WO2023241326A1 - 基于样本协方差矩阵最大特征值变化率的电网异常检测方法 - Google Patents

基于样本协方差矩阵最大特征值变化率的电网异常检测方法 Download PDF

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WO2023241326A1
WO2023241326A1 PCT/CN2023/096035 CN2023096035W WO2023241326A1 WO 2023241326 A1 WO2023241326 A1 WO 2023241326A1 CN 2023096035 W CN2023096035 W CN 2023096035W WO 2023241326 A1 WO2023241326 A1 WO 2023241326A1
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matrix
power grid
anomaly detection
maximum eigenvalue
covariance matrix
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French (fr)
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杨宇辉
吴定会
祁宇轩
杨朝辉
钱加兵
张娟
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无锡隆玛科技股份有限公司
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R31/00—Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
    • G01R31/08—Locating faults in cables, transmission lines, or networks
    • G01R31/088—Aspects of digital computing
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y04—INFORMATION OR COMMUNICATION TECHNOLOGIES HAVING AN IMPACT ON OTHER TECHNOLOGY AREAS
    • Y04S—SYSTEMS INTEGRATING TECHNOLOGIES RELATED TO POWER NETWORK OPERATION, COMMUNICATION OR INFORMATION TECHNOLOGIES FOR IMPROVING THE ELECTRICAL POWER GENERATION, TRANSMISSION, DISTRIBUTION, MANAGEMENT OR USAGE, i.e. SMART GRIDS
    • Y04S10/00—Systems supporting electrical power generation, transmission or distribution
    • Y04S10/50—Systems or methods supporting the power network operation or management, involving a certain degree of interaction with the load-side end user applications
    • Y04S10/52—Outage or fault management, e.g. fault detection or location

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  • the invention belongs to the technical field of power grid anomaly detection, and specifically relates to a power grid anomaly detection method based on the maximum eigenvalue change rate of a sample covariance matrix.
  • the scale of the power grid continues to expand and its structure becomes increasingly complex.
  • the power grid has low risk resistance and is more susceptible to the influence of the external environment during operation, resulting in failure. Therefore, real-time anomaly detection of the power grid is of great significance for reducing the occurrence rate of power grid faults and improving the stability of the power grid.
  • Power grid anomaly detection is mainly a process of determining whether a power grid failure occurs by collecting and analyzing operating data such as power grid electrical quantities and switching quantities.
  • power grid anomaly detection methods are mainly divided into two types: model-driven method and data-driven method. With the diversification of data collection methods, power data continues to be enriched. In the face of today's massive power big data, compared with the model-driven method, the data-driven method has better data processing capabilities and faster data analysis speed, and has gradually become the current mainstream power grid anomaly detection method.
  • power grid anomaly detection methods based on random matrix theory can be roughly divided into two categories: power grid anomaly detection methods based on the single loop theorem and average spectral radius, and power grid anomaly detection methods based on sample covariance matrices.
  • the former does not consider the influence of random changes in noise, resulting in a significant decrease in the sensitivity of anomaly detection in a low signal-to-noise ratio environment.
  • the method based on the sample covariance matrix has high computational efficiency and has obvious advantages in anomaly detection speed, in actual analysis, the signal-to-noise ratio will affect the eigenvalue distribution of the matrix, thereby affecting the anomaly detection accuracy.
  • the present invention proposes a power grid anomaly detection method suitable for different signal-to-noise ratio environments based on the maximum eigenvalue change rate MEVR of the sample covariance matrix, which effectively improves the efficiency of the power grid. Speed and accuracy of anomaly detection.
  • the power grid anomaly detection method of the present invention includes the following steps:
  • Step 1 Collect real-time status data of the power grid, and construct a data source matrix X S based on the status data;
  • Step 2 At each sampling moment, intercept the data source matrix X S through the sliding time window method to obtain the window matrix, and obtain the standard non-Hermitian matrix through data normalization processing.
  • Step 3 Based on the obtained standard non-Hermitian matrix The anomaly detection index ⁇ is calculated through feature decomposition;
  • Step 4 Given the abnormality detection threshold ⁇ , based on the M-P law, detect the abnormal state of the power grid according to the relationship between the constructed abnormality detection index ⁇ and the abnormality detection threshold ⁇ .
  • the real-time status data of the power grid described in step 1 are collected by the wide-area measurement unit, including one or more of node voltage, branch current, load active power, and reactive power. Each status data is analyzed separately.
