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Probabilistic stability of small disturbance in wind power system based on a variational Bayes and Lyapunov theory using PMU data

Probabilistic stability of small disturbance in wind power system based on a variational Bayes... INTRODUCTIONWith the explosive growth of China's total electricity demand and the rapid development of environmental protection technology, energy transition has become a hot research issue so far. New energy, such as wind, solar, and photovoltaic power generation, is the key indicator of China's energy transition. Wind power in the new energy power generation accounts for an important proportion. Due to the uncertain factors of wind speed changing, the output power of wind turbines has the characteristics of fluctuation, randomness, and uncontrollability. Stability and safety will also be affected, so the impact of wind power integration on power system stability of small disturbance has gradually become the attention focus of scholars.During the early research on the small disturbance stability in power system with variable‐speed wind turbines, most of the references were studied by calculating electromechanical oscillation modes of the power system. Firstly, Burchett [1] applied a probability analysis method to the power system with small disturbance stability in 1978. By deriving the average value and the covariance matrix of eigenvalue changing, and according to the relationship between the sensitivity of eigenvalues and the uncertain parameters in the adjoint matrix, the probability that the real parts of all eigenvalues were less than zero was calculated, which established a precedent for the study of the probabilistic stability of small disturbance in power system from the characteristic root. Chen Zhong [2] proposed an analysis method for probabilistic stability of small disturbance considering the influence of random distribution delay, which could quickly calculate the probabilistic stability, and had better effects than the Monte Carlo method. The method for studying the probabilistic stability of the AC‐DC hybrid system in multiple operation modes was used in [3]. Compared with the original AC system, the integration of the DC system of the power grid improved the local oscillation, but worsened the inter‐area oscillation. The proposed probabilistic method by Samundra Gurung et al. [4] was applied to analyse the effect of PV uncertainties arising mainly due to stochastic PV fluctuations and forecast error of small signal stability. The developed framework could provide accurate results to assess PSSS in much less computational time compared with conventional MCS and other analytical techniques.In order to improve the computational efficiency, many scholars have introduced Gram‐Charlier sequence expansion method to probability calculations. Although this method had high computational efficiency, it inevitably brought various mathematical approximations and complex calculations. Therefore, combined with the advantage of the Monte Carlo and analytical method, researchers proposed an approximate method called the Point Estimation Method (PEM), which was an effective method for estimating the normal distribution. The eigenvalues by inputting variable statistics were directly estimated. Especially, expectation and variance had successfully applied to the research on the probabilistic stability of small disturbance in power system [5]. To analyse the probabilistic stability of small disturbance in the three‐machine system, an algorithm of the probabilistic stability analysis of small disturbance based on the discrete point estimation method was proposed in ref. [6]. This method had the characteristics of high precision and high efficiency. However, the interaction among multiple wind farms connected to the grid was not considered. Aiming at the computational efficiency for the uncertainty handling in wind power system, methods of Information‐Gap Decision Theory (IGDT) and Taguchi's Orthogonal Array Testing (TOAT) were proposed in ref. [7]. Using each of these methods, the robust expansion plan for the modified 6‐bus Garver transmission network system was calculated. Furthermore, different uncertainty types could be easily considered in this regard. However, it was proved to be unreasonable from the point of view of mathematical statistic. In some cases, they would bring even wrong judgments of significance factor and lacked some assurances to deal with the effects of interactions. A probabilistic analysis method for small‐signal stability of power systems considering random uncertainty introduced by multiple grid‐connected wind power sources was proposed by Bu [8]. Compared with the non‐analytical method of Monte Carlo simulation, it was very computationally efficient. However, as the penetration rate of wind power increases, the probabilistic instability also increased accordingly. In order to solve the uncertainty of stability margin of the multi‐feed system caused by the randomness of wind power generation, a method was proposed to evaluate the stability margin of the probabilistic stability of small disturbance in electronic power system based on the generalized short‐circuit ratio in ref. [9]. The proposed method has good robustness to probability distribution. Aiming at the power system composed of multiple renewable energy sources, a Probabilistic Small‐Signal Stability Analysis (PSSSA) method based on a Probabilistic Allocation Method (PCM) for wind farms and photovoltaic power generation was proposed in ref. [10]. This method satisfied the accuracy and reduced the computation, but it could have other adverse effects on small‐signal stability when reducing and closing some synchronous units to make the penetration of renewable energy increasing.By deeply studying the above representative papers, the current Monte Carlo methods, analytical methods and approximate methods are all based on the mathematical modelling analysis of the power system. However, each algorithm has its own limitation to applications: Monte Carlo method needs to increase the number of deterministic power flow calculations, which will reduce the calculation efficiency of the algorithm. Analytical rules require complex convolution calculations and the series expansion. The approximation method is difficult to construct the Probability Density Function (PDF) and accurately describe the probability and statistics characteristics when the output random variable is non‐normal distribution. For the above methods, it is hard to solve the problem of computational efficiency when the amount of PMU data is huge. Therefore, the contribution of this paper is to construct a probabilistic analysis method to fully consider various uncertain factors in power system. This method is based on a variational Bayes and Lyapunov theory using PMU data, and introduces the variational Bayesian framework to the data as an association algorithm, which will not only improve the computational efficiency, but also obtain the PDF of the approximation posterior. In order to improve the efficiency of variational Bayesian calculation of the probability whose eigenvalues will fall on the left half‐plane, an RF‐based PMU data processing is proposed to filter out the normal data. Finally, a Bayesian probability based on Lyapunov theory is proposed to analyse the probabilistic stability of small disturbance.The structure of this paper is organized as follows. Section 2 is the basic theory. Section 3 is the PMU data processing based on the Random Forest. Section 4 is the Lyapunov theory of probabilistic stability of small disturbance based on the variational Bayes. Section 5 is the simulation verification, and finally is the conclusion.FUNDERMENTAL THEORYRandom forest theoryRandom forest (RF) [11] is a supervised classification method combining multiple decision trees. It can handle large‐scale data and has high accuracy even when the data are missing. The realization process is relatively simple, and the training speed is fast.There are often hundreds of features in the data set, so it is necessary to reduce eigenvalues when mathematical modelling. How much contribution for each feature of each tree in RF is compared at first, then it will be taken an average, and finally the contribution between eigenvalues is acquired. VIM is used to represent the variable importance measures, and GI is represented as the Gini Index. Assuming there are m features X1, X2, X3,…, Xm, then calculate the GI of each feature Xj score VIMj (Gini), that is, the average change is in the impurity of node splitting of the j‐th feature in all decision trees of RF.Calculate the GI, we can get1GIm=∑k=1|K|∑k′≠kpmkpmk′=1−∑k=1|K|p2mk′\begin{equation}GIm = \sum_{k = 1}^{|K|} {\sum_{k^{\prime} \ne k} {pmkpmk^{\prime}} } = 1 - \sum_{k = 1}^{|K|} {{p}^2mk^{\prime}} \end{equation}In (1), K represents the category, and pmk is the proportion of node m in category K.The calculation formula for the change in GI before and after node m is as follows:2VIMjm(Gini)=GIm−GIl−GIr\begin{equation}VIMj{m}^{(Gini)} = GIm - GIl - GIr\end{equation}In (2), GIl and GIr are Gini indexes of nodes before and after node m.The core of RF is an ensemble operation method based on decision trees. It can also be said to be a tree‐like classifier that selects the most appropriate attribution from internal nodes. Each leaf node contains the data of the same kind of attributions.Variational BayesVariational Bayes [12] is a Bayesian estimation method using an approximate complex integral. It is used mainly in some complex statistical models. The main purpose is to approximate the posterior probability of unobservable variables, so as to make statistical inferences through these variables. For this purpose, the Monte Carlo simulation uses the Markov Chain Monte Carlo (MCMC) algorithm [13] with Gini sampling to approximate complex posterior distributions, which can be well applied to Bayesian statistical inference. However, this method estimates the true posterior through a large number of samples, and the approximation result has a certain degree of randomness. The difference is that variational Bayes uses the local optimality and has a determined solution to maximize a posteriori probability estimation that replaces the complete Bayesian estimation with a single most likely parameter value, and the optimal solution is obtained by continuous iteration through a set of mutual equations. Therefore, this paper uses a single most likely eigenvalue instead of the Bayesian estimation to predict the probability of its eigenvalue falling into a stable interval.For any fixed value x∈[x0, x1], the difference y(x)−y0(x) between the available function y(x) and another available function y0(x) is called the function y(x) in y0(x). The variation at the position or the variation of the function is denoted as δy, then we have,3δy=y(x)−y0(x)=εη(x)\begin{equation}{\delta }_{y} = y( x ) - {y}_0( x ) = \epsilon \eta ( x )\end{equation}δy is a very small number, and η(x) is any parameter of x. The increment of the functional J[y(x)] is:4ΔJ=J[y(x)+δy]−J[y(x)]=δJ+o(δy)\begin{eqnarray} \Delta J &=& J[y( x ) + {\delta }_y] - J[y( x )]\nonumber\\ &=& {\delta }_J + o( {{\delta }_y} ) \end{eqnarray}The difference between the functional increment ΔJ and the variation δJ is an infinitesimal higher order than the first‐order distance, and the variation of the function is the main linear part of the function increment. So there is the following theory [14]: If the function J[y(x)] reaches its extreme value on y = y(x), then its variation δJ on y = y(x) is equal to zero, which is called a variation.Suppose that data x in a model are generated according to the probability distribution of the unknown parameter θ, and there is a priori knowledge about the parameter θ, which can be represented by the probability distribution p(θ). Then, when the data x is observed, we