] is the k-order origin moment of w(n), and K represents the total number of statistical moments used to construct the PDF;

[0105] Introducing the Euler-Lagrange equation in Eq(14), the PDF of w(n) is approximately:

[0106]

[0107] Where λ 0 ,λ 1 ,…λ k is the Lagrange multiplier, which can be determined by the following unconstrained optimization problem:

[0108]

[0109] Where K represents the number of moments used to determine MEM. The larger the K value, the more accurate the MEM estimation is. As K increases, the estimation efficiency of MEM decreases. In order to strike a balance between the accuracy and efficiency of the estimation, the following convergence mechanism is used to reasonably determine the appropriate value of K.

[0110] ‖H e -H e-2 ‖ / ‖H e ‖≤ε H (42)

[0111] H e It can be determined by the estimated probability density function in equation (14), ε H is the relative error, and it is recommended to take 10 -5 ;

[0112] S3. Establish the piecewise translation function of the moment and determine the parameter μ of the piecewise translation function according to the time-varying PDF of w(n) - / + (n), σ - / + (n), and the moment-based HPM model Finally, the standard Gaussian wind speed process z(n) is input into the piecewise translation function to obtain w(n);

[0113] According to the translation process theory, the underlying standard Gaussian random process Z N×1 =[z(1),z(2),…,z(N)] T Perform nonlinear memoryless transformation to obtain the input non-Gaussian white noise process W N×1 :

[0114] w(n)=g[z(n),n]=F W-1 {Φ G [z(n)],n} (43)

[0115] Where g[·]=F W -1 {Φ G [·]} is the transfer function, where Φ G [·] and F W [·] are the cumulative distribution functions (CDFs) of the standard Gaussian white noise process and the non-Gaussian white noise process, respectively. Note that F W and F W -1 They are inverse functions of each other and can be obtained by the following formula:

[0116]

[0117] In order to simulate non-Gaussian processes more intuitively and conveniently, moment-based models such as HPM and JTM can usually be used as translation functions. Through these models, non-Gaussian processes can be explicitly expressed as functions of the underlying standard Gaussian process. However, the parameters of these models are only determined by the first four moments of the target non-Gaussian process, and these models have limitations in the feasible region. When simulating processes with obvious non-Gaussian characteristics, the first four moments cannot well reflect the probability characteristics of the non-Gaussian process, and are affected by the monotonic limit, resulting in unsatisfactory performance of these moment-based models. This study uses a moment-based piecewise translation function and expands it to a time-varying form to adapt to the white noise input generated in non-stationary conditions.

[0118]

[0119] Where μ - (n), σ - (n) and The new process The time-varying mean, standard deviation, and kurtosis of PDF Symmetric around the median of w(n), about w ≤ w mid (n) when it satisfies Similarly + (n), σ + (n) and The new process The time-varying mean, standard deviation, and kurtosis of PDF Symmetric around the median of w(n), with respect to w ≥ w mid (n) when it satisfies and yes Yuwa The moment-based HPM model determined in this study has the first three moments of 0, 1, and 0, respectively, namely:

[0120]

[0121] Where k(n) and l(n) are time-varying polynomial coefficients, which can be calculated as follows:

[0122]

[0123] Next, the newly defined moment μ in equation (20) is given - / + (n), σ - / + (n) and First, the symmetric point w of the PDF of the input process is calculated mid (n):

[0124]

[0125] Then μ - (n), σ - (n) and It can be calculated by the following formula:

[0126] μ - =w mid

[0127]

[0128] Similarly + (n), σ + (n) and It can be calculated by the following formula:

[0129] μ + =w mid

[0130]

[0131] S4. Based on w(n), we further obtain W N×1 =[w(1),w(2),…,w(N)] T , and sum it with the pulse matrix H N×N Input the time-varying autoregressive model to obtain the target non-Gaussian non-stationary wind speed process Y N×1 .

[0132] Comparative Example:

[0133] According to the evolution power spectral density EPSD function of the target wind speed process, the pulse response matrix H is calculated N×N ; Assume that the evolution power spectrum density function of the non-stationary wind speed process is:

[0134]

[0135] in, is the time-varying mean wind, U is the average wind, here it is 50m / s; static wind spectrum The Davenport wind spectrum is used, that is,

[0136]

[0137] Where ζ=0.003, the duration and cutoff frequency of the target spectrum are 600 seconds and 2π respectively. The equation (51) is standardized in one step to obtain:

[0138]

[0139] Then the correlation function of non-stationary wind speed can be expressed as follows

[0140]

[0141] Where E[·] refers to the mathematical expectation; ω represents the circular frequency; e represents the exponential function; Represents an imaginary unit.