  • Step 1 Select the corresponding status data type for the different abnormal status types to be detected to construct the data source matrix X S corresponding to the abnormal status type.
  • the range of anomaly detection threshold ⁇ is generally selected as: 1.1 ⁇ 1.3.
  • step 2 at each sampling moment, the sliding time window method is used to intercept the data source matrix Get the standard non-Hermitian matrix
  • the width T w of the sliding time window is selected from 50 to 300.
  • step 3 the standard non-Hermitian matrix constructed according to step 2 First construct the sample covariance matrix S through equation (2):
  • ⁇ max (t) represents the maximum eigenvalue of the sample covariance matrix S at the current time t; when m ⁇ t, ⁇ max (tm) represents the maximum eigenvalue of the sample covariance matrix S in the m moments before the current time t. .
  • step 4 the M-P law in step 4 is expressed as:
  • ⁇ is the eigenvalue of the covariance matrix S
  • f( ⁇ ) is the empirical spectral distribution of the eigenvalue of the covariance matrix S
  • c is the matrix The ratio of the determinant of , a and b respectively represent the theoretical lower bound and upper bound of the eigenvalue of the sample covariance matrix S
  • represents the absolute value of the eigenvalue of matrix S.
  • the elements of matrix S are independent and identically distributed random variables, and their eigenvalue distribution satisfies the M-P law; when the system is abnormal, the randomness of the matrix elements is destroyed and no longer satisfies the M-P law, and its maximum eigenvalue will is greater than the theoretically correct value; therefore, whether the power grid is abnormal is detected by analyzing the changes in the maximum eigenvalue of matrix S; if ⁇ , it is judged that there is an abnormality, and ⁇ , it is judged that there is no abnormality.
  • the sampling time corresponding to ⁇ is the time when the power grid abnormality occurs.
  • the beneficial effect of the present invention is that the power grid anomaly detection index is set as the change rate of the maximum eigenvalue of the covariance matrix, so that the present invention can be applied to power grid environments with different signal-to-noise ratios.
  • the specific performance is as follows: when the signal-to-noise ratio changes, the maximum eigenvalue of the matrix will change accordingly, but the maximum eigenvalue change rate of the matrix is not affected by the change of the signal-to-noise ratio.
  • the change rate is stable around 1. Only when the maximum characteristic value changes sharply in a short period of time due to an abnormality in the power grid will the change rate increase significantly. Therefore, the present invention can realize power grid anomaly detection under different signal-to-noise ratios, and has higher anomaly detection accuracy and strong adaptability.
  • Figure 1 is a flow chart of the anomaly detection method proposed by the present invention.
  • Figure 2 is an IEEE39 node network topology diagram.
  • Figure 3 is a simulation curve diagram of a short-circuit fault in a low signal-to-noise ratio environment using the power grid anomaly detection method provided by the present invention.
  • Figure 4 is a simulation curve diagram of a short-circuit fault in a high signal-to-noise ratio environment using the power grid anomaly detection method provided by the present invention.
  • Figure 5 shows the simulation curves of short-circuit faults under different signal-to-noise ratio environments using the average spectral radius method.
  • Figure 6 shows the simulation curve of short-circuit faults under different signal-to-noise ratio environments using the sample covariance matrix maximum eigenvalue method.
  • the power grid anomaly detection method provided by the present invention generally includes the following steps:
  • Step 1 Collect real-time status data of the power grid, and construct a data source matrix X S based on the status data;
  • Step 2 At each sampling time t, intercept the data source matrix X S through the sliding time window method to obtain the window matrix, and obtain the standard non-Hermitian matrix through data normalization processing.
  • Step 3 Based on the obtained standard non-Hermitian matrix The anomaly detection index ⁇ is calculated through feature decomposition;
  • Step 4 Given the abnormality detection threshold ⁇ , based on the M-P law, detect the abnormal state of the power grid according to the relationship between the constructed abnormality detection index ⁇ and the abnormality detection threshold ⁇ .
  • the power grid anomaly detection method proposed by the present invention based on the maximum eigenvalue change rate of the sample covariance matrix will be described in detail below using the IEEE39 node network shown in Figure 2 as a specific example.
  • the real-time status data of the power grid mainly include node voltage, branch current, load active power, reactive power, etc.
  • the type of abnormal state we want to detect is a three-phase short circuit to ground fault.