can use Bayesian theorem to update the prior knowledge about the parameter, which is as shown in Equation (5):5P(θ|x)=P(x|θ)P(θ)P(x)\begin{equation}P(\theta |x) = \frac{{P(x|\theta )P(\theta )}}{{P(x)}}\end{equation}The most complicated part of the posterior distribution calculation is to calculate the normalization factor:6p(x)=∫θp(x|θ)p(θ)dθ\begin{equation}p(x) = \int\limits_{\theta }{{p(x|\theta )}}p(\theta )d\theta \end{equation}The integral in Equation (6) can be calculated in low dimensions. It is not feasible to accurately calculate the posterior distribution in high dimensions. Some approximation techniques should be used to obtain the posterior distribution. Therefore, the Variational Inference (VI) method is introduced.PMU DATA PROCESSING BASED ON THE RANDOM FORESTPMU data processingSystem modelAccording to the idea of the node voltage equation [15], assuming that the number of nodes in a certain area of the power system with DFIG is n, then the mathematical model of the node voltage in this area is:7İn=U̇n⋅Fnn⇒İ1⋮İi⋮İj⋮İn0=U̇1⋮U̇i⋮U̇j⋮U̇n0•F11⋯F1i⋯F1j⋯F1n0⋮⋱⋮⋱⋮⋱⋮0Fi1⋯Fii′⋯Fij′⋯Fin0⋮⋱⋮⋱⋮⋱⋮0Fj1⋯Fji′⋯Fjj′⋯Fjn0⋮⋱⋮⋱⋮⋱⋮0Fn1⋯Fni⋯Fnj⋯Fnn000000000\begin{eqnarray} \def\eqcellsep{&}\begin{array}{l} {\rm{ }}\mathop {{{\dot{\bf I}}}}\nolimits_n = \mathop {{\rm{ }}{{\dot{\bf U}}}}\nolimits_n \mathop {\centerdot {{\bf F}}}\nolimits_{nn} \Rightarrow \\[6pt] \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{{\dot{I}}}_1}\\[6pt] \vdots \\[6pt] {{{\dot{I}}}_i}\\[6pt] \vdots \\[6pt] {{{\dot{I}}}_j}\\[6pt] \vdots \\[6pt] {{{\dot{I}}}_n}\\[6pt] 0 \end{array} } \right] = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{{\dot{U}}}_1}\\[6pt] \vdots \\[6pt] {{{\dot{U}}}_i}\\[6pt] \vdots \\[6pt] {{{\dot{U}}}_j}\\[6pt] \vdots \\[6pt] \def\eqcellsep{&}\begin{array}{l} {{\dot{U}}}_n\\[6pt] 0 \end{array} \end{array} } \right] \bullet \left[ { \def\eqcellsep{&}\begin{array}{@{}*{8}{c}@{}} {{F}_{11}}&\quad \cdots &\quad{{F}_{1i}}&\quad \cdots &\quad{{F}_{1j}}&\quad \cdots &\quad{{F}_{1n}}&\quad 0\\[6pt] \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad 0\\[6pt] {{F}_{i1}}&\quad \cdots &\quad{{F}^{\prime}_{ii}}&\quad \cdots &\quad{{F}^{\prime}_{ij}}&\quad \cdots &\quad{{F}_{in}}&\quad 0\\[6pt] \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad 0\\[6pt] {{F}_{j1}}&\quad \cdots &\quad{{F}^{\prime}_{ji}}&\quad \cdots &\quad{{F}^{\prime}_{jj}}&\quad \cdots &\quad{{F}_{jn}}&\quad 0\\[6pt] \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad 0\\[6pt] {{F}_{n1}}&\quad \cdots &\quad{{F}_{ni}}&\quad \cdots &\quad{{F}_{nj}}&\quad \cdots &\quad{{F}_{nn}}&\quad 0\\[6pt] 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0 \end{array} } \right] \end{array} \nonumber\\ \end{eqnarray}where i,j = 1,2,…, n are node numbers; Fnn is the node admittance matrix; Fij and Fji are mutual admittances between nodes i and j; Fii and Fjj are self‐admittances of nodes i and j, respectively;U̇n${\dot{U}}_n$andİn${\dot{I}}_n$represent node voltage and node injection current;U̇i,U̇j${\dot{U}}_i,{\dot{U}}_j$andİi,İj${\dot{I}}_i,{\dot{I}}_j$are voltage and injection current of nodes i and j, respectively.Due to the influence of wind speed, the equilibrium state of the wind power system will fluctuate over a large range. Set the injection current of the previous equilibrium state ƒ to 0, and add the node ƒ to the admittance matrix. When entering the next equilibrium state, the current change is ΔIf, and the voltage equation of the area with n+1 nodes is:8İn+1=U̇n+1•F(n+1)(n+1)=İ1⋮İi⋮İj⋮İnΔIf=U̇1⋮U̇i⋮U̇j⋮U̇nU̇f•F11⋯F1i⋯F1j⋯F1n0⋮⋱⋮⋱⋮⋱⋮0Fi1⋯Fii′⋯Fij′⋯FinFif′⋮⋱⋮⋱⋮⋱⋮0Fj1⋯Fji′⋯Fjj′⋯FjnFjf′⋮⋱⋮⋱⋮⋱⋮0Fn1⋯Fni⋯Fnj⋯Fnn000Ffi′0Ffj′00Fff′\begin{eqnarray} \def\eqcellsep{&}\begin{array}{l} \mathop {\dot{I}}\nolimits_{n + 1} = \mathop {\dot{U}}\nolimits_{n + 1} \bullet \mathop F\nolimits_{(n + 1)(n + 1)} = \\[6pt] \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{{\dot{I}}}_1}\\[6pt] \vdots \\[6pt] {{{\dot{I}}}_i}\\[6pt] \vdots \\[6pt] {{{\dot{I}}}_j}\\[6pt] \vdots \\[6pt] {{{\dot{I}}}_n}\\[6pt] {\Delta {I}_f} \end{array} } \right] = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{{\dot{U}}}_1}\\[6pt] \vdots \\[6pt] {{{\dot{U}}}_i}\\[6pt] \vdots \\[6pt] {{{\dot{U}}}_j}\\[6pt] \vdots \\[6pt] \def\eqcellsep{&}\begin{array}{l} {{\dot{U}}}_n\\[6pt] {{\dot{U}}}_f \end{array} \end{array} } \right] \bullet \left[ { \def\eqcellsep{&}\begin{array}{@{}*{8}{c}@{}} {{F}_{11}}&\quad \cdots &\quad{{F}_{1i}}&\quad \cdots &\quad{{F}_{1j}}&\quad \cdots &\quad{{F}_{1n}}&\quad 0\\[6pt] \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad 0\\[6pt] {{F}_{i1}}&\quad \cdots &\quad{{F}^{\prime}_{ii}}&\quad \cdots &\quad{{F}^{\prime}_{ij}}&\quad \cdots &\quad{{F}_{in}}&\quad{{F}^{\prime}_{if}}\\[6pt] \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad 0\\[6pt] {{F}_{j1}}&\quad \cdots &\quad{{F}^{\prime}_{ji}}&\quad \cdots &\quad{{F}^{\prime}_{jj}}&\quad \cdots &\quad{{F}_{jn}}&\quad{{F}^{\prime}_{jf}}\\[6pt] \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad 0\\[6pt] {{F}_{n1}}&\quad \cdots &\quad{{F}_{ni}}&\quad \cdots &\quad{{F}_{nj}}&\quad \cdots &\quad{{F}_{nn}}&\quad 0\\[6pt] 0&\quad 0&\quad{{F}^{\prime}_{fi}}&\quad 0&\quad{{F}^{\prime}_{fj}}&\quad 0&\quad 0&\quad{{F}^{\prime}_{ff}} \end{array} } \right] \end{array} \nonumber\\ \end{eqnarray}F'if, F'jf, F'fi, F'fj and F'ff are newly introduced admittance elements after the joining node ƒ; F'ii, F'jj, F'ji and F'ij are admittance elements that change after the node ƒ is added; Uƒ is the node voltage at the previous ƒ equilibrium point.PMU feature data extractionIt can be seen from formula (8) that when the equilibrium state changes, the admittance element of the regional node voltage equation of the wind power system will also change accordingly. According to the eigenvalue calculation formula, there is:9λiI−F(n+1)(n+1)=0\begin{equation}\left| {{\lambda }_i{{\bf I}} - \mathop F\nolimits_{(n + 1)(n + 1)} } \right| = 0\end{equation}It can be calculated that its eigenvalue λi = xi+jyi will also change immediately, where xi and yi are coordinates of the eigenvalue on the imaginary axis.For an n‐dimensional random matrix F(Fi,j), the trace of the matrix is defined as:10trF=∑i=1nFi,i=∑i=1n|λi|,(i=1,2,3,…,n)\begin{equation}trF = \sum_{i = 1}^n {{F}_{i,i}} = \sum_{i = 1}^n {|{\lambda }_i|} ,(i = 1,2,3, \ldots ,n)\end{equation}Fi,i is the main diagonal element of matrix F; λi​ is the eigenvalue of matrix F. It can be seen from Equation (10) that the sum of all eigenvalues is called the matrix trace, which can reflect the statistical characteristics of the matrix F, so the trace of the matrix can be used to reduce the amount of calculation.The statistics of linear eigenvalues can be defined as:11P(μ)=∑i=1nμ(λi),(i=1,2,3,…,n)\begin{equation}P(\mu ) = \sum_{i = 1}^n {\mu ({\lambda }_i)} ,(i = 1,2,3, \ldots ,n)\end{equation}μ(λi) is the test function. Then we can use Linear Eigenvalue Statistical (LES) [16] indicators to extract features from the PMU data matrix. Finally, based on the extracted PMU feature data matrix, the RF algorithm is used to classify the feature value data. At the same time, the normal data are screened out, so as to obtain the probability of its eigenvalue falling on the left half plane according to the variational Bayes.PMU data classificationBy analysing the changing trend of the measured data of the PMU, and the PMU data measured in a period of time can be used as the key wide‐area data. Such data are relatively easy to obtain in the actual operation of the system.Consider the regional node voltage equation of the system at the n+1 node, the eigenvalueλi=λ1,λ2,λ3,…,λnk+1${\lambda }_i = {\lambda }_1,{\lambda }_2,{\lambda }_3, \ldots ,{\lambda }_{{n}_{k + 1}}$ at the equilibrium state of k+1 can be calculated. nk+1 is the total number of eigenvalue samples of the PMU data set, and the normal data in the measured data processed by the PMU data are treated as the related data. The abnormal data are treated as the unrelated data.12pΩλi=pΩ1|λiPMUabnormaldatapΩ2λiPMUnormaldata\begin{equation}p\left( {\left. {{\Omega }} \right|{\lambda }_i} \right) = \left\{ \def\eqcellsep{&}\begin{array}{l} p\left( {{\Omega }_1|{\lambda }_i} \right) {\rm{PMU abnormal data}}\\[6pt] p\left( {\left. {{\Omega }_2} \right|{\lambda }_i} \right){\rm{ PMU normal data}} \end{array} \right.\end{equation}In (12), Ω1 is normal data, and Ω2 is abnormal data. However, in the case of the large number of PMU fundamental currents, they are node voltage data. As the process of updating the state continues, the Gaussian component of the traditional method calculating the posterior PDF will increase geometrically. This will lead to excessive computational complexity and the low computational efficiency. Therefore, the posterior PDF needs to be introduced into the Bayesian framework to obtain the state approximate posterior PDF, which will provide the convenience in the subsequent state updating process.PMU data classification based on random forestThis paper uses the Gini index as a contribution evaluation index [14]. In this case, VIM is represented by W(Gini), and the Gini index is represented by G. Calculate the Gini index score Wi corresponding to each eigenvalue λi. That is, the average change in the impurity of the node split of the i‐th feature is in all decision trees of RF.The calculation formula [17] of the Gini index is as follows:13Gm=∑k=1|K|∑k′≠kpmkpmk′=1−∑k=1|K|p2mk′\begin{equation}Gm = \sum_{k = 1}^{|K|} {\sum_{k^{\prime} \ne k} {pmkpmk^{\prime}} } = 1 - \sum_{k = 1}^{|K|} {{p}^2mk^{\prime}} \end{equation}where K indicates that there are K categories; pmk indicates the proportion of category k in node m.The importance of eigenvalue λi at node m, that is, the change in the Gini index before and after the node m branch is:14Wim(Gini)=Gm−Gl−Gr\begin{equation}Wi{m}^{(Gini)} = Gm - Gl - Gr\end{equation}Gl and Gr represent Gini indexes of two new nodes after branching.If the node of eigenvalue λi in decision tree j is set M, then the importance of λi in the j‐th tree is15Wim(Gini)=∑m∈MWim(Gini)\begin{equation}Wi{m}^{(Gini)} = \sum_{m \in M} {Wi{m}^{(Gini)}} \end{equation}Assuming that there are q trees in RF, then we have, 16Wim(Gini)=∑j=1qWim(Gini)\begin{equation}Wi{m}^{(Gini)} = \sum_{j = 1}^q {Wi{m}^{(Gini)}} \end{equation}Finally, normalize all the obtained importance scores, then we can get,17Wi=Wi∑j=1cWj\begin{equation}Wi = \frac{{Wi}}{{\sum\nolimits_{j = 1}^c {Wj} }}\end{equation}According to the importance score obtained by Equation(14), all eigenvalues are classified by using Equation (17). The eigenvalues whose distance from the coordinate origin are less than τ are recorded as the normal data Ω1, and the remaining eigenvalues as abnormal data are recorded as Ω2.A VARIATIONAL BAYES AND LYAPUNOV THEORY OF PROBABILITY STABILITY OF SMALL DISTURBANCEBayesian probability and stability analysisAssume that the eigenvalues obtained from the voltage equation of the system in the area of n nodes are used as experimental samples, the probability Φk${\Phi }_k$ of the occurrence of m+1 events can be predicted according to the probability Equation (6).18P(Ym+1=ym+1D,ε)=∫P(ωD,ε)P(Ym+1=ym+1ω,ε)dω=∫ωP(ωD,ε)dω=E[P(ωD,ε)]\begin{equation} \def\eqcellsep{&}\begin{array}{l} P(\left. {{Y}_{m + 1} = {y}_{m + 1}} \right|D,\varepsilon )\\[6pt] = \int{{P(\left. \omega \right|D,\varepsilon )P}}(\left. {{Y}_{m + 1} = {y}_{m + 1}} \right|\omega ,\varepsilon )d\omega \\[6pt] = \int{{\omega P(\left. \omega \right|D,\varepsilon )d\omega }}\\[6pt] = E[P(\left. \omega \right|D,\varepsilon )] \end{array} \end{equation}where D is an experimental sample. Y is the event variable; y is the value of the state variable; The Bayesian probability of event Ym+1 is the expected value of the prior probability ω relative to the posterior probability. According to the Bayesian equation, the equation for calculating the posterior probability density from the prior probability is:19P(ωD,ε)=P(ωε)P(D|ε)P(Dω,ε)=P(ω|ε)∫P(D|ω,ε)P(ωε)dωP(Dω,ε)\begin{eqnarray} P(\left. \omega \right|D,\varepsilon ) &=& \frac{{P(\left. \omega \right|\varepsilon )}}{{P(D|\varepsilon )}}P(\left. D \right|\omega ,\varepsilon )\nonumber\\ && = \frac{{P(\omega |\varepsilon )}}{{\int{{P(D|\omega ,\varepsilon )P(\left. \omega \right|\varepsilon )}}d\omega }}P(\left. D \right|\omega ,\varepsilon )\end{eqnarray}where ω is the prior probability of the event occurrence; P is the PDF; ε is the prior knowledge of the observer.Assume that y = y1, y2,…, yr have r possible states, Ai is the number of the eigenvalues falling on the left half plane in the sample space, and it is denoted as event y = yi, then the Dirichlet distribution of the prior probability is:20P(ωε)=Dir(ω|ζ1,ζ2,…,ζr)=∏ωkζk−1∏k=1rψ(ζk)ψ(ζ)\begin{equation} \def\eqcellsep{&}\begin{array}{l} P(\left. \omega \right|\varepsilon ) = Dir(\omega |{\zeta }_1,{\zeta }_2, \ldots ,{\zeta }_r)\\[6pt] {\rm{ }} = \dfrac{{\prod {\omega _k^{{\zeta }_k - 1}} }}{{\prod\limits_{k = 1}^r {\psi ({\zeta }_k)} }}\psi (\zeta ) \end{array} \end{equation}whereζk=P(y=yk|ω,ε),k=1,2,…,r${\zeta }_k = P( {y = {y}^k} |\omega ,\varepsilon ),k = 1,2, \ldots ,r$,ζ=∑k=1rζk$\zeta = \sum_{k = 1}^r {{\zeta }_k} $, andζk > 0, k = 1,2,…, r. The PDF is:21P(ωnk+1,ε)=Dir(ω|ζ1+A1,ζ2+A2,…,ζr+Ar)\begin{equation}P(\left. \omega \right|{n}_{k + 1},\varepsilon ) = Dir(\omega |{\zeta }_1 + {A}_1,{\zeta }_2 + {A}_2, \ldots ,{\zeta }_r + {A}_r)\end{equation}Substituting Equation (21) into Equation (18), the probability that the eigenvalue falling on the left half plane can be calculated as:22P(Ym+1=yknk+1,ε)=∫ωkDir(ω|ζ1+A1,ζ2+A2,…,ζr+Ar)dω=ζk+Akζ+A\begin{equation} \def\eqcellsep{&}\begin{array}{l} P(\left. {{Y}_{m + 1} = {y}^k} \right|{n}_{k + 1},\varepsilon )\\[12pt] = \int{{{\omega }_k}}Dir(\omega |{\zeta }_1 + {A}_1,{\zeta }_2 + {A}_2, \ldots ,{\zeta }_r + {A}_r)d\omega \\[12pt] = \dfrac{{{\zeta }_k + {A}_k}}{{\zeta + A}} \end{array} \end{equation}Now calculate the probability of n+1 eigenvalues falling on the left half plane when the system re‐enters a new equilibrium state.For the Bayesian network structure of the variable F = (F1,F2,…, Fn) in the state transition matrix in Equation (7), it includes a one‐to‐one mapping and a directed non‐direction composed of the eigenvalue λi in variable F. The ring graph O and the probability distribution set P are corresponding to their variables.First of all, each characteristic variable is independent, that is,23P(λO,D)=∏i=1nP(λi|O,D)\begin{equation}P(\left. \lambda \right|O,D) = \prod\limits_{i = 1}^n {P({\lambda }_i} |O,D)\end{equation}P(λi|O,D)$P( {{\lambda }_{^i}} |O,D)$is the prior probability density of the i‐th eigenvalue. Then Equation (21) of Dirichlet distribution is satisfied, and the probability whose the n+1th eigenvalue in D that is predicted to fall on the left half plane is:24Φk(λn+1)=p(λn+1D,O)=E(∏i=1nωijk)=∏i=1n∫ωijkp(ωs|D,O)dω=∏i=1nζijk+Aijkζij+Aij\begin{equation} \def\eqcellsep{&}\begin{array}{c} {\Phi }_k({\lambda }^{n + 1}) = p(\left. {{\lambda }^{n + 1}} \right|D,O) = E(\prod\limits_{i = 1}^n {{\omega }_{ijk}} )\\[6pt] = \prod\limits_{i = 1}^n {\int{{{\omega }_{ijk}p({\omega }_s}}|D,O)} d\omega \\[6pt] = \prod\limits_{i = 1}^n {\dfrac{{{\zeta }_{ijk} + {A}_{ijk}}}{{{\zeta }_{ij} + {A}_{ij}}}} \end{array} \end{equation}whereζij=∑k=1riζijk,Aij=∑k=1riAijk${\zeta }_{ij} = \sum_{k = 1}^{ri} {{\zeta }_{ijk}} ,{A}_{ij} = \sum_{k = 1}^{ri} {{A}_{ijk}} $ and ζijk=P(λ=λijk|ω,ε),k=1,2,…,r${\zeta }_{ijk} = P( {\lambda = {\lambda }^{ijk}} |\omega ,\varepsilon ),k = 1,2, \ldots ,r$.System asymptotically stable based on the Lyapunov theoryConsidering that the time and space correlation between different connected power supplies will affect the probability and stability of the wind power system, the time and space correlation of the power supplies are closely related to their geographical locations according to a certain time difference. Assume that at different times, the correlation parameter between two power supplies that are more than 100 km together is 0, and the correlation parameter between two power supplies that are less than 100 km together is 1. Then, the spatiotemporal correlation coefficient can reflect the geographical distance among the power supplies at different time, and m grid‐connected wind power supplies can be constructed. Then, according to the Lyapunov theory, the proof that the asymptotically stable of the system in a wide range is as follows.First, prove that the system is asymptotically stable, and a mathematical model based on the two equilibrium node voltage equations of (7) and (8) is established as follows:25İn+1=Vİn\begin{equation}{\dot{I}}_{n + 1} = V{\dot{I}}_n\end{equation}where V is a non‐singular matrix of order n×n. The matrix V is transformed into the following non‐singular diagonal matrix and we can get VΛ:26VΛ=τ10⋯00τ2⋯0⋮⋮⋱0000τn\begin{equation}{V}_\Lambda = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{4}{c}@{}} {{\tau }_1}&\quad 0&\quad \cdots &\quad 0\\[6pt] 0&\quad{{\tau }_2}&\quad \cdots &\quad 0\\[6pt] \vdots &\quad \vdots &\quad \ddots &\quad 0\\[6pt] 0&\quad 0&\quad 0&\quad{{\tau }_n} \end{array} } \right]\end{equation}Theorem 1: Let the state equation of the discrete system be:27İ(k+1)=τ10⋯00τ2⋯0⋮⋮⋱0000τnİ(k)\begin{equation}\dot{I}(k + 1) = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{4}{c}@{}} {{\tau }_1}&\quad 0&\quad \cdots &\quad 0\\[6pt] 0&\quad{{\tau }_2}&\quad \cdots &\quad 0\\[6pt] \vdots &\quad \vdots &\quad \ddots &\quad 0\\[6pt] 0&\quad 0&\quad 0&\quad{{\tau }_n} \end{array} } \right]\dot{I}(k)\end{equation}Then the necessary and sufficient condition for the system to be asymptotically stable at the equilibrium point is that the characteristic root of the system lies in the unit circle.Proof: According to the Lyapunov stability theorem, the necessary and sufficient condition for the system to be stable at an equilibrium point is that for any given positive symmetric matrix Q, there exists a positive symmetric matrix P, which satisfies:28VTPV−P=−Q\begin{equation}{V}^TPV - P = - Q\end{equation}and29V(İ(k))=İT(k)Pİ(k)\begin{equation}V(\dot{I}(k)) = {\dot{I}}^{\rm{T}}(k)P\dot{I}(k)\end{equation}is the Lyapunov function of the system.Choose Q = I, I is the identity matrix, and substitute it into Equation (28), then we can get30τ10⋯00τ2⋯0⋮⋮⋱0000τnp11p12⋯p1np21p22⋯p2n⋮⋮⋱⋮pn1pn2⋯pnnτ10⋯00τ2⋯0⋮⋮⋱0000τn−p11p12⋯p1np21p22⋯p2n⋮⋮⋱⋮pn1pn2⋯pnn=−10⋯001⋯0⋮⋮⋱00001\begin{equation} \def\eqcellsep{&}\begin{array}{l} \left[ { \def\eqcellsep{&}\begin{array}{@{}*{4}{c}@{}} {{\tau }_1}&\quad 0&\quad \cdots &\quad 0\\[6pt] 0&\quad{{\tau }_2}&\quad \cdots &\quad 0\\[6pt] \vdots &\quad \vdots &\quad \ddots &\quad 0\\[6pt] 0&\quad 0&\quad 0&\quad{{\tau }_n} \end{array} } \right]\left[ { \def\eqcellsep{&}\begin{array}{@{}*{4}{c}@{}} {{p}_{11}}&\quad{{p}_{12}}&\quad \cdots &\quad{{p}_{1n}}\\[6pt] {{p}_{21}}&\quad{{p}_{22}}&\quad \cdots &\quad{{p}_{2n}}\\[6pt] \vdots &\quad \vdots &\quad \ddots &\quad \vdots \\[6pt] {{p}_{n1}}&\quad{{p}_{n2}}&\quad \cdots &\quad{{p}_{nn}} \end{array} } \right]\left[ { \def\eqcellsep{&}\begin{array}{@{}*{4}{c}@{}} {{\tau }_1}&\quad 0&\quad \cdots &\quad 0\\[6pt] 0&\quad{{\tau }_2}&\quad \cdots &\quad 0\\[16pt] \vdots &\quad \vdots &\quad \ddots &\quad 0\\[6pt] 0&\quad 0&\quad 0&\quad{{\tau }_n} \end{array} } \right]\\[6pt] {\rm{ }} - \left[ { \def\eqcellsep{&}\begin{array}{@{}*{4}{c}@{}} {{p}_{11}}&\quad{{p}_{12}}&\quad \cdots &\quad{{p}_{1n}}\\[6pt] {{p}_{21}}&\quad{{p}_{22}}&\quad \cdots &\quad{{p}_{2n}}\\[6pt] \vdots &\quad \vdots &\quad \ddots &\quad \vdots \\[6pt] {{p}_{n1}}&\quad{{p}_{n2}}&\quad \cdots &\quad{{p}_{nn}} \end{array} } \right] = - \left[ { \def\eqcellsep{&}\begin{array}{@{}*{4}{c}@{}} 1&\quad 0&\quad \cdots &\quad 0\\[6pt] 0&\quad 1&\quad \cdots &\quad 0\\[6pt] \vdots &\quad \vdots &\quad \ddots &\quad 0\\[6pt] 0&\quad 0&\quad 0&\quad 1 \end{array} } \right] \end{array} \end{equation}that is,31p11(1−λ12)=1p21(1−λ2λ1)=0⋮pn1(1−λnλ1)=0p12(1−λ1λ2)=0p22(1−λ22)=1⋮pn2(1−λnλ2)=0p23(1−λ2λ3)=0⋮pn(n−1)1−λnλn−1=0⋱⋯pnn(1−λn2)=1\begin{equation} \def\eqcellsep{&}\begin{array}{@{}*{5}{c}@{}} { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{p}_{11}(1 - {\lambda }_1^2) = 1}\\[15pt] {{p}_{21}(1 - {\lambda }_2{\lambda }_1) = 0}\\[15pt] \vdots \\[15pt] {{p}_{n1}(1 - {\lambda }_n{\lambda }_1) = 0} \end{array} }&{ \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{p}_{12}(1 - {\lambda }_1{\lambda }_2) = 0}\\[15pt] {{p}_{22}(1 - {\lambda }_2^2) = 1}\\[15pt] \vdots \\[15pt] {{p}_{n2}(1 - {\lambda }_n{\lambda }_2) = 0} \end{array} }&{ \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {}\\[15pt] {{p}_{23}(1 - {\lambda }_2{\lambda }_3) = 0}\\[15pt] \vdots \\[15pt] {{p}_{n(n - 1)}1 - {\lambda }_n{\lambda }_{n - 1} = 0} \end{array} }&{ \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {}\\[15pt] {}\\[15pt] \ddots \\[15pt] \cdots \end{array} }&{ \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {}\\[15pt] {}\\[15pt] {}\\[15pt] {{p}_{nn}(1 - {\lambda }_n^2) = 1} \end{array} } \end{array} \end{equation}From Equation (31), we have,32P=11−λ1211−λ22⋱11−λn2\begin{equation}P = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{4}{c}@{}} {\dfrac{1}{{1 - {\lambda }_1^2}}}&\quad{}&\quad{}&\quad{}\\[15pt] {}&\quad{\dfrac{1}{{1 - {\lambda }_2^2}}}&\quad{}&\quad{}\\[15pt] {}&\quad{}&\quad \ddots &\quad{}\\[15pt] {}&\quad{}&\quad{}&\quad{\dfrac{1}{{1 - {\lambda }_n^2}}} \end{array} } \right]\end{equation}To guarantee that matrix P is a positive symmetrical matrix, it requires to satisfy the following Equation (33):33|λ1|<1|λ2|<1⋮|λn|<1\begin{equation} \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {|{\lambda }_1| &lt; 1}\\[12pt] {|{\lambda }_2| &lt; 1}\\[15pt] \vdots \\[15pt] {|{\lambda }_n| &lt; 1} \end{array} \end{equation}In other words, when the characteristic root of the system is located in the unit circle, the equilibrium point of the system is asymptotically stable.Then we will prove that the system probability is asymptotically stable, and we let34Γk(λn+1)=[Φk1(λ),Φk2(λ),…,Φkn(λ)]T=G(n+1,n)Γk(λn)\begin{equation} \def\eqcellsep{&}\begin{array}{l} {\Gamma }_k({\lambda }^{n + 1})\\[15pt] = {[\Phi _k^1(\lambda ),\Phi _k^2(\lambda ), \ldots ,\Phi _k^n(\lambda )]}^{\rm{T}}\\[15pt] = G(n + 1,n){\Gamma }_k({\lambda }^n) \end{array} \end{equation}where G(n+1,n)$G(n + 1,n)$ is the transition matrix, and Φkn(λ)$\Phi _k^n(\lambda )$ represents the probability that the nth eigenvalue λn falls on the left half plane.We select the Lyapunov function as follows:35Vk(Φk(λn),n)=ΓkT(λn)ΩmΓk(λn)\begin{equation}{V}_k({\Phi }_k({\lambda }^n),n) = \Gamma _k^{\rm{T}}({\lambda }^n){\Omega }_m{\Gamma }_k({\lambda }^n)\end{equation}Then we take the Lyapunov first‐order difference equation [18] as:36ΔV(Φk(λn),n)=V(Φk(λn+1),n+1)−V(Φk(λn),n)=ΓkT(λn+1)ΩmΓk(λn+1)−ΓkT(λn)ΩmΓk(λn)=ΓkT(λn)GT(n+1,n)ΩmG(n+1,n)Γk(λn)−ΓkT(λn)ΩmΓk(λn)=ΓkT(λn)[GT(n+1,n)ΩmG(n+1,n)−Ωm]Γk(λn)\begin{equation} \def\eqcellsep{&}\begin{array}{l} \Delta V({\Phi }_k({\lambda }^n),n)\\[6pt] = V({\Phi }_k({\lambda }^{n + 1}),n + 1) - V({\Phi }_k({\lambda }^n),n)\\[6pt] {\rm{ = }}\Gamma _k^{\rm{T}}({\lambda }^{n + 1}){\Omega }_m{\Gamma }_k({\lambda }^{n + 1}) - \Gamma _k^{\rm{T}}({\lambda }^n){\Omega }_m{\Gamma }_k({\lambda }^n)\\[6pt] {\rm{ = }}\Gamma _k^{\rm{T}}({\lambda }^n){G}^T(n + 1,n){\Omega }_mG(n + 1,n){\Gamma }_k({\lambda }^n)\\[6pt] - \Gamma _k^{\rm{T}}({\lambda }^n){\Omega }_m{\Gamma }_k({\lambda }^n)\\[6pt] {\rm{ = }}\Gamma _k^{\rm{T}}({\lambda }^n)[{G}^T(n + 1,n){\Omega }_mG(n + 1,n) - {\Omega }_m]{\Gamma }_k({\lambda }^n)\end{array} \end{equation}For any positive definite symmetric matrix Q(n), according to the Lyapunov discrete system judgment method, there is:37GT(n+1,n)ΩmG(n+1,n)−Ωm=−Qn\begin{equation}{G}^T(n + 1,n){\Omega }_mG(n + 1,n) - {\Omega }_m = - Q\left( n \right)\end{equation}And then, we can find that:38Ωk=GT(0,n+1)ΩmG(0,n+1)−∑i=0nGT(i,n+1)Q(i)G(i,n+1)>0\begin{eqnarray} {\Omega }_k &=& {G}^T(0,n + 1){\Omega }_mG(0,n + 1)\nonumber\\ && - \sum_{i = 0}^n {{G}^T(i,n + 1)Q(i)G(i,n + 1)} &gt; 0\end{eqnarray}where Ωm is positive definite, so Vk(Φk(λn),n)${V}_k({\Phi }_k({\lambda }^n),n)$ is positive definite. According to the Lyapunov stability theorem of time‐varying discrete system, the system is asymptotically stable.Variational Bayesian probabilistic analysis based on Lyapunov theoryFrom the above theoretical reasoning and the proposed stability analysis method, this paper proposes the variational Bayesian probabilistic analysis based on the Lyapunov theory and the design process of probabilistic stability of small disturbance is as follows.Step 1: According to the mathematical model of the node voltage in a certain area, calculating its characteristic root and linear characteristic value statistics of the matrix trace, and performing the characteristic extraction on the PMU matrix by using the Linear Eigenvalue Statistical (LES) index.Step 2: Use the Gini index as a contribution evaluation index to measure the change in the Gini index of the eigenvalue λi before and after the branch of node m, which is as the importance score index, and finally normalize the characteristics according to the importance score. The eigenvalue is classified as the positive abnormal data.Step 3: The obtained normal eigenvalue data are used as the experimental sample D, and then the posterior probability densityp(ω|D,ε)$p( \omega |D,\varepsilon )$is calculated by the prior probability p(ω|ε)$p( \omega |\varepsilon )$, and the probability of the eigenvalue falling on the left half plane is calculated according to the Dirichlet distribution of the prior probability.Step 4: According to formula (23), each characteristic variable is independent and satisfies the Dirichlet distribution of Equation (21), and the probability of the n+1th characteristic value in D falling on the left half plane can be predicted.Step 5: Establish a mathematical model according to the two equilibrium node voltage Equations (7) and (8):39İn+1=Vİn\begin{equation}{\dot{I}}_{n + 1} = V{\dot{I}}_n\end{equation}The matrix V is transformed into the following non‐singular diagonal matrix form VΛ. According to Theorem 1, judge whether or not the equilibrium point of the system is asymptotically stable.Step 6: Choose the Lyapunov function (35) to prove that the system probability is asymptotically stable. According to the method of Lyapunov discrete system, judge whether or not Ωm is positive definite, that is whether the probability system is asymptotically stable.The schematic diagram of the entire algorithm operation is shown in Figure 1.1FIGUREFlow chart of the proposed algorithmSIMULATIONSystem modelThe wind farm is connected to the IEEE New England 10‐machine 39‐node system [15], and PMUs are installed at each node, which is shown in Figure 2. The system voltage level is 345 kV, the frequency is 60 Hz, and the sampling frequency is 3 kHz. The load of each bus in the system is set to a constant power model. Consider the random fluctuation of wind speed, and then analyse the probability of small disturbance in the system. PMUs are added at node 28 and node 29, the branch is separated, and multiple DFIG wind turbines are equivalent to the wind farm. The wind farm has a capacity of 60 MW and consists of 45 double‐fed induction wind turbines with a rated power of 1.5 W.2FIGUREIEEE New England 10‐machine 39‐node system with a wind farmBy actively changing the output of the wind turbines to simulate the change of the external wind speed, and the system entering a new equilibrium state, the PMU data measured by the system are analysed, and the PMU characteristic data are extracted as the sample space. Using the RF method, according to the important score of each eigenvalue, the positive anomaly data are classified as shown in Figure 3. The blue areas are normal data and denoted as Ω1, and the red areas are abnormal data and denoted as Ω2.3FIGUREClassification results of PMU feature dataFrom Figure 3, the sample set is randomly generated and the segmentation surfaces are approximately circular and square. Compared with the segmentation of the sample space between Naive Bayes and RF, from the perspective of accuracy rate, it can be seen that RF is better than a single decision tree of this test set. It can be seen intuitively from the feature space that RF has stronger segmentation ability than the naive Bayes’.Statistical characteristics and probability distribution curves of system eigenvalues under different wind turbine outputsAccording to the normal data obtained, the statistical characteristics (expectation and standard deviation) of the probability distribution of the system eigenvalues are obtained through simulations, which are shown in Table 1.1TABLEStatistical characteristics of the probability distribution of system eigenvaluesReal partImaginary partWind turbine output powerMathematical expectationsStandard deviationMathematical expectationsStandard deviation15 MW30 MW45 MW60 MW−0.98720.01982.1520.0192−0.90270.08822.1840.1852−0.82810.21461.5910.2192−0.70440.22461.7130.303For the above system, stable calculations of small disturbance are performed. Take the real part of the eigenvalues as an example, the Monte Carlo simulation of 11,700 random samples are as the accurate values. Compared with the method of calculating the posterior probability density by the prior probability according to the method in this paper, based on the last feature data processing, two different methods fitting the real probability density curves of the eigenvalues are shown in Figure 4. When the wind turbine output is 45 MW operation mode, calculate the PDF of the real part of the eigenvalue and the probability of falling in the stable interval (negative real part), respectively.4FIGUREThe probability density of the real part of eigenvaluesFrom the probability density curves of the real part of the eigenvalue by the 45 MW wind turbine output in Figure 4, it can be found that in this equilibrium state, whether it is Monte Carlo probability density or Bayesian posterior probability density, the probability density distribution of the real part of the eigenvalue can have the probability of a positive real part. However, it can be clearly seen that the probability of a positive real part of the eigenvalues of Bayesian posterior probability density is much lower than Monte Carlo probability density, so the accuracy of analysing the probabilistic stability of small disturbance by the method of Bayesian posterior probability density will be higher.In Figures 5 and 6, according to the accuracy curves of the negative real part of the eigenvalue and the probability loss curves of the abnormal eigenvalue under different wind speeds over 60 times, it can be seen that with the increase of the eigenvalue, the probability of the negative real part of the eigenvalue appears gradually increasing. When the number of eigenvalues increases from 0 to about 500, the stability probability increases rapidly to 60%, and the number of losses of abnormal eigenvalues also decreases rapidly to about 1. It can be seen that the proposed method is highly computationally efficient. When the number of eigenvalues increases from about 500 to 1400, the stability probability gradually increases to 70%. During the process of the number of eigenvalues increasing to 11,700, the stability probability will increase steadily starting from 70%. It can be read from Figure 5 that the final stability probability is about 81.85% when the number of eigenvalues is 11,700 at that time. Correspondingly, the loss of abnormal eigenvalues gradually decays to around 0.5. It can be concluded that the proposed method is stable and reliable for processing a large number of wide‐area time series PMU data.5FIGUREThe accuracy of eigenvalues falls on the left half plane6FIGURELoss curves of abnormal eigenvaluesComparative experimentIn this part, the prediction method of Monte Carlo simulation probability based on the Markov chain is used as a comparative simulation method. It can be seen from Figure 7 that the probability of its eigenvalue falling on the negative real part fluctuates around 45%, and its stability probability fluctuates drastically. The system probabilistic stability is much lower than that used by the method proposed in this paper. The result shows that the method proposed in this paper is more in line with the description of the probabilistic stability of small disturbance when the number of wide‐area time series PMU data is large.7FIGUREMonte Carlo method based on the Markov chain predicts the probability that eigenvalues have negative real partsCONCLUSIONIn this paper, aiming at the probabilistic stability of small disturbance in power system, a probabilistic stability method of variational Bayes based on the PMU data and Lyapunov theory is proposed. In view of the existed probabilistic stability analysis method of small disturbance, it is limited to the problem of the low calculation efficiency when the number of wide‐area time series PMU data is large. The PMU feature data processing based on RF is proposed to filter out the normal data in order to improve the efficiency of the variational Bayes and calculating the probability of its eigenvalue falling on the left half plane. Then, from the Bayesian probabilistic perspective, considering the time‐space correlation in the connected power sources will affect the probabilistic stability of the wind power system, a system asymptotic stability theorem based on the Lyapunov theory is proposed, and also its proof procedure is given. It proves that the probability of wind power system is asymptotically stable. Finally, it is predicted to make the system enter the next equilibrium state, and the probabilistic stability of the system is close to 81.85%. Compared with the traditional Monte Carlo algorithm based on the Markov chain, the approximation prediction of the probabilistic stability of small disturbance has a significant improvement.AUTHOR CONTRIBUTIONSM.Y.: Funding acquisition; Supervision; Validation; Writing – review & editing. J.L.: Investigation; Methodology; Software; Visualization; Writing – original draft. S.Z.: Formal analysis; Methodology; Validation; Writing – review & editingACKNOWLEDGEMENTSThis work was supported by the Pyramid Talent Training Project of Beijing University of Civil Engineering and Architecture (JDYC20200324); Post Graduate Education and Teaching Quality Improvement Project of Beijing University of Civil Engineering and Architecture (J2022007); BUCEA Post Graduate Innovation Project (PG2022132); Security Control and Simulation for Power System and Large Power Generation Equipment (SKLD20M17); National Innovation and Entrepreneurship Training Program for College Students (202110016052,S202110016122,S202110016123); Project of Beijing Association of Higher Education (YB2021131); National Natural Science Foundation of China (51407201); Education, Teaching and Scientific Research Project of China Construction Education Association (2021051).CONFLICT OF INTERESTThe authors declare no conflict of interest.DATA AVAILABILITY STATEMENTData sharing not applicable to this article as no datasets were generated or analysed during the current study.REFERENCESBurchett, R.C., Heydt, G.T.: Probabilistic methods for power system dynamic stability studies. 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IET Radar, Sonar Navig. 13(8), 1389–1399 (2019)Xiaoyang, T., Zhang, S., Zhang, G.: Wide‐area backup protection algorithm based on regional phasor measurement unit and nadal fault‐injection current. Autom. Electr. Power Syst. 45(15), 158–165 (2021)Xiaoli, L., Xianghui, Z., Shuaidong, Z., Haotian, Z.: An FCM clustering method based on PMU data feature extraction. Eng. J. Wuhan Univ. 54(03), 232–238 (2021)Mazenc, F., Ito, H., Pepe, P.: Construction of Lyapunov functionals for coupled differential and continuous time difference equations. In: The 52nd IEEE Conference on Decision and Control. Florence, Italy, pp. 2245–2250 (2013)Yang, F., Piao, P., Lai, Y., Pei, L.: Margin based permutation variable importance:a stable importance measure for random forest. In: The 12th International Conference on Intelligent Systems and Knowledge Engineering (ISKE). Nanjing, China, pp. 1–8 (2017) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "IET Generation, Transmission & Distribution" Wiley

