CN112861074B - Hankel-DMD-based method for extracting electromechanical parameters of power system - Google Patents
Hankel-DMD-based method for extracting electromechanical parameters of power system Download PDFInfo
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Abstract
A Hankel-DMD-based method for extracting electromechanical parameters of an electric power system belongs to the field of operation and maintenance of the electric power system. The method firstly excavates the electromechanical oscillation mode existing in the random response data under the excitation of the power system environment, and then extracts electromechanical oscillation parameters (oscillation frequency and damping ratio) of the power system on line from the random response data by utilizing a Hankel-DMD method. The method realizes the extraction of the electromechanical oscillation parameters under the condition of normal operation of the power system; compared with the traditional method for extracting the electromechanical oscillation parameters by using a longer analysis window, the method only needs a very short analysis time window, so that the real-time extraction of the oscillation parameters can be realized; compared with the traditional method, the method has higher extraction precision and extraction speed, and real-time tracking performance, thereby having higher practical application value.
Description
Technical Field
The invention belongs to the technical field of operation and maintenance of power systems, and particularly relates to a Hankel-DMD dynamic mode decomposition method based on Hankel matrix enhancement for extracting electromechanical oscillation parameters of a power system.
Background
With the interconnection of modern large power grids and the large-scale development and utilization of new energy, the complexity of the analysis of the low-frequency electromechanical oscillation problem is increased. In recent years, with the development of wide-area measurement technology, an important technical means is provided for online extraction of electromechanical oscillation mode parameters of a power system. In an actual system, the occurrence probability of obvious disturbance is relatively small, and transient oscillation signals after disturbance cannot be acquired in real time, so that the electromechanical parameter extraction method based on transient oscillation response signals after disturbance has a large limitation on online application. And random response signals under the action of environmental excitation such as load random fluctuation and the like exist at all times under the condition of normal operation of the system, so that the random response signals are easy to obtain, and the extraction result can reflect the small interference stability level of the normal operation condition of the system. Therefore, the safety online monitoring based on the random response under the normal operation condition of the system receives wide attention.
At present, the method for extracting the electromechanical oscillation parameters based on the random response under the normal operation condition of the system needs to utilize more observation data and a longer analysis time window for calculation, has long calculation time and is not suitable for real-time monitoring of an electric power system; in addition, the existing method has poor tracking and extracting performance for the mode, the extracting time delay is long, and the existing method is difficult to adapt to the modern power system with variable operation modes.
Therefore, a new technical solution is needed in the prior art to solve the problem.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the method for extracting the electromechanical parameters of the power system based on the Hankel-DMD is provided for solving the technical problems that the existing method for extracting the electromechanical oscillation parameters is poor in mode tracking and extracting performance, long in extraction time delay and difficult to adapt to the modern power system with variable operation modes.
The method for extracting the electromechanical parameters of the power system based on the Hankel-DMD comprises the following steps which are sequentially carried out,
step one, constructing an observation matrix by utilizing observation sampling data obtained by a wide area measurement system
Wherein the content of the first and second substances,
is composed of t 1 To t N An observation matrix formed by the observation values of the moments; x (c) m ,t N ) Representing the sampling data at the sampling point m at the time t; x is the number of l Representing a vector formed by all observation sampling data values at the first moment;
step two, utilizing the self-observed quantity x (c) in the mth row in the constructed observation matrix 1 ,t 1 ) To x (c) 1 ,t N-1 ) Forming a Hankel matrix of p rows and q columns:
wherein the content of the first and second substances,
also shown is the m-th row observed quantity x using the observation matrix m (t 1 ) To x m (t n-1 ) The formed Hankel matrix is used as a matrix,
meanwhile, the self-observed quantity x (c) in the mth row of the observation matrix 1 ,t 2 ) To x (c) 1 ,t N ) The Hankel matrix forming p rows and q columns is:
wherein, the first and the second end of the pipe are connected with each other,
also shown is the m-th row observed quantity x using the observation matrix m (t 2 ) To x m (t n ) The formed Hankel matrix is used as a matrix,
constructing a scale factor alpha by utilizing the two Hankel matrixes m ，
Reconstructing the data matrix by using Hankel matrixes at different sampling points:
wherein the content of the first and second substances,
is composed of t 1 Time to t N-1 An observation matrix formed by the observation values of the moments;