  • Step 1 Set the abnormal status type to three-phase short circuit and ground fault.
  • the IEEE39 node network shown in Figure 2 contains a total of 39 buses.
  • the sampling interval is 1 ms.
  • Step 2 First use the sliding time window method to construct the window matrix X.
  • T w is the number of matrix columns of the window matrix X, Represents the width of the sliding time window.
  • T w is generally selected from tens to hundreds, and T w can generally be selected from 50 to 300 based on experimental effect feedback.
  • the smaller T w means that the matrix contains fewer samples, and the matrix is more sensitive to noise signals. Therefore, in order to reduce the impact of noise on state data, the asymptotic assumption condition (N, T w ⁇ and N/T w On the basis of ⁇ [0, ⁇ )), a larger T w should be selected as much as possible.
  • x i,j represents the elements in the window matrix X, is the standard non-Hermitian matrix after normalization elements in; is the row vector of the window matrix X, represents a standard non-Hermitian matrix The row vector of ; ⁇ ( xi ) is the matrix row average of the window matrix X, represents a standard non-Hermitian matrix The matrix row average of ; ⁇ ( xi ) is the matrix row standard deviation of the window matrix X, represents a standard non-Hermitian matrix The matrix row standard deviation.
  • a standard non-Hermitian matrix of 39 ⁇ 100 is obtained according to the above data processing steps.
  • the feature decomposition process in step 3 is:
  • ⁇ max (t) represents the maximum eigenvalue of the sample covariance matrix S at the current time t; when m ⁇ t, ⁇ max (tm) represents the maximum eigenvalue of the sample covariance matrix S in the m moments before the current time t. .
  • ⁇ is the eigenvalue of the covariance matrix S
  • f( ⁇ ) is the empirical spectral distribution of the eigenvalue of the covariance matrix S
  • c is the matrix The ratio of the determinant of , a and b respectively represent the theoretical lower bound and upper bound of the eigenvalue of the sample covariance matrix S
  • represents the absolute value of the eigenvalue of matrix S.
  • the elements of matrix S are independent and identically distributed random variables, and their eigenvalue distribution satisfies the M-P law; when the system is abnormal, the randomness of the matrix elements is destroyed and no longer satisfies the M-P law, and its maximum eigenvalue will is greater than the theoretical exact value of the matrix. Therefore, whether the power grid is abnormal can be detected by analyzing the changes in the maximum eigenvalue of matrix S.
  • the maximum eigenvalue of matrix S is 2.349; at the 12th sampling point, the maximum eigenvalue of matrix S is 2.457, both of which are smaller than the theoretically correct values.
  • an abnormality occurred in the power grid.
  • the maximum eigenvalue of matrix S was 3.019, which was much larger than its theoretical correct value.
  • its maximum eigenvalue change rate ⁇ 1.207.
  • Equation (5) The relationship between the anomaly detection index ⁇ and the anomaly detection threshold ⁇ in step 4 is shown in Equation (5):
  • Equation (6) The power grid anomaly detection results are shown in Equation (6):
  • the embodiment is shown in Figure 3.
  • the solid straight line represents the anomaly detection threshold ⁇
  • the solid curve represents the anomaly detection index ⁇ .
  • the curve fluctuates below the straight line, that is, when ⁇ ⁇ ⁇ , it means that no abnormality occurs.
  • the fluctuation amplitude of the curve becomes larger, that is, when ⁇ ⁇ ⁇ , it indicates an abnormal situation in the power grid.
  • the sampling time corresponding to ⁇ is the time when the power grid abnormality occurs.
  • the grid protection device will respond immediately and perform corresponding fault protection operations.
  • engineers can judge whether to take protective measures based on the type of abnormal status and manual experience.
  • the method proposed by the present invention is compared with two other currently widely used power grid anomaly detection methods, specifically including: mean spectral radius (MSR) method and sample covariance matrix Maximum Eigenvalue (MESCM) method.
  • MSR mean spectral radius
  • MESCM sample covariance matrix Maximum Eigenvalue
  • the simulation curves of ground short-circuit faults in different signal-to-noise ratio environments using the sample covariance matrix maximum eigenvalue change rate (MEVR) method proposed by the present invention are shown in Figures 3 and 4.
  • the simulation curves of ground short-circuit faults of MSR method and MESCM method in different signal-to-noise ratio environments are shown in Figure 5 and Figure 6 respectively.