[0142] According to the TVAR model definition, the simulation coefficient and the correlation function have the following relationship

[0143]

[0144] p represents the model order. Combining equations (55) and (56), the model coefficient a i (n) can be solved and further combined into A N×N Matrix form:

[0145]

[0146] Finally, the impulse matrix of the VAR model is equivalent to H N×N =A -1 N×N , H N×N The sum of the i-th row and column j elements in the matrix h j (i) (i = 1, ..., N; j = 1, ..., N), represents the impulse response of time lag j at instantaneous time i.

[0147] Specify the first 6 statistical moments E[y k (n)], k = 1, ..., 6 to solve the first 6 time-varying statistical moments E[w k (n)], and then based on the obtained E[w k (n)] uses the maximum entropy method (MEM) to reconstruct the time-varying PDF of w(n).

[0148] This patent uses the specified PDF to calculate the specified moment. The specified PDF can be obtained by the mixed model Constructed. Where i represents the i-th component of the mixed model, Indicates the proportion of this component, satisfying The mixed PDF is composed of partial Gaussian distribution (SG) and partial hyperbolic secant distribution (SHS). The two distribution functions are respectively given by

[0149]

[0150] in, and Φ(·) are the PDF and CDF of the standard Gaussian process respectively; ξ, θ, α are the location parameter, scale parameter and skewness parameter of the two distributions respectively. In order to ensure the accuracy of wind speed simulation, the first two moments must match the potential moment information of the standardized process. Therefore, the constructed PDF will be further standardized as follows:

[0151] where γ 1 and γ 2 Denote the mixed PDF f H The mean and variance of (y).

[0152] By determining the model parameters, the specified PDF is constructed, and the specified first six moments obtained from the specified PDF are shown in Table 1

[0153] Table 1 Mixed model parameters and the corresponding first six moments

[0154]

[0155] where β k represents E[y k (n)],k=1,...,6

[0156] According to the specified E[y k (n)], k = 1, ..., 6, the first six time-varying moments E[w(n)] can be obtained by the following formula: k (n)],k=1,…,6:

[0157]

[0158] Once the time-varying statistical moment E[w k (n)], k = 1, ..., 6, and its corresponding time-varying PDF can be obtained by the maximum entropy method through the following optimization problem:

[0159]

[0160] Where fW (w,n) is the time-varying PDF of w(n); H(n) is the time-varying information entropy of w(n).

[0161] Introducing the Euler-Lagrange equation in equation (65), the PDF of w(n) is approximately:

[0162]

[0163] Where λ 0 ,λ 1 ,…λ k is the Lagrange multiplier, which can be determined by the following unconstrained optimization problem:

[0164]

[0165] Establish the piecewise shift function of the moment and determine the parameter μ of the piecewise shift function according to the time-varying PDF of w(n) - / + (n), σ - / + (n), and the moment-based HPM model Finally, the standard Gaussian wind speed process z(n) is input into the piecewise translation function to obtain w(n):

[0166] Get f W After (w,n), the piecewise function of the moment is established as follows:

[0167]

[0168] where z(n) is the standard Gaussian wind speed process, and Is and The moment-based HPM model determined in this study has the first three moments of 0, 1, and 0, respectively, namely:

[0169]

[0170] Where k(n) and l(n) are time-varying polynomial coefficients, which can be calculated as follows:

[0171]

[0172] Next, the newly defined moment μ in equation (68) is given - / + (n), σ - / + (n) and First, the symmetric point w of the PDF of the input process is calculated mid (n):

[0173]

[0174] Then μ - (n), σ - (n) and It can be calculated by the following formula:

[0175] μ - (n) = w mid (n)

[0176]

[0177] Similarly + (n), σ + (n) and It can be calculated by the following formula:

[0178] μ + (n) = w mid (n)

[0179]

[0180] Based on w(n), we further get W N×1 =[w(1),w(2),…,w(N)] T , and sum it with the pulse matrix H N×N Input time-varying autoregressive model Y N×1 =H N×N W N×1 Get the target non-Gaussian non-stationary wind speed process Y N×1 .