is composed of t 2 To t N An observation matrix formed by the observation values of the time instants,
for the last column of the Hankel matrix formed at sample point m,
the first column of a Hankel matrix formed at a sampling point m, | | · | | | is the Euclidean norm of the matrix;
step three, performing linear mapping on the discrete sampling data in the data matrix reconstructed in the step two, and then bringing the observation matrix into the linear mapping to obtain a relationship between the Krlov subspace sequence and two adjacent subspace sequences as follows:
wherein A is a high-dimensional matrix of the wide area measurement system;
is composed of t 1 Time to t N-1 An observation matrix formed by the observation values of the moments;
is composed of t 2 To t N Observation moment formed by observation values of timeArraying;
step four, passing the observation matrix
Orthogonal projection Decomposition (POD) mode matrix of (1)
The high-dimensional matrix A of the original system and the approximate low-dimensional approximate matrix thereof are combined
And (3) associating, namely:
wherein the matrix
The matrix is a low-dimensional approximate matrix of the matrix A, and T is expressed as the transposition of the matrix;
step five, mode matrix
By means of a pair matrix
Singular Value Decomposition (SVD) was performed to obtain:
wherein the mode matrix
Left singular value vector, matrix, obtained for singular value decomposition
For its right singular value vector, modeMatrix array
Sum matrix
Are all unitary matrices and are used as a matrix,
is a singular value diagonal matrix;
step six, approximating the state matrix of the system in low dimension
Solving by a minimization problem formula to obtain a system low-dimensional approximate matrix
The calculation formula of (2) is as follows:
step seven, approximating a system low-dimensional state matrix
The characteristic value decomposition is carried out to obtain:
wherein, w i For low-dimensional approximation of the eigenvalues, omega, of a matrix of a discrete system i Eigenvectors corresponding to low-dimensional approximation matrices and eigenvalues for discrete systems,
converting the discrete eigenvalues to eigenvalues under a continuous system using the following equation:
λ i ＝log(ω i )/Δt
wherein λ is i The characteristic value is the characteristic value of the continuous system; Δ t is the sampling interval;
and step eight, calculating and obtaining the oscillation frequency and the damping ratio of the electromechanical oscillation by using the characteristic value of the continuous system obtained in the step seven through an oscillation frequency and damping ratio formula of the electromechanical oscillation, and thus, completing the calculation of the electromechanical parameter extraction method of the power system based on Hankel matrix enhanced dynamic mode decomposition Hankel-DMD.
The gram Lei Luofu Krylov subspace sequence obtained in the third step is:
X 1 N-1 ＝[x 1 Ax 1 … A N-2 x 1 ]＝[x 1 … x N-1 ]
wherein A is a high-dimensional matrix of the wide area measurement system;
is composed of t 1 To t N An observation matrix formed by the observation values of the moments;
is composed of t 1 Time to t N-1 An observation matrix formed by the observation values of the moments;
is composed of t 2 To t N And the observation matrix is formed by the observation values of the time.
The linear mapping of the discrete sample data in the third step is represented as:
x i+1 ＝Ax i
wherein A is a high-dimensional matrix of the wide-area measurement system, x i Sampled data for time i, x i+1 The sampled data at time i + 1.
The minimization problem formula in the sixth step is as follows:
wherein the content of the first and second substances,
is a Fibonacci Frobenius norm.
The formula of the oscillation frequency and the damping ratio of the electromechanical oscillation in the step eight is as follows:
wherein λ is i I.e. the characteristic value of the continuous system, f i To the oscillation frequency, σ i Is the damping ratio.
Through the design scheme, the invention can bring the following beneficial effects:
the method is based on the random response data of the power system under the normal operation condition, takes the real-time performance of online monitoring into consideration, extracts the core parameters (frequency and damping ratio) of the electromechanical oscillation interval mode by using the Hankel-DMD method, has high calculation speed and high accuracy, and realizes the extraction of the electromechanical oscillation parameters under the normal operation condition of the power system; compared with the traditional method for extracting the electromechanical oscillation parameters by using a longer analysis window, the method only needs a very short analysis time window, so that the real-time extraction of the oscillation parameters can be realized; compared with the traditional method, the method has higher extraction precision and extraction speed, and good tracking performance, thereby having higher practical application value.
Drawings
The invention is further described with reference to the following figures and detailed description:
fig. 1 is a flow chart of the method for extracting electromechanical parameters of the power system based on the Hankel-DMD.
Fig. 2 is a four-machine two-area system wiring diagram in the embodiment of the method for extracting electromechanical parameters of the power system based on the Hankel-DMD.
Fig. 3 is a diagram of active power time series data of a tie line between two regions in an embodiment of the method for extracting electromechanical parameters of the power system based on Hankel-DMD.
Fig. 4 is a frequency parameter extraction result diagram in the embodiment of the method for extracting electromechanical parameters of the power system based on the Hankel-DMD.