  • the detection index of the MEVR method is the maximum eigenvalue change rate ⁇ of the matrix, and the threshold is ⁇ ;
  • the detection index of the MSR method is r MSR , and the threshold is Inner Radius;
  • the detection index of the MESCM method is the maximum eigenvalue ⁇ max of the matrix, and the threshold is ⁇ .
  • the intersection between the detection index and the threshold is the abnormal moment of the power grid.
  • the present invention uses an anomaly detection method based on the sample covariance matrix maximum eigenvalue change rate (MEVR) and the MSR method.
  • MEVR sample covariance matrix maximum eigenvalue change rate
  • the power grid anomaly detection performance of the MESCM method is summarized in Table 1.
  • the abnormality detection of the maximum eigenvalue change rate of the sample covariance matrix proposed by the present invention is The method has the earliest abnormality detection time and the lowest voltage drop ratio, which can effectively realize high-precision power grid anomaly detection under different signal-to-noise ratios, and can also detect power grid anomalies earlier.
  • the present invention sets the power grid anomaly detection index to the maximum eigenvalue change rate of the covariance matrix that is less affected by changes in the signal-to-noise ratio. Therefore, power grid anomaly detection under different signal-to-noise ratios can be achieved, and has a higher Anomaly detection accuracy and strong adaptability.

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Abstract

一种基于样本协方差矩阵最大特征值变化率的电网异常检测方法。首先，采集电网实时状态数据组成数据源矩阵。其次，通过滑动时间窗方法对数据源矩阵进行截取，得到窗口矩阵，并通过数据标准化处理得到标准非厄米特矩阵。然后，依据标准非厄米特矩阵构建样本协方差矩阵并对其进行特征分解，计算得到样本协方差矩阵的最大特征值变化率，即异常检测指标。最后，基于M-P律和给定的异常检测阈值，通过比较异常检测指标与异常检测阈值的大小关系实现电网的异常状态检测。协方差矩阵最大特征值变化率受信噪比变化影响较小，因此可以实现不同信噪比下的电网异常检测，具有较高的异常检测精度和较强的适应性。