[0181] The above method can be used to simulate the non-stationary non-Gaussian wind speed process with specified EPSD and specified statistical moment, and the results calculated by the proposed method are compared with those calculated by the traditional TVAR method. Figure 1 The wind speed simulation results of the method of the present invention and the traditional method are given, and the relevant functions and PDF are further shown. Figure 2 As shown in the figure, the correlation functions estimated by the two methods are highly consistent with the target correlation function, indicating that both methods can simulate the corresponding non-stationary characteristics well. However, when the non-Gaussianity is strong, there is a deviation between the PDF of the simulation results obtained by the traditional method and the target PDF, while the estimation results of the method of the present invention can still match the target PDF well, which can illustrate the effectiveness and rationality of the calculation method.

[0182] Furthermore, the ability of estimating statistical moments of the proposed method is compared with that of the traditional TVAR-based method as shown in Table III. The results show that the proposed method has higher accuracy than the traditional TVAR-based method, especially for high-order moments. It can be seen that the proposed method is significantly superior to the traditional TVAR-based method in simulating non-stationary strong non-Gaussian distributions.

[0183] Table 2 Comparison of statistical moments of simulated samples by two methods

[0184]

[0185]

[0186] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A wind speed simulation method based on a time-varying autoregressive model and a maximum entropy method, characterized in that: include: S1. First, the non-stationary characteristics of the target wind speed are determined by specifying the evolution power spectral density, i.e., EPSD function, and the impulse matrix of the time-varying autoregressive, i.e., TVAR model is calculated; Consider a univariate non-Gaussian non-stationary wind speed process x(t), whose mean and standard deviation over time are represented by μ x (t) and σ x (t), where t represents time, the normalized stationary non-Gaussian process vector y(t) with zero mean and unit STD can be easily obtained by the following formula: y(t)=[x(t)-μ x (t)] / σ x (t) (1) The evolution power spectral density (EPSD) of y(t) can be obtained from the EPSDS of x(t) x (ω,t) determines: Where ω represents the circular frequency, and with loss of generality, the simulation target in this patent is y(t); The time-varying autoregressive model is a parametric model that can be expressed as Where n represents the nth moment, i.e., t = nΔt; N represents the total number of time points; p is the model order; a i (n) is the model coefficient, a i (0) = 1; w(n) is the model input, which is a zero-mean white noise; y(n) is the model output, a zero-mean discrete non-stationary process; Furthermore, the TVAR model can be expressed in vector matrix form as the output Y N×1 Between the input N×1 Relationship, that is Y N×1 =H N×N W N×1 (4) Among them, H N×N is the pulse matrix of the TVAR model, and the non-stationarity of the target wind speed is reflected here. The specific solution is as follows: After determining the model order p, multiply both sides of equation (3) by y(nj), j = 1, 2, ..., p and take the expectation to obtain Where R yy is the correlation function of the non-stationary wind speed y(t), which can be expressed as follows where E[·] refers to the mathematical expectation; S(ω,t) is the evolving power spectrum of y(t), ω represents the circular frequency; e represents the exponential function; Represents an imaginary unit. Combining the formula and the formula, the model coefficient a i (n) can be solved and further combined into A N×N In matrix form: The impulse matrix H of the final TVAR model is N×N Equivalent to H N×N =A -1 N×N ; S2, based on the first k-order time-varying statistical moments E[y k (n)] to solve the first k-order time-varying statistical moments E[w k (n)], and then the maximum entropy method (MEM) is used to reconstruct the time-varying PDF. N×N When the matrix is ​​determined, it is only necessary to simulate the corresponding non-Gaussian non-stationary white noise process W N×1 Input into the TVAR model, the target non-Gaussian non-stationary random process can be simulated. First, the simulation of the non-Gaussian non-stationary white noise process must be completed. The input process W needs to be determined. N×1 That is, the statistical moment of non-Gaussian non-stationary white noise. Due to the linear transformation of the TVAR model, the output process Y N×1 The specified target statistical moment is related to the input process W N×1 The statistical moments of the input process and the output process of the TVAR model are not the same, so it is necessary to determine the mapping relationship between the statistical moments of the input process and the output process. N×1 The time-varying statistical moment of , the corresponding time-varying PDF of the non-Gaussian white noise process can be easily constructed; Based on the target power spectrum, time-varying variance The standard deviation σ can be calculated by formula (6) y (t), then, the higher-order statistical moments of y(n) can be calculated as follows: Where k represents the order of the statistical moment; γ k Represents the target skewness