Fig. 5 is a diagram of a damping ratio parameter extraction result in an embodiment of the Hankel-DMD based power system electromechanical parameter extraction method of the present invention.
Fig. 6 is a graph comparing the mode tracking performance of the Hankel-DMD-based electric power system electromechanical parameter extraction method and the traditional random subspace method (SSI) in the invention.
Fig. 7 is a comparison graph of the time for extracting the electromechanical parameters of the power system based on the Hankel-DMD and the time for extracting the mode parameters of the conventional stochastic subspace method (SSI) according to the present invention.
Detailed Description
The method for extracting the electromechanical parameters of the power system based on the Hankel-DMD is characterized in that a Hankel matrix is constructed by utilizing a tiny analysis time window, and online extraction of the electromechanical oscillation parameters is achieved. The method comprises the following steps:
step one, establishing an observation matrix by using data of a wide area measurement system as follows:
wherein, x (c) m ,t N ) Representing that data is sampled at a sampling point m at the time t; x is the number of l Representing a vector of all observable samples at time i.
In the original DMD method, the constructed observation matrix is directly processed as follows to obtain a system low-dimensional approximate matrix A':
(1) the discrete sampled metrology data can be represented by a linear map as:
x i+1 ＝Ax i
wherein A is a high-dimensional matrix of the wide-area measurement system, x i Sampled data for time i, x i+1 Is i +1Sampled data of time instants.
(2) And (3) substituting the observation matrix into the linear mapping to obtain a Krylov subspace sequence represented as:
X 1 N-1 ＝[x 1 Ax 1 … A N-2 x 1 ]＝[x 1 … x N-1 ]
wherein A is a high-dimensional matrix of the wide area measurement system,
is composed of t 1 To t N An observation matrix formed by the observation values of the moments;
is composed of t 1 Time to t N-1 An observation matrix formed by the observation values of the moments;
is composed of t 2 To t N And the observation matrix is formed by the observation values of the time.
Thereby, it is possible to obtain:
(3) further, by the original data matrix
The normal Orthogonal projection Decomposition (POD) mode matrix U of (a) associates the original system high-dimensional matrix a with its approximate low-dimensional approximation matrix a', i.e.:
A＝UA'U T
wherein, the matrix A' is a low-dimensional approximate matrix of the matrix A. T is denoted as the transpose of the matrix.
(4) The modal matrix mode U can be obtained by pairing the matrix
Singular Value Decomposition (SVD) was performed to obtain:
the mode matrix U is a left singular value vector, the matrix V is a right singular value vector, the mode matrix U and the matrix V are unitary matrices, and sigma is a singular value diagonal matrix.
(5) The low-dimensional approximate state matrix a' of the system can be solved by a minimization problem:
wherein, the first and the second end of the pipe are connected with each other,
is Frobenius norm.
In summary, the calculation formula of the system low-dimensional approximation matrix a' can be obtained as follows:
the system low-dimensional approximate matrix A' obtained by the method can extract the electromechanical oscillation parameters of the disturbance signals with obvious oscillation characteristics, but the oscillation characteristics in the random response signals are not obvious, so that the method cannot be used for extracting the electromechanical oscillation parameters of the random response signals.
In the invention, the observation matrix is constructed in the first step, and then the following steps are carried out to carry out dimension expansion processing on the observation matrix.
Step two, forming a Hankel matrix with p rows and q columns by using the mth row of data in the observation matrix as follows:
wherein the content of the first and second substances,
representing the m-th row observed quantity x (c) using the observation matrix 1 ,t 1 ) (i.e., x) m (t 1 ) To x (c) 1 ,t N-1 ) (i.e. x) m (t n-1 ) ) formed Hankel matrix.
Simultaneously observing the self-observed quantity x (c) in the mth row of the matrix 1 ,t 2 ) To x (c) 1 ,t N ) The resulting p rows and q columns of the Hankel matrix can be written as:
the scale factor is constructed using two newly formed Hankel matrices, which are calculated as:
wherein the content of the first and second substances,
for the last column of the Hankel matrix formed at sample point m,
the first column of the Hankel matrix formed at the sampling point m; and | l | · | is the euclidean norm of the matrix.
Reconstructing the data matrix by using the Hankel matrix at different sampling points:
carrying out SVD on the new data matrix to obtain:
approximate state matrix to system low dimension
Solving by minimizing a problem formula to calculate a system low-dimensional approximate matrix
Wherein the mode matrix
Left singular value vector, matrix, obtained for singular value decomposition
For its right singular value vector, the mode matrix
Sum matrix
Are all unitary matrices and are used as a matrix,
is a singular value diagonal matrix;
the minimization problem formula is as follows:
wherein, the first and the second end of the pipe are connected with each other,
is a Fibonacci Frobenius norm.