Description

基于样本协方差矩阵最大特征值变化率的电网异常检测方法 技术领域
本发明属于电网异常检测技术领域，具体涉及一种基于样本协方差矩阵最大特征值变化率的电网异常检测方法。
背景技术
随着新型电力系统的发展，电网规模不断扩大，结构日趋复杂。在这种背景下，电网抗风险能力较低，在运行过程中较易受到外界环境的影响，从而发生故障。因此，对电网进行实时的异常检测，对于降低电网故障发生率、提高电网稳定性具有重要意义。
电网异常检测主要是通过采集和分析电网电气量、开关量等运行数据来判断电网是否发生故障的过程。目前，电网异常检测的方法主要分为模型驱动法和数据驱动法两种。随着数据采集手段的多样化，电力数据不断丰富。面对如今海量的电力大数据，相较于模型驱动法，数据驱动法有更好的数据处理能力和更快的数据分析速度，逐渐成为目前主流的电网异常检测方法。
现有的数据驱动法主要以随机矩阵理论为基础，通过分析由电网状态数据构建的状态矩阵实现故障检测。目前，基于随机矩阵理论的电网异常检测方法可大致分为两类：基于单环定理和平均谱半径的电网异常检测方法，以及基于样本协方差矩阵的电网异常检测方法。前者由于未考虑噪声的随机变化影响，导致在低信噪比环境下，异常检测的灵敏度会明显下降。基于样本协方差矩阵的方法虽然计算效率高，在异常检测速度上有明显的优势，但是在实际分析中，信噪比会影响矩阵的特征值分布，进而影响异常检测精度。
发明内容
由于信噪比对矩阵特征值有较大的影响，本发明基于样本协方差矩阵的最大特征值变化率MEVR，提出了一种适用于不同信噪比环境的电网异常检测方法，有效提高了电网异常检测的速度和精度。
为实现上述目的，本发明所述的电网异常检测方法包括以下步骤：
步骤1：采集电网实时状态数据，基于状态数据构造数据源矩阵XS；
步骤2：在每个采样时刻，通过滑动时间窗方法对数据源矩阵XS进行截取，得到窗口矩阵，并通过数据标准化处理得到标准非厄米特矩阵
步骤3：根据得到的标准非厄米特矩阵通过特征分解计算得出异常检测指标α；
步骤4：给定异常检测阈值ε，基于M-P律，根据所构建的异常检测指标α与异常检测阈值ε的大小关系对电网异常状态进行检测。
步骤1所述电网实时状态数据由广域测量单元采集得到，包括节点电压、支路电流、负荷有功功率、无功功率中的一种或多种，每一种状态数据单独进行分析。
步骤1针对要检测的不同异常状态类型选择对应的状态数据种类，来构造该异常状态类型对应的数据源矩阵XS。异常检测阈值ε的范围一般选择：1.1＜ε＜1.3。
具体的，步骤2中，在每个采样时刻,运用滑动时间窗方法截取数据源矩阵XS，构建窗口矩阵X，再通过式(1)的行变换将窗口矩阵X中的实时状态数据标准化，得到标准非厄米特矩阵
其中，i＝1,2,…,N,N为窗口矩阵X的矩阵行数，即窗口矩阵X中每种状态变量的具体个数；j＝1,2,…,Tw,Tw为窗口矩阵X的矩阵列数，代表滑动时间窗的宽度；xi,j代表窗口矩阵X中的元素，是经过标准化后标准非厄米特矩阵中的元素；是窗口矩阵X的行向量，代表标准非厄米特矩阵的行向量；α(xi)是窗口矩阵X的矩阵行平均值，代表标准非厄米特矩阵的矩阵行平均值；σ(xi)是窗口矩阵X的矩阵行标准差，代表标准非厄米特矩阵的矩阵行标准差。滑动时间窗的宽度Tw选取范围为50～300。
具体的，步骤3的特征分解过程为：依据步骤2构造的标准非厄米特矩阵先通过式(2)构建样本协方差矩阵S：
其中，为的共轭转置矩阵；
然后通过特征分解手段得到样本协方差矩阵S的最大特征值λmax(t)；最后由式(3)计算出矩阵S的最大特征值变化率，即异常检测指标α：
其中，λmax(t)表示当前时刻t样本协方差矩阵S的最大特征值；当m＜t时，λmax(t-m)表示当前时刻t前m个时刻中样本协方差矩阵S的最大特征值。
具体的，步骤4中的M-P律表述为：
对于N×Tw阶的矩阵当矩阵内所有元素满足独立同分布且各元素满足均值为0、方差为1，且矩阵的行列式之比保持不变时，其协方差矩阵S的所有特征值满足式(4)所示：
其中，λ为协方差矩阵S的特征值，f(λ)为协方差矩阵S特征值的经验谱分布；c为矩阵的行列式之比，a和b分别表示样本协方差矩阵S特征值的理论下确界和上确界， |λ|表示矩阵S特征值的绝对值。