Probabilistic stability of small disturbance in wind power system based on a variational Bayes and Lyapunov theory using PMU data

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Abstract

INTRODUCTIONWith the explosive growth of China's total electricity demand and the rapid development of environmental protection technology, energy transition has become a hot research issue so far. New energy, such as wind, solar, and photovoltaic power generation, is the key indicator of China's energy transition. Wind power in the new energy power generation accounts for an important proportion. Due to the uncertain factors of wind speed changing, the output power of wind turbines has the characteristics of fluctuation, randomness, and uncontrollability. Stability and safety will also be affected, so the impact of wind power integration on power system stability of small disturbance has gradually become the attention focus of scholars.During the early research on the small disturbance stability in power system with variable‐speed wind turbines, most of the references were studied by calculating electromechanical oscillation modes of the power system. Firstly, Burchett [1] applied a probability analysis method to the power system with small disturbance stability in 1978. By deriving the average value and the covariance matrix of eigenvalue changing, and according to the relationship between the sensitivity of eigenvalues and the uncertain parameters in the adjoint matrix, the probability that the real parts of all eigenvalues were less than zero was calculated, which established a precedent for the study of the probabilistic stability of small disturbance in power system from the characteristic root. Chen Zhong [2] proposed an analysis method for probabilistic stability of small disturbance considering the influence of random distribution delay, which could quickly calculate the probabilistic stability, and had better effects than the Monte Carlo method. The method for studying the probabilistic stability of the AC‐DC hybrid system in multiple operation modes was used in [3]. Compared with the original AC system, the integration of the DC system of the power grid improved the local oscillation, but worsened the inter‐area oscillation. The proposed probabilistic method by Samundra Gurung et al. [4] was applied to analyse the effect of PV uncertainties arising mainly due to stochastic PV fluctuations and forecast error of small signal stability. The developed framework could provide accurate results to assess PSSS in much less computational time compared with conventional MCS and other analytical techniques.In order to improve the computational efficiency, many scholars have introduced Gram‐Charlier sequence expansion method to probability calculations. Although this method had high computational efficiency, it inevitably brought various mathematical approximations and complex calculations. Therefore, combined with the advantage of the Monte Carlo and analytical method, researchers proposed an approximate method called the Point Estimation Method (PEM), which was an effective method for estimating the normal distribution. The eigenvalues by inputting variable statistics were directly estimated. Especially, expectation and variance had successfully applied to the research on the probabilistic stability of small disturbance in power system [5]. To analyse the probabilistic stability of small disturbance in the three‐machine system, an algorithm of the probabilistic stability analysis of small disturbance based on the discrete point estimation method was proposed in ref. [6]. This method had the characteristics of high precision and high efficiency. However, the interaction among multiple wind farms connected to the grid was not considered. Aiming at the computational efficiency for the uncertainty handling in wind power system, methods of Information‐Gap Decision Theory (IGDT) and Taguchi's Orthogonal Array Testing (TOAT) were proposed in ref. [7]. Using each of these methods, the robust expansion plan for the modified 6‐bus Garver transmission network system was calculated. Furthermore, different uncertainty types could be easily considered in this regard. However, it was proved to be unreasonable from the point of view of mathematical statistic. In some cases, they would bring even wrong judgments of significance factor and lacked some assurances to deal with the effects of interactions. A probabilistic analysis method for small‐signal stability of power systems considering random uncertainty introduced by multiple grid‐connected wind power sources was proposed by Bu [8]. Compared with the non‐analytical method of Monte Carlo simulation, it was very computationally efficient. However, as the penetration rate of wind power increases, the probabilistic instability also increased accordingly. In order to solve the uncertainty of stability margin of the multi‐feed system caused by the randomness of wind power generation, a method was proposed to evaluate the stability margin of the probabilistic stability of small disturbance in electronic power system based on the generalized short‐circuit ratio in ref. [9]. The proposed method has good robustness to probability distribution. Aiming at the power system composed of multiple renewable energy sources, a Probabilistic Small‐Signal Stability Analysis (PSSSA) method based on a Probabilistic Allocation Method (PCM) for wind farms and photovoltaic power generation was proposed in ref. [10]. This method satisfied the accuracy and reduced the computation, but it could have other adverse effects on small‐signal stability when reducing and closing some synchronous units to make the penetration of renewable energy increasing.By deeply studying the above representative papers, the current Monte Carlo methods, analytical methods and approximate methods are all based on the mathematical modelling analysis of the power system. However, each algorithm has its own limitation to applications: Monte Carlo method needs to increase the number of deterministic power flow calculations, which will reduce the calculation efficiency of the algorithm. Analytical rules require complex convolution calculations and the series expansion. The approximation method is difficult to construct the Probability Density Function (PDF) and accurately describe the probability and statistics characteristics when the output random variable is non‐normal distribution. For the above methods, it is hard to solve the problem of computational efficiency when the amount of PMU data is huge. Therefore, the contribution of this paper is to construct a probabilistic analysis method to fully consider various uncertain factors in power system. This method is based on a variational Bayes and Lyapunov theory using PMU data, and introduces the variational Bayesian framework to the data as an association algorithm, which will not only improve the computational efficiency, but also obtain the PDF of the approximation posterior. In order to improve the efficiency of variational Bayesian calculation of the probability whose eigenvalues will fall on the left half‐plane, an RF‐based PMU data processing is proposed to filter out the normal data. Finally, a Bayesian probability based on Lyapunov theory is proposed to analyse the probabilistic stability of small disturbance.The structure of this paper is organized as follows. Section 2 is the basic theory. Section 3 is the PMU data processing based on the Random Forest. Section 4 is the Lyapunov theory of probabilistic stability of small disturbance based on the variational Bayes. Section 5 is the simulation verification, and finally is the conclusion.FUNDERMENTAL THEORYRandom forest theoryRandom forest (RF) [11] is a supervised classification method combining multiple decision trees. It can handle large‐scale data and has high accuracy even when the data are missing. The realization process is relatively simple, and the training speed is fast.There are often hundreds of features in the data set, so it is necessary to reduce eigenvalues when mathematical modelling. How much contribution for each feature of each tree in RF is compared at first, then it will be taken an average, and finally the contribution between eigenvalues is acquired. VIM is used to represent the variable importance measures, and GI is represented as the Gini Index. Assuming there are m features X1, X2, X3,…, Xm, then calculate the GI of each feature Xj score VIMj (Gini), that is, the average change is in the impurity of node splitting of the j‐th feature in all decision trees of RF.Calculate the GI, we can get1GIm=∑k=1|K|∑k′≠kpmkpmk′=1−∑k=1|K|p2mk′\begin{equation}GIm = \sum_{k = 1}^{|K|} {\sum_{k^{\prime} \ne k} {pmkpmk^{\prime}} } = 1 - \sum_{k = 1}^{|K|} {{p}^2mk^{\prime}} \end{equation}In (1), K represents the category, and pmk is the proportion of node m in category K.The calculation formula for the change in GI before and after node m is as follows:2VIMjm(Gini)=GIm−GIl−GIr\begin{equation}VIMj{m}^{(Gini)} = GIm - GIl - GIr\end{equation}In (2), GIl and GIr are Gini indexes of nodes before and after node m.The core of RF is an ensemble operation method based on decision trees. It can also be said to be a tree‐like classifier that selects the most appropriate attribution from internal nodes. Each leaf node contains the data of the same kind of attributions.Variational BayesVariational Bayes [12] is a Bayesian estimation method using an approximate complex integral. It is used mainly in some complex statistical models. The main purpose is to approximate the posterior probability of unobservable variables, so as to make statistical inferences through these variables. For this purpose, the Monte Carlo simulation uses the Markov Chain Monte Carlo (MCMC) algorithm [13] with Gini sampling to approximate complex posterior distributions, which can be well applied to Bayesian statistical inference. However, this method estimates the true posterior through a large number of samples, and the approximation result has a certain degree of randomness. The difference is that variational Bayes uses the local optimality and has a determined solution to maximize a posteriori probability estimation that replaces the complete Bayesian estimation with a single most likely parameter value, and the optimal solution is obtained by continuous iteration through a set of mutual equations. Therefore, this paper uses a single most likely eigenvalue instead of the Bayesian estimation to predict the probability of its eigenvalue falling into a stable interval.For any fixed value x∈[x0, x1], the difference y(x)−y0(x) between the available function y(x) and another available function y0(x) is called the function y(x) in y0(x). The variation at the position or the variation of the function is denoted as δy, then we have,3δy=y(x)−y0(x)=εη(x)\begin{equation}{\delta }_{y} = y( x ) - {y}_0( x ) = \epsilon \eta ( x )\end{equation}δy is a very small number, and η(x) is any parameter of x. The increment of the functional J[y(x)] is:4ΔJ=J[y(x)+δy]−J[y(x)]=δJ+o(δy)\begin{eqnarray} \Delta J &=& J[y( x ) + {\delta }_y] - J[y( x )]\nonumber\\ &=& {\delta }_J + o( {{\delta }_y} ) \end{eqnarray}The difference between the functional increment ΔJ and the variation δJ is an infinitesimal higher order than the first‐order distance, and the variation of the function is the main linear part of the function increment. So there is the following theory [14]: If the function J[y(x)] reaches its extreme value on y = y(x), then