Step three, obtaining a system approximate low-dimensional state matrix in the extraction process
And then, decomposing the characteristic values to obtain:
wherein, w i And ω i Respectively, the eigenvalue and corresponding eigenvector of the discrete system low-dimensional approximation matrix.
And converting the discrete characteristic value into a characteristic value under a continuous system:
λ i ＝log(ω i )/Δt
wherein λ is i The characteristic value is the characteristic value of the continuous system; Δ t is the sampling interval.
Further calculating to obtain the oscillation frequency f of the electromechanical oscillation i And damping ratio sigma i Expressed as:
and the method for extracting the electromechanical parameters of the power system based on the Hankel-DMD is completed through calculation.
The invention utilizes the mth row of data in the constructed observation matrix to form a Hankel matrix with p rows and q columns, and the step is dimension expansion processing of an original observation matrix, aiming at extracting oscillation characteristics from random response signals with unobvious electromechanical oscillation characteristics, and using an original DMD algorithm for the oscillation signals in the aspect of extracting the oscillation characteristic parameters of the random response signals to realize the on-line extraction of the electromechanical oscillation parameters driven by random data and the stable evaluation of electromechanical small interference.
The present invention is further illustrated in detail below with reference to examples:
in the four-machine two-zone system shown in fig. 2, the active power P of the tie line 3-101 between the two zones is selected, and the time sequence data chart of the signal is shown in fig. 3. As can be seen from fig. 3, the signal is a noise-like signal since it is the active power of the tie-line of the system under normal operating conditions.
The electromechanical oscillation modes actually present at this time of the system were obtained using the small interference stability analysis (SSSA) and the parameters are listed in table 1. Meanwhile, the signal is used as an input signal of the method, and because the signal has certain randomness, an analysis window of 10s and a sliding window of 0.1s are selected to extract electromechanical oscillation parameters of the system under the operation condition. The results of the SSI and the extraction by the method of the invention were statistically analyzed and are also given in Table 1.
TABLE 1 comparison of parameter extraction results between the inventive method and the SSI method
The parameter extraction results (frequency and damping ratio) are shown in fig. 4 and 5, and it can be seen from fig. 4 and 5 that the statistical distribution of the extraction results has a certain randomness. Further analyzing fig. 4 and 5, it can be seen that the extracted frequency and damping ratio results are concentrated near the mean value, and the fitting curve of the probability distribution has an obvious peak near the mean value, further verifying the feasibility and effectiveness of the method of the present invention in the aspect of extracting the characteristic parameters of the electromechanical mode of the power system.
The change of the operation condition of the system is simulated, the comparison result of the electromechanical parameter extraction method of the power system and the mode tracking performance of the SSI is shown in figure 6. As can be seen from FIG. 6, when the system damping changes due to sudden changes of the system operation mode, the method can timely sense the changes of the system damping level, accurately track the relatively stable mode extraction result through the shortest transition time, and further verify the effectiveness of the method in the aspect of extraction of the electromechanical oscillation parameters of the power system. Meanwhile, under two operation modes before and after the operation mode is changed, the damping ratio extraction results of the two methods fluctuate slightly by taking the mean value as the center, but the extraction result of the method is closest to the small-interference reference value, the standard deviation of the result is smaller, and the extraction result has better reliability.
As can be seen from the comparison of the parameter extraction times of the two methods shown in FIG. 7, the method of the present invention has a shorter time, and is more suitable for real-time monitoring of the power system.