当电网状态正常时，矩阵S的元素为独立同分布的随机变量，其特征值分布满足M-P律；当系统异常时，矩阵元素的随机性被破坏，不再满足M-P律，其最大特征值会大于理论上确值；因此，通过分析矩阵S最大特征值的变化情况来检测电网是否异常；若α≥ε，判断为有异常情况，α＜ε，判断为无异常情况。α所对应的采样时刻则为电网异常状况产生时刻。
本发明的有益效果为：将电网异常检测指标设定为协方差矩阵最大特征值的变化率，使本发明可以适用于不同信噪比的电网环境中。具体表现为：信噪比变化时，矩阵的最大特征值会随之改变，但是矩阵的最大特征值变化率却不受信噪比变化的影响。电网稳态时变化率稳定在1附近，只有当电网出现异常导致的最大特征值短时间急剧变化才会引起变化率的显著增加。因此，本发明可以实现不同信噪比下的电网异常检测，且具有较高的异常检测精度和较强的适应性。
附图说明
图1为本发明提出的异常检测方法的流程图。
图2为IEEE39节点网络拓扑结构图。
图3为本发明提供的电网异常检测方法在低信噪比环境下短路故障的仿真曲线图。
图4为本发明提供的电网异常检测方法在高信噪比环境下短路故障的仿真曲线图。
图5为平均谱半径法在不同信噪比环境下短路故障的仿真曲线图。
图6为样本协方差矩阵最大特征值法在不同信噪比环境下短路故障的仿真曲线图。
具体实施方式
下面结合附图和实施例对本发明做进一步的说明。
如图1所示，本发明提供的电网异常检测方法总体包括以下几个步骤:
步骤1：采集电网实时状态数据，基于状态数据构造数据源矩阵XS；
步骤2：在每个采样时刻t，通过滑动时间窗方法对数据源矩阵XS进行截取，得到窗口矩阵，并通过数据标准化处理得到标准非厄米特矩阵
步骤3：根据得到的标准非厄米特矩阵通过特征分解计算得出异常检测指标α；
步骤4：给定异常检测阈值ε，基于M-P律，根据所构建的异常检测指标α与异常检测阈值ε的大小关系对电网异常状态进行检测。
下面将通过图2所示的IEEE39节点网络为具体实例，详细说明本发明提出的基于样本协方差矩阵最大特征值变化率的电网异常检测方法。电网实时状态数据主要有节点电压、支路电流、负荷有功功率、无功功率等，我们可以根据案例实际需求选取合适的数据种类，同时被选取的每种状态数据需要单独进行分析。实施例中我们要检测的异常状态类型为三相短路接地故障。
步骤1：设置异常状态类型为三相短路接地故障。具体地，实验中设置9号母线的负荷在采样时刻t＝500～600ms(采样起始时t＝0)进行三相短路接地操作。同时，为了验证所提方法在不同信噪比下对短路故障检测的有效性，分别在ρ＝45dB的高信噪比环境和ρ＝20dB的低信噪比环境下独立进行实验。
图2所示的IEEE39节点网络共包含39条母线。选取每条母线的电压为测量状态变量，即N＝39。利用现有的广域测量单元对母线电压进行时长2s的采样，采样间隔为1ms，则共有2000个采样时刻(采样点)，得到39×2000的母线电压矩阵，也就是步骤1所说的数据源矩阵XS。
步骤2首先运用滑动时间窗方法构建窗口矩阵X，N＝39为窗口矩阵X的矩阵行数，即窗口矩阵X中每种状态变量的具体个数；Tw为窗口矩阵X的矩阵列数，代表滑动时间窗的宽度。Tw一般取几十到几百，根据实验效果反馈Tw一般可选取50～300。Tw越小意味着矩阵所包含的样本数越少，矩阵对噪声信号越为敏感，因此为了降低噪声对状态数据的影响，在满足渐进假设条件(N,Tw→∞且N/Tw∈[0，∞))的基础上，应该尽量选取较大Tw。
然后，通过式(1)进行行变换，将窗口矩阵X中的实时状态数据标准化，得到标准非厄米特矩阵
其中，i＝1,2,…,N；j＝1,2,…,Tw,；xi,j代表窗口矩阵X中的元素，是经过标准化后标准非厄米特矩阵中的元素；是窗口矩阵X的行向量，代表标准非厄米特矩阵的行向量；μ(xi)是窗口矩阵X的矩阵行平均值，代表标准非厄米特矩阵的矩阵行平均值；σ(xi)是窗口矩阵X的矩阵行标准差，代表标准非厄米特矩阵的矩阵行标准差。
实施例中，选取滑动时间窗口宽度Tw＝100，在每个采样时刻，按照上述的数据处理步骤得到39×100的标准非厄米特矩阵
步骤3的特征分解过程为：
依据步骤2构造的标准非厄米特矩阵先通过式(2)构建样本协方差矩阵S：
其中，为的共轭转置矩阵；
然后通过特征分解手段得到样本协方差矩阵S的最大特征值λmax(t)；最后由式(3)可计算出矩阵S的最大特征值变化率，即异常检测指标α：
其中，λmax(t)表示当前时刻t样本协方差矩阵S的最大特征值；当m＜t时，λmax(t-m)表示当前时刻t前m个时刻中样本协方差矩阵S的最大特征值。