its variation δJ on y = y(x) is equal to zero, which is called a variation.Suppose that data x in a model are generated according to the probability distribution of the unknown parameter θ, and there is a priori knowledge about the parameter θ, which can be represented by the probability distribution p(θ). Then, when the data x is observed, we can use Bayesian theorem to update the prior knowledge about the parameter, which is as shown in Equation (5):5P(θ|x)=P(x|θ)P(θ)P(x)\begin{equation}P(\theta |x) = \frac{{P(x|\theta )P(\theta )}}{{P(x)}}\end{equation}The most complicated part of the posterior distribution calculation is to calculate the normalization factor:6p(x)=∫θp(x|θ)p(θ)dθ\begin{equation}p(x) = \int\limits_{\theta }{{p(x|\theta )}}p(\theta )d\theta \end{equation}The integral in Equation (6) can be calculated in low dimensions. It is not feasible to accurately calculate the posterior distribution in high dimensions. Some approximation techniques should be used to obtain the posterior distribution. Therefore, the Variational Inference (VI) method is introduced.PMU DATA PROCESSING BASED ON THE RANDOM FORESTPMU data processingSystem modelAccording to the idea of the node voltage equation [15], assuming that the number of nodes in a certain area of the power system with DFIG is n, then the mathematical model of the node voltage in this area is:7İn=U̇n⋅Fnn⇒İ1⋮İi⋮İj⋮İn0=U̇1⋮U̇i⋮U̇j⋮U̇n0•F11⋯F1i⋯F1j⋯F1n0⋮⋱⋮⋱⋮⋱⋮0Fi1⋯Fii′⋯Fij′⋯Fin0⋮⋱⋮⋱⋮⋱⋮0Fj1⋯Fji′⋯Fjj′⋯Fjn0⋮⋱⋮⋱⋮⋱⋮0Fn1⋯Fni⋯Fnj⋯Fnn000000000\begin{eqnarray} \def\eqcellsep{&}\begin{array}{l} {\rm{ }}\mathop {{{\dot{\bf I}}}}\nolimits_n = \mathop {{\rm{ }}{{\dot{\bf U}}}}\nolimits_n \mathop {\centerdot {{\bf F}}}\nolimits_{nn} \Rightarrow \\[6pt] \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{{\dot{I}}}_1}\\[6pt] \vdots \\[6pt] {{{\dot{I}}}_i}\\[6pt] \vdots \\[6pt] {{{\dot{I}}}_j}\\[6pt] \vdots \\[6pt] {{{\dot{I}}}_n}\\[6pt] 0 \end{array} } \right] = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{{\dot{U}}}_1}\\[6pt] \vdots \\[6pt] {{{\dot{U}}}_i}\\[6pt] \vdots \\[6pt] {{{\dot{U}}}_j}\\[6pt] \vdots \\[6pt] \def\eqcellsep{&}\begin{array}{l} {{\dot{U}}}_n\\[6pt] 0 \end{array} \end{array} } \right] \bullet \left[ { \def\eqcellsep{&}\begin{array}{@{}*{8}{c}@{}} {{F}_{11}}&\quad \cdots &\quad{{F}_{1i}}&\quad \cdots &\quad{{F}_{1j}}&\quad \cdots &\quad{{F}_{1n}}&\quad 0\\[6pt] \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad 0\\[6pt] {{F}_{i1}}&\quad \cdots &\quad{{F}^{\prime}_{ii}}&\quad \cdots &\quad{{F}^{\prime}_{ij}}&\quad \cdots &\quad{{F}_{in}}&\quad 0\\[6pt] \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad 0\\[6pt] {{F}_{j1}}&\quad \cdots &\quad{{F}^{\prime}_{ji}}&\quad \cdots &\quad{{F}^{\prime}_{jj}}&\quad \cdots &\quad{{F}_{jn}}&\quad 0\\[6pt] \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad 0\\[6pt] {{F}_{n1}}&\quad \cdots &\quad{{F}_{ni}}&\quad \cdots &\quad{{F}_{nj}}&\quad \cdots &\quad{{F}_{nn}}&\quad 0\\[6pt] 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0 \end{array} } \right] \end{array} \nonumber\\ \end{eqnarray}where i,j = 1,2,…, n are node numbers; Fnn is the node admittance matrix; Fij and Fji are mutual admittances between nodes i and j; Fii and Fjj are self‐admittances of nodes i and j, respectively;U̇n${\dot{U}}_n$andİn${\dot{I}}_n$represent node voltage and node injection current;U̇i,U̇j${\dot{U}}_i,{\dot{U}}_j$andİi,İj${\dot{I}}_i,{\dot{I}}_j$are voltage and injection current of nodes i and j, respectively.Due to the influence of wind speed, the equilibrium state of the wind power system will fluctuate over a large range. Set the injection current of the previous equilibrium state ƒ to 0, and add the node ƒ to the admittance matrix. When entering the next equilibrium state, the current change is ΔIf, and the voltage equation of the area with n+1 nodes is:8İn+1=U̇n+1•F(n+1)(n+1)=İ1⋮İi⋮İj⋮İnΔIf=U̇1⋮U̇i⋮U̇j⋮U̇nU̇f•F11⋯F1i⋯F1j⋯F1n0⋮⋱⋮⋱⋮⋱⋮0Fi1⋯Fii′⋯Fij′⋯FinFif′⋮⋱⋮⋱⋮⋱⋮0Fj1⋯Fji′⋯Fjj′⋯FjnFjf′⋮⋱⋮⋱⋮⋱⋮0Fn1⋯Fni⋯Fnj⋯Fnn000Ffi′0Ffj′00Fff′\begin{eqnarray} \def\eqcellsep{&}\begin{array}{l} \mathop {\dot{I}}\nolimits_{n + 1} = \mathop {\dot{U}}\nolimits_{n + 1} \bullet \mathop F\nolimits_{(n + 1)(n + 1)} = \\[6pt] \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{{\dot{I}}}_1}\\[6pt] \vdots \\[6pt] {{{\dot{I}}}_i}\\[6pt] \vdots \\[6pt] {{{\dot{I}}}_j}\\[6pt] \vdots \\[6pt] {{{\dot{I}}}_n}\\[6pt] {\Delta {I}_f} \end{array} } \right] = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{{\dot{U}}}_1}\\[6pt] \vdots \\[6pt] {{{\dot{U}}}_i}\\[6pt] \vdots \\[6pt] {{{\dot{U}}}_j}\\[6pt] \vdots \\[6pt] \def\eqcellsep{&}\begin{array}{l} {{\dot{U}}}_n\\[6pt] {{\dot{U}}}_f \end{array} \end{array} } \right] \bullet \left[ { \def\eqcellsep{&}\begin{array}{@{}*{8}{c}@{}} {{F}_{11}}&\quad \cdots &\quad{{F}_{1i}}&\quad \cdots &\quad{{F}_{1j}}&\quad \cdots &\quad{{F}_{1n}}&\quad 0\\[6pt] \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad 0\\[6pt] {{F}_{i1}}&\quad \cdots &\quad{{F}^{\prime}_{ii}}&\quad \cdots &\quad{{F}^{\prime}_{ij}}&\quad \cdots &\quad{{F}_{in}}&\quad{{F}^{\prime}_{if}}\\[6pt] \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad 0\\[6pt] {{F}_{j1}}&\quad \cdots &\quad{{F}^{\prime}_{ji}}&\quad \cdots &\quad{{F}^{\prime}_{jj}}&\quad \cdots &\quad{{F}_{jn}}&\quad{{F}^{\prime}_{jf}}\\[6pt] \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad \ddots &\quad \vdots &\quad 0\\[6pt] {{F}_{n1}}&\quad \cdots &\quad{{F}_{ni}}&\quad \cdots &\quad{{F}_{nj}}&\quad \cdots &\quad{{F}_{nn}}&\quad 0\\[6pt] 0&\quad 0&\quad{{F}^{\prime}_{fi}}&\quad 0&\quad{{F}^{\prime}_{fj}}&\quad 0&\quad 0&\quad{{F}^{\prime}_{ff}} \end{array} } \right] \end{array} \nonumber\\ \end{eqnarray}F'if, F'jf, F'fi, F'fj and F'ff are newly introduced admittance elements after the joining node ƒ; F'ii, F'jj, F'ji and F'ij are admittance elements that change after the node ƒ is added; Uƒ is the node voltage at the previous ƒ equilibrium point.PMU feature data extractionIt can be seen from formula (8) that when the equilibrium state changes, the admittance element of the regional node voltage equation of the wind power system will also change accordingly. According to the eigenvalue calculation formula, there is:9λiI−F(n+1)(n+1)=0\begin{equation}\left| {{\lambda }_i{{\bf I}} - \mathop F\nolimits_{(n + 1)(n + 1)} } \right| = 0\end{equation}It can be calculated that its eigenvalue λi = xi+jyi will also change immediately, where xi and yi are coordinates of the eigenvalue on the imaginary axis.For an n‐dimensional random matrix F(Fi,j), the trace of the matrix is defined as:10trF=∑i=1nFi,i=∑i=1n|λi|,(i=1,2,3,…,n)\begin{equation}trF = \sum_{i = 1}^n {{F}_{i,i}} = \sum_{i = 1}^n {|{\lambda }_i|} ,(i = 1,2,3, \ldots ,n)\end{equation}Fi,i is the main diagonal element of matrix F; λi​ is the eigenvalue of matrix F. It can be seen from Equation (10) that the sum of all eigenvalues is called the matrix trace, which can reflect the statistical characteristics of the matrix F, so the trace of the matrix can be used to reduce the amount of calculation.The statistics of linear eigenvalues can be defined as:11P(μ)=∑i=1nμ(λi),(i=1,2,3,…,n)\begin{equation}P(\mu ) = \sum_{i = 1}^n {\mu ({\lambda }_i)} ,(i = 1,2,3, \ldots ,n)\end{equation}μ(λi) is the test function. Then we can use Linear Eigenvalue Statistical (LES) [16] indicators to extract features from the PMU data matrix. Finally, based on the extracted PMU feature data matrix, the RF algorithm is used to classify the feature value data. At the same time, the normal data are screened out, so as to obtain the probability of its eigenvalue falling on the left half plane according to the variational Bayes.PMU data classificationBy analysing the changing trend of the measured data of the PMU, and the PMU data measured in a period of time can be used as the key wide‐area data. Such data are relatively easy to obtain in the actual operation of the system.Consider the regional node voltage equation of the system at the n+1 node, the eigenvalueλi=λ1,λ2,λ3,…,λnk+1${\lambda }_i = {\lambda }_1,{\lambda }_2,{\lambda }_3, \ldots ,{\lambda }_{{n}_{k + 1}}$ at the equilibrium state of k+1 can be calculated. nk+1 is the total number of eigenvalue samples of the PMU data set, and the normal data in the measured data processed by the PMU data are treated as the related data. The abnormal data are treated as the unrelated data.12pΩλi=pΩ1|λiPMUabnormaldatapΩ2λiPMUnormaldata\begin{equation}p\left( {\left. {{\Omega }} \right|{\lambda }_i} \right) = \left\{ \def\eqcellsep{&}\begin{array}{l} p\left( {{\Omega }_1|{\lambda }_i} \right) {\rm{PMU abnormal data}}\\[6pt] p\left( {\left. {{\Omega }_2} \right|{\lambda }_i} \right){\rm{ PMU normal data}} \end{array} \right.\end{equation}In (12), Ω1 is normal data, and Ω2 is abnormal data. However, in the case of the large number of PMU fundamental currents, they are node voltage data. As the process of updating the state continues, the Gaussian component of the traditional method calculating the posterior PDF will increase geometrically. This will lead to excessive computational complexity and the low computational efficiency. Therefore, the posterior PDF needs to be introduced into the Bayesian framework to obtain the state approximate posterior PDF, which will provide the convenience in the subsequent state updating process.PMU data classification based on random forestThis paper uses the Gini index as a contribution evaluation index [14]. In this case, VIM is represented by W(Gini), and the Gini index is represented by G. Calculate the Gini index score Wi corresponding to each eigenvalue λi. That is, the average change in the impurity of the node split of the i‐th feature is in all decision trees of RF.The calculation formula [17] of the Gini index is as follows:13Gm=∑k=1|K|∑k′≠kpmkpmk′=1−∑k=1|K|p2mk′\begin{equation}Gm = \sum_{k = 1}^{|K|} {\sum_{k^{\prime} \ne k} {pmkpmk^{\prime}} } = 1 - \sum_{k = 1}^{|K|} {{p}^2mk^{\prime}} \end{equation}where K indicates that there are K categories; pmk indicates the proportion of category k in node m.The importance of eigenvalue λi at node m, that is, the change in the Gini index before and after the node m branch is:14Wim(Gini)=Gm−Gl−Gr\begin{equation}Wi{m}^{(Gini)} = Gm - Gl - Gr\end{equation}Gl and Gr represent Gini indexes of two new nodes after branching.If the node of eigenvalue λi in decision tree j is set M, then the importance of λi in the j‐th tree is15Wim(Gini)=∑m∈MWim(Gini)\begin{equation}Wi{m}^{(Gini)} = \sum_{m \in M} {Wi{m}^{(Gini)}} \end{equation}Assuming that there are q trees in RF, then we have, 16Wim(Gini)=∑j=1qWim(Gini)\begin{equation}Wi{m}^{(Gini)} = \sum_{j = 1}^q {Wi{m}^{(Gini)}} \end{equation}Finally, normalize all the obtained importance scores, then we can get,17Wi=Wi∑j=1cWj\begin{equation}Wi = \frac{{Wi}}{{\sum\nolimits_{j = 1}^c {Wj} }}\end{equation}According to the importance score obtained by Equation(14), all eigenvalues are classified by using Equation (17). The eigenvalues whose distance from the coordinate origin are less than τ are recorded as the normal data Ω1, and the remaining eigenvalues as abnormal data are recorded as Ω2.A VARIATIONAL BAYES AND LYAPUNOV THEORY OF PROBABILITY STABILITY OF SMALL DISTURBANCEBayesian probability and stability analysisAssume that the eigenvalues obtained from the voltage equation of the system in the area of n nodes are used as experimental samples, the probability Φk${\Phi }_k$ of the occurrence of m+1 events can be predicted according to the probability Equation (6).18P(Ym+1=ym+1D,ε)=∫P(ωD,ε)P(Ym+1=ym+1ω,ε)dω=∫ωP(ωD,ε)dω=E[P(ωD,ε)]\begin{equation} \def\eqcellsep{&}\begin{array}{l} P(\left. {{Y}_{m + 1} = {y}_{m + 1}} \right|D,\varepsilon )\\[6pt] = \int{{P(\left. \omega \right|D,\varepsilon )P}}(\left. {{Y}_{m + 1} = {y}_{m + 1}} \right|\omega ,\varepsilon )d\omega \\[6pt] = \int{{\omega P(\left. \omega \right|D,\varepsilon )d\omega }}\\[6pt] = E[P(\left. \omega \right|D,\varepsilon )] \end{array} \end{equation}where D is an experimental sample. Y