Claims (5)
1. The method for extracting the electromechanical parameters of the power system based on the Hankel-DMD is characterized by comprising the following steps: comprises the following steps which are sequentially carried out,
step one, constructing an observation matrix by utilizing observation sampling data obtained by a wide area measurement system
Wherein the content of the first and second substances,
is composed of t 1 To t N An observation matrix formed by the observation values of the moments; x (c) m ,t N ) Representing the sampling data at the sampling point m at the time t; x is the number of l Representing a vector formed by all observation sampling data values at the first moment;
step two, utilizing the self-observed quantity x (c) in the mth row in the constructed observation matrix 1 ,t 1 ) To x (c) 1 ,t N-1 ) Forming a Hankel matrix of p rows and q columns:
wherein the content of the first and second substances,
also shown is the m-th row observed quantity x using the observation matrix m (t 1 ) To x m (t n-1 ) The formed Hankel matrix is used as a matrix,
meanwhile, the self-observed quantity x (c) in the mth row of the observation matrix 1 ,t 2 ) To x (c) 1 ,t N ) The Hankel matrix forming p rows and q columns is:
wherein, the first and the second end of the pipe are connected with each other,
also shown is the m-th row observed quantity x using the observation matrix m (t 2 ) To x m (t n ) The formed Hankel matrix is used as a matrix,
constructing a scale factor alpha by utilizing the two Hankel matrixes m ，
Reconstructing the data matrix by using Hankel matrixes at different sampling points:
wherein the content of the first and second substances,
is composed of t 1 Time to t N-1 An observation matrix formed by the observation values of the moments;
is composed of t 2 To t N Observation of timeThe measurement matrix is used for measuring the matrix,
for the last column of the Hankel matrix formed at sample point m,
is the first column of the Hankel matrix formed at the sampling point m, and | | · | | is the Euclidean norm of the matrix;
step three, performing linear mapping on the discrete sampling data in the data matrix reconstructed in the step two, and bringing the observation matrix into the linear mapping to obtain a relationship between the Krlov Krylov subspace sequence and two adjacent subspace sequences as follows:
wherein A is a high-dimensional matrix of the wide area measurement system;
is composed of t 1 Time to t N-1 An observation matrix formed by the observation values of the moments;
is composed of t 2 To t N An observation matrix formed by the observation values of the moments;
step four, passing the observation matrix
Orthogonal projection Decomposition (POD) mode matrix of (1)
Heighten the original systemDimension matrix A and its approximate low dimension approximate matrix
And (3) associating, namely:
wherein, the matrix
The matrix is a low-dimensional approximate matrix of the matrix A, and T is expressed as the transposition of the matrix;
step five, mode matrix
By means of a pair matrix
Singular Value Decomposition (SVD) was performed to obtain:
wherein the mode matrix
Left singular value vector, matrix, obtained for singular value decomposition
For its right singular value vector, the mode matrix
Sum matrix
Are all unitaryThe matrix is a matrix of a plurality of matrices,
is a singular value diagonal matrix;
step six, approximating the state matrix of the system in low dimension
Solving by a minimization problem formula to obtain a system low-dimensional approximate matrix
The calculation formula of (2) is as follows:
step seven, approximating a system low-dimensional state matrix
The characteristic value decomposition is carried out to obtain:
wherein w i For low-dimensional approximation of the eigenvalues, omega, of a matrix of a discrete system i Eigenvectors corresponding to low-dimensional approximation matrices and eigenvalues for discrete systems,
converting the discrete eigenvalues to eigenvalues under a continuous system using the following equation:
λ i ＝log(ω i )/Δt
wherein λ is i The characteristic value is the characteristic value of the continuous system; Δ t is the sampling interval;
and step eight, calculating and obtaining the oscillation frequency and the damping ratio of the electromechanical oscillation by using the characteristic value of the continuous system obtained in the step seven through an oscillation frequency and damping ratio formula of the electromechanical oscillation, and thus, completing the calculation of the electromechanical parameter extraction method of the power system based on Hankel matrix enhanced dynamic mode decomposition Hankel-DMD.
2. The Hankel-DMD based power system electromechanical parameter extraction method as claimed in claim 1, wherein: the gram Lei Luofu Krylov subspace sequence obtained in the third step is as follows:
X 1 N-1 ＝[x 1 Ax 1 …A N-2 x 1 ]＝[x 1 … x N-1 ]
wherein A is a high-dimensional matrix of the wide area measurement system;
is composed of t 1 To t N An observation matrix formed by the observation values of the moments;
is composed of t 1 Time to t N-1 An observation matrix formed by the observation values of the moments;
is composed of t 2 To t N And the observation matrix is formed by the observation values of the time.
3. The Hankel-DMD based power system electromechanical parameter extraction method as claimed in claim 1, wherein: the linear mapping of the discrete sample data in the third step is represented as:
x i+1 ＝Ax i
wherein A is the high dimension of the wide area measurement systemMatrix, x i For sampled data at time i, x i+1 Is the sampled data at time i + 1.
4. The Hankel-DMD based power system electromechanical parameter extraction method as claimed in claim 1, wherein: the minimization problem formula in the sixth step is as follows:
wherein the content of the first and second substances,
is a Fibonacci Frobenius norm.
5. The Hankel-DMD based power system electromechanical parameter extraction method as claimed in claim 1, wherein: the formula of the oscillation frequency and the damping ratio of the electromechanical oscillation in the step eight is as follows:
wherein λ is i I.e. the characteristic value of the continuous system, f i To the oscillation frequency, σ i Is the damping ratio.
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