实施例中，第7个采样点矩阵S的最大特征值为2.349，其最大特征值变化率α＝1.064；第12个采样点矩阵S的最大特征值为2.457，其最大特征值变化率α＝1.017；第514个采样点矩阵S的最大特征值为3.019，其最大特征值变化率α＝1.207。
步骤4中的M-P律表述为：
对于N×Tw阶的矩阵当矩阵内所有元素满足独立同分布且各元素满足均值为0、方差为1，且矩阵的行列式之比保持不变时，其协方差矩阵S的所有特征值满足式(4)所示：
其中，λ为协方差矩阵S的特征值，f(λ)为协方差矩阵S特征值的经验谱分布；c为矩阵的行列式之比，a和b分别表示样本协方差矩阵S特征值的理论下确界和上确界， |λ|表示矩阵S特征值的绝对值。
当电网状态正常时，矩阵S的元素为独立同分布的随机变量，其特征值分布满足M-P律；当系统异常时，矩阵元素的随机性被破坏，不再满足M-P律，其最大特征值会大于矩阵理论上确值。因此，可以通过分析矩阵S最大特征值的变化情况来检测电网是否异常。
在本实例中，计算得到矩阵S理论上确值b＝2.639。电网正常运行状态下，如第7个采样点，矩阵S最大特征值为2.349；第12个采样点，矩阵S最大特征值为2.457，两者皆小于理论上确值。在第514个采样点，电网出现异常情况，矩阵S最大特征值为3.019，远大于其理论上确值，同时其最大特征值变化率α＝1.207。
步骤4中异常检测指标α与异常检测阈值ε的大小关系如式(5)所示：
其中，异常检测阈值ε一般根据实际电网环境来给定，异常检测阈值ε过低会增加误检的概率，过高则可能会导致漏检，因此在工程实际中，为了保证较高的异常检测精度，ε应设定相对较高，一般应设定1.1＜ε＜1.3。本实施案例考虑一定的裕度，给定异常检测阈值ε＝1.2。
电网异常检测结果如式(6)所示：
实施例如图3所示，实直线代表异常检测阈值ε，实曲线代表异常检测指标α。电网正常运行情况下，曲线在直线下方波动，即α＜ε时，表示无异常情况发生。在第514个采样时刻，曲线波动幅度变大，即α≥ε时，表示电网出现异常情况。
α所对应的采样时刻则为电网异常状况产生时刻。电网保护装置会立即响应，执行相应的故障保护操作。在工程实际中，工程人员可以根据异常状态种类，依据人工经验判断是否采取保护措施。
为了验证本发明提出的电网异常检测方法的有效性，将本发明提出的方法与另外两种目前应用广泛的电网异常检测方法进行对比，具体包括：平均谱半径(MSR)法和样本协方差矩阵最大特征值(MESCM)法。本发明提出的样本协方差矩阵最大特征值变化率(MEVR)法在不同信噪比环境中接地短路故障的仿真曲线如图3，图4所示。MSR法和MESCM法在不同信噪比环境中接地短路故障的仿真曲线分别如图5，图6所示。具体的，MEVR法的检测指标为矩阵最大特征值变化率α，阈值为ε；MSR法的检测指标为rMSR，阈值为Inner Radius；MESCM法的检测指标为矩阵的最大特征值λmax，阈值为γ。以上异常检测方法中，检测指标与阈值的交点即为电网异常时刻。
由图3，图4可知，在不同信噪比下，所提MEVR方法在故障时刻附近都表现出了明显的曲线变化，异常检测指标α在t＝501时刻附近突破阈值；观察图5发现，高信噪比下λmax在整个分析过程中完全大于阈值，表示电网一直处于异常状态，这表明MESCM方法存在虚警问题；观察图6发现，低信噪比下，rMSR在整个分析过程中完全大于阈值，表示电网一直处于正常状态，这表明MSR方法存在漏警问题。
在不同信噪比环境的电网接地短路故障中，本发明基于样本协方差矩阵最大特征值变化率(MEVR)的异常检测方法和MSR法，MESCM法的电网异常检测性能总结如表1所示。
表1异常检出时刻和节点电压下降比
对比表1中具体的异常检出时刻和节点电压下降比可以看出，不同信噪比下，相较于MSR法和MESCM法，本发明提出的样本协方差矩阵最大特征值变化率的异常检测方法异常检出时刻最早、电压下降比最低，能有效实现不同信噪比下的高精度电网异常检测，同时还可以较早地检测出电网异常。
可以看出，本发明将电网异常检测指标设定为受信噪比变化影响较小的协方差矩阵最大特征值变化率，因此，可以实现不同信噪比下的电网异常检测，且具有较高的异常检测精度和较强的适应性。