is the event variable; y is the value of the state variable; The Bayesian probability of event Ym+1 is the expected value of the prior probability ω relative to the posterior probability. According to the Bayesian equation, the equation for calculating the posterior probability density from the prior probability is:19P(ωD,ε)=P(ωε)P(D|ε)P(Dω,ε)=P(ω|ε)∫P(D|ω,ε)P(ωε)dωP(Dω,ε)\begin{eqnarray} P(\left. \omega \right|D,\varepsilon ) &=& \frac{{P(\left. \omega \right|\varepsilon )}}{{P(D|\varepsilon )}}P(\left. D \right|\omega ,\varepsilon )\nonumber\\ && = \frac{{P(\omega |\varepsilon )}}{{\int{{P(D|\omega ,\varepsilon )P(\left. \omega \right|\varepsilon )}}d\omega }}P(\left. D \right|\omega ,\varepsilon )\end{eqnarray}where ω is the prior probability of the event occurrence; P is the PDF; ε is the prior knowledge of the observer.Assume that y = y1, y2,…, yr have r possible states, Ai is the number of the eigenvalues falling on the left half plane in the sample space, and it is denoted as event y = yi, then the Dirichlet distribution of the prior probability is:20P(ωε)=Dir(ω|ζ1,ζ2,…,ζr)=∏ωkζk−1∏k=1rψ(ζk)ψ(ζ)\begin{equation} \def\eqcellsep{&}\begin{array}{l} P(\left. \omega \right|\varepsilon ) = Dir(\omega |{\zeta }_1,{\zeta }_2, \ldots ,{\zeta }_r)\\[6pt] {\rm{ }} = \dfrac{{\prod {\omega _k^{{\zeta }_k - 1}} }}{{\prod\limits_{k = 1}^r {\psi ({\zeta }_k)} }}\psi (\zeta ) \end{array} \end{equation}whereζk=P(y=yk|ω,ε),k=1,2,…,r${\zeta }_k = P( {y = {y}^k} |\omega ,\varepsilon ),k = 1,2, \ldots ,r$,ζ=∑k=1rζk$\zeta = \sum_{k = 1}^r {{\zeta }_k} $, andζk > 0, k = 1,2,…, r. The PDF is:21P(ωnk+1,ε)=Dir(ω|ζ1+A1,ζ2+A2,…,ζr+Ar)\begin{equation}P(\left. \omega \right|{n}_{k + 1},\varepsilon ) = Dir(\omega |{\zeta }_1 + {A}_1,{\zeta }_2 + {A}_2, \ldots ,{\zeta }_r + {A}_r)\end{equation}Substituting Equation (21) into Equation (18), the probability that the eigenvalue falling on the left half plane can be calculated as:22P(Ym+1=yknk+1,ε)=∫ωkDir(ω|ζ1+A1,ζ2+A2,…,ζr+Ar)dω=ζk+Akζ+A\begin{equation} \def\eqcellsep{&}\begin{array}{l} P(\left. {{Y}_{m + 1} = {y}^k} \right|{n}_{k + 1},\varepsilon )\\[12pt] = \int{{{\omega }_k}}Dir(\omega |{\zeta }_1 + {A}_1,{\zeta }_2 + {A}_2, \ldots ,{\zeta }_r + {A}_r)d\omega \\[12pt] = \dfrac{{{\zeta }_k + {A}_k}}{{\zeta + A}} \end{array} \end{equation}Now calculate the probability of n+1 eigenvalues falling on the left half plane when the system re‐enters a new equilibrium state.For the Bayesian network structure of the variable F = (F1,F2,…, Fn) in the state transition matrix in Equation (7), it includes a one‐to‐one mapping and a directed non‐direction composed of the eigenvalue λi in variable F. The ring graph O and the probability distribution set P are corresponding to their variables.First of all, each characteristic variable is independent, that is,23P(λO,D)=∏i=1nP(λi|O,D)\begin{equation}P(\left. \lambda \right|O,D) = \prod\limits_{i = 1}^n {P({\lambda }_i} |O,D)\end{equation}P(λi|O,D)$P( {{\lambda }_{^i}} |O,D)$is the prior probability density of the i‐th eigenvalue. Then Equation (21) of Dirichlet distribution is satisfied, and the probability whose the n+1th eigenvalue in D that is predicted to fall on the left half plane is:24Φk(λn+1)=p(λn+1D,O)=E(∏i=1nωijk)=∏i=1n∫ωijkp(ωs|D,O)dω=∏i=1nζijk+Aijkζij+Aij\begin{equation} \def\eqcellsep{&}\begin{array}{c} {\Phi }_k({\lambda }^{n + 1}) = p(\left. {{\lambda }^{n + 1}} \right|D,O) = E(\prod\limits_{i = 1}^n {{\omega }_{ijk}} )\\[6pt] = \prod\limits_{i = 1}^n {\int{{{\omega }_{ijk}p({\omega }_s}}|D,O)} d\omega \\[6pt] = \prod\limits_{i = 1}^n {\dfrac{{{\zeta }_{ijk} + {A}_{ijk}}}{{{\zeta }_{ij} + {A}_{ij}}}} \end{array} \end{equation}whereζij=∑k=1riζijk,Aij=∑k=1riAijk${\zeta }_{ij} = \sum_{k = 1}^{ri} {{\zeta }_{ijk}} ,{A}_{ij} = \sum_{k = 1}^{ri} {{A}_{ijk}} $ and ζijk=P(λ=λijk|ω,ε),k=1,2,…,r${\zeta }_{ijk} = P( {\lambda = {\lambda }^{ijk}} |\omega ,\varepsilon ),k = 1,2, \ldots ,r$.System asymptotically stable based on the Lyapunov theoryConsidering that the time and space correlation between different connected power supplies will affect the probability and stability of the wind power system, the time and space correlation of the power supplies are closely related to their geographical locations according to a certain time difference. Assume that at different times, the correlation parameter between two power supplies that are more than 100 km together is 0, and the correlation parameter between two power supplies that are less than 100 km together is 1. Then, the spatiotemporal correlation coefficient can reflect the geographical distance among the power supplies at different time, and m grid‐connected wind power supplies can be constructed. Then, according to the Lyapunov theory, the proof that the asymptotically stable of the system in a wide range is as follows.First, prove that the system is asymptotically stable, and a mathematical model based on the two equilibrium node voltage equations of (7) and (8) is established as follows:25İn+1=Vİn\begin{equation}{\dot{I}}_{n + 1} = V{\dot{I}}_n\end{equation}where V is a non‐singular matrix of order n×n. The matrix V is transformed into the following non‐singular diagonal matrix and we can get VΛ:26VΛ=τ10⋯00τ2⋯0⋮⋮⋱0000τn\begin{equation}{V}_\Lambda = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{4}{c}@{}} {{\tau }_1}&\quad 0&\quad \cdots &\quad 0\\[6pt] 0&\quad{{\tau }_2}&\quad \cdots &\quad 0\\[6pt] \vdots &\quad \vdots &\quad \ddots &\quad 0\\[6pt] 0&\quad 0&\quad 0&\quad{{\tau }_n} \end{array} } \right]\end{equation}Theorem 1: Let the state equation of the discrete system be:27İ(k+1)=τ10⋯00τ2⋯0⋮⋮⋱0000τnİ(k)\begin{equation}\dot{I}(k + 1) = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{4}{c}@{}} {{\tau }_1}&\quad 0&\quad \cdots &\quad 0\\[6pt] 0&\quad{{\tau }_2}&\quad \cdots &\quad 0\\[6pt] \vdots &\quad \vdots &\quad \ddots &\quad 0\\[6pt] 0&\quad 0&\quad 0&\quad{{\tau }_n} \end{array} } \right]\dot{I}(k)\end{equation}Then the necessary and sufficient condition for the system to be asymptotically stable at the equilibrium point is that the characteristic root of the system lies in the unit circle.Proof: According to the Lyapunov stability theorem, the necessary and sufficient condition for the system to be stable at an equilibrium point is that for any given positive symmetric matrix Q, there exists a positive symmetric matrix P, which satisfies:28VTPV−P=−Q\begin{equation}{V}^TPV - P = - Q\end{equation}and29V(İ(k))=İT(k)Pİ(k)\begin{equation}V(\dot{I}(k)) = {\dot{I}}^{\rm{T}}(k)P\dot{I}(k)\end{equation}is the Lyapunov function of the system.Choose Q = I, I is the identity matrix, and substitute it into Equation (28), then we can get30τ10⋯00τ2⋯0⋮⋮⋱0000τnp11p12⋯p1np21p22⋯p2n⋮⋮⋱⋮pn1pn2⋯pnnτ10⋯00τ2⋯0⋮⋮⋱0000τn−p11p12⋯p1np21p22⋯p2n⋮⋮⋱⋮pn1pn2⋯pnn=−10⋯001⋯0⋮⋮⋱00001\begin{equation} \def\eqcellsep{&}\begin{array}{l} \left[ { \def\eqcellsep{&}\begin{array}{@{}*{4}{c}@{}} {{\tau }_1}&\quad 0&\quad \cdots &\quad 0\\[6pt] 0&\quad{{\tau }_2}&\quad \cdots &\quad 0\\[6pt] \vdots &\quad \vdots &\quad \ddots &\quad 0\\[6pt] 0&\quad 0&\quad 0&\quad{{\tau }_n} \end{array} } \right]\left[ { \def\eqcellsep{&}\begin{array}{@{}*{4}{c}@{}} {{p}_{11}}&\quad{{p}_{12}}&\quad \cdots &\quad{{p}_{1n}}\\[6pt] {{p}_{21}}&\quad{{p}_{22}}&\quad \cdots &\quad{{p}_{2n}}\\[6pt] \vdots &\quad \vdots &\quad \ddots &\quad \vdots \\[6pt] {{p}_{n1}}&\quad{{p}_{n2}}&\quad \cdots &\quad{{p}_{nn}} \end{array} } \right]\left[ { \def\eqcellsep{&}\begin{array}{@{}*{4}{c}@{}} {{\tau }_1}&\quad 0&\quad \cdots &\quad 0\\[6pt] 0&\quad{{\tau }_2}&\quad \cdots &\quad 0\\[16pt] \vdots &\quad \vdots &\quad \ddots &\quad 0\\[6pt] 0&\quad 0&\quad 0&\quad{{\tau }_n} \end{array} } \right]\\[6pt] {\rm{ }} - \left[ { \def\eqcellsep{&}\begin{array}{@{}*{4}{c}@{}} {{p}_{11}}&\quad{{p}_{12}}&\quad \cdots &\quad{{p}_{1n}}\\[6pt] {{p}_{21}}&\quad{{p}_{22}}&\quad \cdots &\quad{{p}_{2n}}\\[6pt] \vdots &\quad \vdots &\quad \ddots &\quad \vdots \\[6pt] {{p}_{n1}}&\quad{{p}_{n2}}&\quad \cdots &\quad{{p}_{nn}} \end{array} } \right] = - \left[ { \def\eqcellsep{&}\begin{array}{@{}*{4}{c}@{}} 1&\quad 0&\quad \cdots &\quad 0\\[6pt] 0&\quad 1&\quad \cdots &\quad 0\\[6pt] \vdots &\quad \vdots &\quad \ddots &\quad 0\\[6pt] 0&\quad 0&\quad 0&\quad 1 \end{array} } \right] \end{array} \end{equation}that is,31p11(1−λ12)=1p21(1−λ2λ1)=0⋮pn1(1−λnλ1)=0p12(1−λ1λ2)=0p22(1−λ22)=1⋮pn2(1−λnλ2)=0p23(1−λ2λ3)=0⋮pn(n−1)1−λnλn−1=0⋱⋯pnn(1−λn2)=1\begin{equation} \def\eqcellsep{&}\begin{array}{@{}*{5}{c}@{}} { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{p}_{11}(1 - {\lambda }_1^2) = 1}\\[15pt] {{p}_{21}(1 - {\lambda }_2{\lambda }_1) = 0}\\[15pt] \vdots \\[15pt] {{p}_{n1}(1 - {\lambda }_n{\lambda }_1) = 0} \end{array} }&{ \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{p}_{12}(1 - {\lambda }_1{\lambda }_2) = 0}\\[15pt] {{p}_{22}(1 - {\lambda }_2^2) = 1}\\[15pt] \vdots \\[15pt] {{p}_{n2}(1 - {\lambda }_n{\lambda }_2) = 0} \end{array} }&{ \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {}\\[15pt] {{p}_{23}(1 - {\lambda }_2{\lambda }_3) = 0}\\[15pt] \vdots \\[15pt] {{p}_{n(n - 1)}1 - {\lambda }_n{\lambda }_{n - 1} = 0} \end{array} }&{ \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {}\\[15pt] {}\\[15pt] \ddots \\[15pt] \cdots \end{array} }&{ \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {}\\[15pt] {}\\[15pt] {}\\[15pt] {{p}_{nn}(1 - {\lambda }_n^2) = 1} \end{array} } \end{array} \end{equation}From Equation (31), we have,32P=11−λ1211−λ22⋱11−λn2\begin{equation}P = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{4}{c}@{}} {\dfrac{1}{{1 - {\lambda }_1^2}}}&\quad{}&\quad{}&\quad{}\\[15pt] {}&\quad{\dfrac{1}{{1 - {\lambda }_2^2}}}&\quad{}&\quad{}\\[15pt] {}&\quad{}&\quad \ddots &\quad{}\\[15pt] {}&\quad{}&\quad{}&\quad{\dfrac{1}{{1 - {\lambda }_n^2}}} \end{array} } \right]\end{equation}To guarantee that matrix P is a positive symmetrical matrix, it requires to satisfy the following Equation (33):33|λ1|<1|λ2|<1⋮|λn|<1\begin{equation} \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {|{\lambda }_1| &lt; 1}\\[12pt] {|{\lambda }_2| &lt; 1}\\[15pt] \vdots \\[15pt] {|{\lambda }_n| &lt; 1} \end{array} \end{equation}In other words, when the characteristic root of the system is located in the unit circle, the equilibrium point of the system is asymptotically stable.Then we will prove that the system probability is asymptotically stable, and we let34Γk(λn+1)=[Φk1(λ),Φk2(λ),…,Φkn(λ)]T=G(n+1,n)Γk(λn)\begin{equation} \def\eqcellsep{&}\begin{array}{l} {\Gamma }_k({\lambda }^{n + 1})\\[15pt] = {[\Phi _k^1(\lambda ),\Phi _k^2(\lambda ), \ldots ,\Phi _k^n(\lambda )]}^{\rm{T}}\\[15pt] = G(n + 1,n){\Gamma }_k({\lambda }^n) \end{array} \end{equation}where G(n+1,n)$G(n + 1,n)$ is the transition matrix, and Φkn(λ)$\Phi _k^n(\lambda )$ represents the probability that the nth eigenvalue λn falls on the left half plane.We select the Lyapunov function as follows:35Vk(Φk(λn),n)=ΓkT(λn)ΩmΓk(λn)\begin{equation}{V}_k({\Phi }_k({\lambda }^n),n) = \Gamma _k^{\rm{T}}({\lambda }^n){\Omega }_m{\Gamma }_k({\lambda }^n)\end{equation}Then we take the Lyapunov first‐order difference equation [18] as:36ΔV(Φk(λn),n)=V(Φk(λn+1),n+1)−V(Φk(λn),n)=ΓkT(λn+1)ΩmΓk(λn+1)−ΓkT(λn)ΩmΓk(λn)=ΓkT(λn)GT(n+1,n)ΩmG(n+1,n)Γk(λn)−ΓkT(λn)ΩmΓk(λn)=ΓkT(λn)[GT(n+1,n)ΩmG(n+1,n)−Ωm]Γk(λn)\begin{equation} \def\eqcellsep{&}\begin{array}{l} \Delta V({\Phi }_k({\lambda }^n),n)\\[6pt] = V({\Phi }_k({\lambda }^{n + 1}),n + 1) - V({\Phi }_k({\lambda }^n),n)\\[6pt] {\rm{ = }}\Gamma _k^{\rm{T}}({\lambda }^{n + 1}){\Omega }_m{\Gamma }_k({\lambda }^{n + 1}) - \Gamma _k^{\rm{T}}({\lambda }^n){\Omega }_m{\Gamma }_k({\lambda }^n)\\[6pt] {\rm{ = }}\Gamma _k^{\rm{T}}({\lambda }^n){G}^T(n + 1,n){\Omega }_mG(n + 1,n){\Gamma }_k({\lambda }^n)\\[6pt] - \Gamma _k^{\rm{T}}({\lambda }^n){\Omega }_m{\Gamma }_k({\lambda }^n)\\[6pt] {\rm{ = }}\Gamma _k^{\rm{T}}({\lambda }^n)[{G}^T(n + 1,n){\Omega }_mG(n + 1,n) - {\Omega }_m]{\Gamma }_k({\lambda }^n)\end{array} \end{equation}For any positive definite symmetric