value, kurtosis value, excess skewness value, and excess kurtosis value; The first four order moments are sufficient to represent common probability distributions. For some non-Gaussian processes, the first four order moments may not be sufficient to characterize their probability distributions, so higher order statistical moments are needed to summarize their distribution characteristics. In order to better simulate non-Gaussian non-stationary wind speeds, after comprehensively considering accuracy and efficiency, it is necessary to further derive the relationship between the higher order moments of the input process and the output process. Assuming that the input process and the output process have zero mean and are unrelated at any two different moments, the relationship between their second to sixth order marginal moments can be expressed as Once the input process W is obtained N×1 The time-varying statistical moment of , the corresponding time-varying PDF of the non-Gaussian white noise process can be constructed using a moment-based model. The maximum entropy method is used to construct the PDF to ensure that the estimated PDF has high-order moment information and can better reflect the probability distribution of high-order moments. According to the time-varying statistical moment of , the corresponding time-varying PDF can be obtained by the maximum entropy method through the following optimization problem: Where f W (w,n) is the time-varying PDF of w(n); H(n) is the time-varying information entropy of w(n); E[w k (n)] is the k-order origin moment of w(n), and K represents the total number of statistical moments used to construct the PDF; Introducing the Euler-Lagrange equation in Eq(14), the PDF of w(n) is approximately: Where λ0,λ1,…λ k is the Lagrange multiplier, which can be determined by the following unconstrained optimization problem: Where K represents the number of moments used to determine MEM. The larger the K value, the more accurate the MEM estimation is. As K increases, the estimation efficiency of MEM decreases. In order to strike a balance between the accuracy and efficiency of the estimation, the following convergence mechanism is used to reasonably determine the appropriate value of K. ‖H e -H e-2 ‖ / ‖H e ‖≤ε H (17) H e It can be determined by the estimated probability density function in equation (14), ε H is the relative error, and it is recommended to take 10 -5 ; S3. Establish the piecewise translation function of the moment and determine the parameter μ of the piecewise translation function according to the time-varying PDF of w(n) - / + (n), σ - / + (n), and the moment-based HPM model Finally, the standard Gaussian wind speed process z(n) is input into the piecewise translation function to obtain w(n); According to the translation process theory, the underlying standard Gaussian random process Z N×1 =[z(1),z(2),…,z(N)] T Perform nonlinear memoryless transformation to obtain the input non-Gaussian white noise process W N×1 : w(n)=g[z(n),n]=F W -1 {Φ G [z(n)],n} (18) Where g[·]=F W -1 {Φ G [·]} is the transfer function, where Φ G [·] and F W [·] are the cumulative distribution functions (CDFs) of the standard Gaussian white noise process and the non-Gaussian white noise process, respectively. Note that F W and F W -1 They are inverse functions of each other and can be obtained by the following formula: In order to simulate non-Gaussian processes more intuitively and conveniently, moment-based models such as HPM and JTM can usually be used as translation functions. Through these models, non-Gaussian processes can be explicitly expressed as functions of the underlying standard Gaussian process. However, the parameters of these models are only determined by the first four moments of the target non-Gaussian process, and these models have limitations in the feasible region. When simulating processes with obvious non-Gaussian characteristics, the first four moments cannot well reflect the probability characteristics of the non-Gaussian process, and are affected by the monotonic limit, resulting in unsatisfactory performance of these moment-based models. This study uses a moment-based piecewise translation function and expands it to a time-varying form to adapt to the white noise input generated in non-stationary conditions. Where μ - (n), σ - (n) and The new process The time-varying mean, standard deviation, and kurtosis of of Symmetric around the median of w(n), about w ≤ w mid (n) when it satisfies Similarly + (n), σ + (n) and The new process The time-varying mean, standard deviation, and kurtosis of of Symmetric around the median of w(n), with respect to w ≥ w mid (n) when it satisfies and yes Yuwa The moment-based HPM model determined in this study has the first three moments of 0, 1, and 0, respectively, namely: Where k(n) and l(n) are time-varying polynomial coefficients, which can be calculated as follows: Next, the newly defined moment μ in equation (20) is given - / + (n), σ - / + (n) and First, the symmetric point w of the PDF of the input process is calculated mid (n): Then μ - (n), σ - (n) and It can be calculated by the following formula: m - =w mid Similarly + (n), σ + (n) and It can be calculated by the following formula: m + =w mid S4. Based on w(n), we further obtain W N×1 =[w(1),w(2),…,w(N)] T , and sum it with the pulse matrix H N×N Input the time-varying autoregressive model to obtain the target non-Gaussian non-stationary wind speed process Y N×1 .

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