以上的实施例仅用于说明本发明，而并非作为对本发明的限定，所应用的对象不限于IEEE39节点网络，所设置的电网异常状态类型不限于短路故障，只要在本发明的范围内，对上述实施实例的变化、变型都将落在本发明的保护范围内。

Claims (10)

  1. 基于样本协方差矩阵最大特征值变化率的电网异常检测方法，其特征是，包括以下步骤：
    步骤1：采集电网实时状态数据，基于状态数据构造数据源矩阵XS；
    步骤2：在每个采样时刻，通过滑动时间窗方法对数据源矩阵XS进行截取，得到窗口矩阵，并通过数据标准化处理得到标准非厄米特矩阵
    步骤3：根据得到的标准非厄米特矩阵通过特征分解计算得出异常检测指标α；
    步骤4：给定异常检测阈值ε，基于M-P律，根据所构建的异常检测指标α与异常检测阈值ε的大小关系对电网异常状态进行检测。
  2. 如权利要求1所述的基于样本协方差矩阵最大特征值变化率的电网异常检测方法，其特征是，所述异常检测阈值ε的范围为：1.1<ε<1.3。
  3. 如权利要求1所述的基于样本协方差矩阵最大特征值变化率的电网异常检测方法，其特征是，步骤1所述电网实时状态数据由广域测量单元采集得到，包括节点电压、支路电流、负荷有功功率、无功功率中的一种或多种，每一种状态数据单独进行分析。
  4. 如权利要求1所述的基于样本协方差矩阵最大特征值变化率的电网异常检测方法，其特征是，步骤1针对要检测的不同异常状态类型选择对应的状态数据种类，来构造该异常状态类型对应的数据源矩阵XS。
  5. 如权利要求1所述的基于样本协方差矩阵最大特征值变化率的电网异常检测方法，其特征是，步骤2中，在每个采样时刻,运用滑动时间窗方法截取数据源矩阵XS，构建窗口矩阵X，再通过式(1)的行变换将窗口矩阵X中的实时状态数据标准化，得到标准非厄米特矩阵
    其中，i＝1,2,…,N,N为窗口矩阵X的矩阵行数，即窗口矩阵X中每种状态变量的具体个数；j＝1,2,…,Tw,Tw为窗口矩阵X的矩阵列数，代表滑动时间窗的宽度；xi,j代表窗口矩阵X中的元素，是经过标准化后标准非厄米特矩阵中的元素；是窗口矩阵X的行向量，代表标准非厄米特矩阵的行向量；μ(xi)是窗口矩阵X的矩阵行平均值，
    代表标准非厄米特矩阵的矩阵行平均值；σ(xi)是窗口矩阵X的矩阵行标准差，代表标准非厄米特矩阵的矩阵行标准差。
  6. 如权利要求5所述的基于样本协方差矩阵最大特征值变化率的电网异常检测方法，其特征是，步骤3的特征分解过程为：依据步骤2构造的标准非厄米特矩阵先通过式(2)构建样本协方差矩阵S：
    其中，为的共轭转置矩阵；
    然后通过特征分解手段得到样本协方差矩阵S的最大特征值λmax(t)；最后由式(3)计算出矩阵S的最大特征值变化率，即异常检测指标α：
    其中，λmax(t)表示当前时刻t样本协方差矩阵S的最大特征值；当m<t时，λmax(t-m)表示当前时刻t前m个时刻中样本协方差矩阵S的最大特征值。
  7. 如权利要求6所述的基于样本协方差矩阵最大特征值变化率的电网异常检测方法，其特征是，步骤4中的M-P律表述为：
    对于N×Tw阶的矩阵当矩阵内所有元素满足独立同分布且各元素满足均值为0、方
    差为1，且矩阵的行列式之比保持不变时，其协方差矩阵S的所有特征值满足式(4)所示：
    其中，λ为协方差矩阵S的特征值，f(λ)为协方差矩阵S特征值的经验谱分布；c为矩阵的行列式之比，a和b分别表示样本协方差矩阵S特征值的理论下确界和上确界， |λ|表示矩阵S特征值的绝对值。
  8. 如权利要求7所述的基于样本协方差矩阵最大特征值变化率的电网异常检测方法，其特征是，当电网状态正常时，矩阵S的元素为独立同分布的随机变量，其特征值分布满足M-P律；当系统异常时，矩阵元素的随机性被破坏，不再满足M-P律，其最大特征值会大于理论上确值；因此，通过分析矩阵S最大特征值的变化情况来检测电网是否异常；若α≥ε，判断为有异常情况，α<ε，判断为无异常情况。
  9. 如权利要求8所述的基于样本协方差矩阵最大特征值变化率的电网异常检测方法，其特征是，α所对应的采样时刻为电网异常状况产生时刻。
  10. 如权利要求5所述的基于样本协方差矩阵最大特征值变化率的电网异常检测方法，其特征是，步骤2中滑动时间窗的宽度Tw选取50～300。
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