matrix Q(n), according to the Lyapunov discrete system judgment method, there is:37GT(n+1,n)ΩmG(n+1,n)−Ωm=−Qn\begin{equation}{G}^T(n + 1,n){\Omega }_mG(n + 1,n) - {\Omega }_m = - Q\left( n \right)\end{equation}And then, we can find that:38Ωk=GT(0,n+1)ΩmG(0,n+1)−∑i=0nGT(i,n+1)Q(i)G(i,n+1)>0\begin{eqnarray} {\Omega }_k &=& {G}^T(0,n + 1){\Omega }_mG(0,n + 1)\nonumber\\ && - \sum_{i = 0}^n {{G}^T(i,n + 1)Q(i)G(i,n + 1)} &gt; 0\end{eqnarray}where Ωm is positive definite, so Vk(Φk(λn),n)${V}_k({\Phi }_k({\lambda }^n),n)$ is positive definite. According to the Lyapunov stability theorem of time‐varying discrete system, the system is asymptotically stable.Variational Bayesian probabilistic analysis based on Lyapunov theoryFrom the above theoretical reasoning and the proposed stability analysis method, this paper proposes the variational Bayesian probabilistic analysis based on the Lyapunov theory and the design process of probabilistic stability of small disturbance is as follows.Step 1: According to the mathematical model of the node voltage in a certain area, calculating its characteristic root and linear characteristic value statistics of the matrix trace, and performing the characteristic extraction on the PMU matrix by using the Linear Eigenvalue Statistical (LES) index.Step 2: Use the Gini index as a contribution evaluation index to measure the change in the Gini index of the eigenvalue λi before and after the branch of node m, which is as the importance score index, and finally normalize the characteristics according to the importance score. The eigenvalue is classified as the positive abnormal data.Step 3: The obtained normal eigenvalue data are used as the experimental sample D, and then the posterior probability densityp(ω|D,ε)$p( \omega |D,\varepsilon )$is calculated by the prior probability p(ω|ε)$p( \omega |\varepsilon )$, and the probability of the eigenvalue falling on the left half plane is calculated according to the Dirichlet distribution of the prior probability.Step 4: According to formula (23), each characteristic variable is independent and satisfies the Dirichlet distribution of Equation (21), and the probability of the n+1th characteristic value in D falling on the left half plane can be predicted.Step 5: Establish a mathematical model according to the two equilibrium node voltage Equations (7) and (8):39İn+1=Vİn\begin{equation}{\dot{I}}_{n + 1} = V{\dot{I}}_n\end{equation}The matrix V is transformed into the following non‐singular diagonal matrix form VΛ. According to Theorem 1, judge whether or not the equilibrium point of the system is asymptotically stable.Step 6: Choose the Lyapunov function (35) to prove that the system probability is asymptotically stable. According to the method of Lyapunov discrete system, judge whether or not Ωm is positive definite, that is whether the probability system is asymptotically stable.The schematic diagram of the entire algorithm operation is shown in Figure 1.1FIGUREFlow chart of the proposed algorithmSIMULATIONSystem modelThe wind farm is connected to the IEEE New England 10‐machine 39‐node system [15], and PMUs are installed at each node, which is shown in Figure 2. The system voltage level is 345 kV, the frequency is 60 Hz, and the sampling frequency is 3 kHz. The load of each bus in the system is set to a constant power model. Consider the random fluctuation of wind speed, and then analyse the probability of small disturbance in the system. PMUs are added at node 28 and node 29, the branch is separated, and multiple DFIG wind turbines are equivalent to the wind farm. The wind farm has a capacity of 60 MW and consists of 45 double‐fed induction wind turbines with a rated power of 1.5 W.2FIGUREIEEE New England 10‐machine 39‐node system with a wind farmBy actively changing the output of the wind turbines to simulate the change of the external wind speed, and the system entering a new equilibrium state, the PMU data measured by the system are analysed, and the PMU characteristic data are extracted as the sample space. Using the RF method, according to the important score of each eigenvalue, the positive anomaly data are classified as shown in Figure 3. The blue areas are normal data and denoted as Ω1, and the red areas are abnormal data and denoted as Ω2.3FIGUREClassification results of PMU feature dataFrom Figure 3, the sample set is randomly generated and the segmentation surfaces are approximately circular and square. Compared with the segmentation of the sample space between Naive Bayes and RF, from the perspective of accuracy rate, it can be seen that RF is better than a single decision tree of this test set. It can be seen intuitively from the feature space that RF has stronger segmentation ability than the naive Bayes’.Statistical characteristics and probability distribution curves of system eigenvalues under different wind turbine outputsAccording to the normal data obtained, the statistical characteristics (expectation and standard deviation) of the probability distribution of the system eigenvalues are obtained through simulations, which are shown in Table 1.1TABLEStatistical characteristics of the probability distribution of system eigenvaluesReal partImaginary partWind turbine output powerMathematical expectationsStandard deviationMathematical expectationsStandard deviation15 MW30 MW45 MW60 MW−0.98720.01982.1520.0192−0.90270.08822.1840.1852−0.82810.21461.5910.2192−0.70440.22461.7130.303For the above system, stable calculations of small disturbance are performed. Take the real part of the eigenvalues as an example, the Monte Carlo simulation of 11,700 random samples are as the accurate values. Compared with the method of calculating the posterior probability density by the prior probability according to the method in this paper, based on the last feature data processing, two different methods fitting the real probability density curves of the eigenvalues are shown in Figure 4. When the wind turbine output is 45 MW operation mode, calculate the PDF of the real part of the eigenvalue and the probability of falling in the stable interval (negative real part), respectively.4FIGUREThe probability density of the real part of eigenvaluesFrom the probability density curves of the real part of the eigenvalue by the 45 MW wind turbine output in Figure 4, it can be found that in this equilibrium state, whether it is Monte Carlo probability density or Bayesian posterior probability density, the probability density distribution of the real part of the eigenvalue can have the probability of a positive real part. However, it can be clearly seen that the probability of a positive real part of the eigenvalues of Bayesian posterior probability density is much lower than Monte Carlo probability density, so the accuracy of analysing the probabilistic stability of small disturbance by the method of Bayesian posterior probability density will be higher.In Figures 5 and 6, according to the accuracy curves of the negative real part of the eigenvalue and the probability loss curves of the abnormal eigenvalue under different wind speeds over 60 times, it can be seen that with the increase of the eigenvalue, the probability of the negative real part of the eigenvalue appears gradually increasing. When the number of eigenvalues increases from 0 to about 500, the stability probability increases rapidly to 60%, and the number of losses of abnormal eigenvalues also decreases rapidly to about 1. It can be seen that the proposed method is highly computationally efficient. When the number of eigenvalues increases from about 500 to 1400, the stability probability gradually increases to 70%. During the process of the number of eigenvalues increasing to 11,700, the stability probability will increase steadily starting from 70%. It can be read from Figure 5 that the final stability probability is about 81.85% when the number of eigenvalues is 11,700 at that time. Correspondingly, the loss of abnormal eigenvalues gradually decays to around 0.5. It can be concluded that the proposed method is stable and reliable for processing a large number of wide‐area time series PMU data.5FIGUREThe accuracy of eigenvalues falls on the left half plane6FIGURELoss curves of abnormal eigenvaluesComparative experimentIn this part, the prediction method of Monte Carlo simulation probability based on the Markov chain is used as a comparative simulation method. It can be seen from Figure 7 that the probability of its eigenvalue falling on the negative real part fluctuates around 45%, and its stability probability fluctuates drastically. The system probabilistic stability is much lower than that used by the method proposed in this paper. The result shows that the method proposed in this paper is more in line with the description of the probabilistic stability of small disturbance when the number of wide‐area time series PMU data is large.7FIGUREMonte Carlo method based on the Markov chain predicts the probability that eigenvalues have negative real partsCONCLUSIONIn this paper, aiming at the probabilistic stability of small disturbance in power system, a probabilistic stability method of variational Bayes based on the PMU data and Lyapunov theory is proposed. In view of the existed probabilistic stability analysis method of small disturbance, it is limited to the problem of the low calculation efficiency when the number of wide‐area time series PMU data is large. The PMU feature data processing based on RF is proposed to filter out the normal data in order to improve the efficiency of the variational Bayes and calculating the probability of its eigenvalue falling on the left half plane. Then, from the Bayesian probabilistic perspective, considering the time‐space correlation in the connected power sources will affect the probabilistic stability of the wind power system, a system asymptotic stability theorem based on the Lyapunov theory is proposed, and also its proof procedure is given. It proves that the probability of wind power system is asymptotically stable. Finally, it is predicted to make the system enter the next equilibrium state, and the probabilistic stability of the system is close to 81.85%. Compared with the traditional Monte Carlo algorithm based on the Markov chain, the approximation prediction of the probabilistic stability of small disturbance has a significant improvement.AUTHOR CONTRIBUTIONSM.Y.: Funding acquisition; Supervision; Validation; Writing – review & editing. J.L.: Investigation; Methodology; Software; Visualization; Writing – original draft. S.Z.: Formal analysis; Methodology; Validation; Writing – review & editingACKNOWLEDGEMENTSThis work was supported by the Pyramid Talent Training Project of Beijing University of Civil Engineering and Architecture (JDYC20200324); Post Graduate Education and Teaching Quality Improvement Project of Beijing University of Civil Engineering and Architecture (J2022007); BUCEA Post Graduate Innovation Project (PG2022132); Security Control and Simulation for Power System and Large Power Generation Equipment (SKLD20M17); National Innovation and Entrepreneurship Training Program for College Students (202110016052,S202110016122,S202110016123); Project of Beijing Association of Higher Education (YB2021131); National Natural Science Foundation of China (51407201); Education, Teaching and Scientific Research Project of China Construction Education Association (2021051).CONFLICT OF INTERESTThe authors declare no conflict of interest.DATA AVAILABILITY STATEMENTData sharing not applicable to this article as no datasets were generated or analysed during the current study.REFERENCESBurchett, R.C., Heydt, G.T.: Probabilistic methods for power system dynamic stability studies. 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Journal

"IET Generation, Transmission & Distribution"Wiley

Published: Dec 1, 2022

Keywords: PMU data; probabilistic stability; small disturbance; variational Bayes; wind power system

References