CN113517686A - Low-frequency oscillation analysis method based on Givens orthogonal similarity transformation - Google Patents

Low-frequency oscillation analysis method based on Givens orthogonal similarity transformation Download PDF

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CN113517686A
CN113517686A CN202110488277.3A CN202110488277A CN113517686A CN 113517686 A CN113517686 A CN 113517686A CN 202110488277 A CN202110488277 A CN 202110488277A CN 113517686 A CN113517686 A CN 113517686A
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刘尚伟
吴玲
赵友国
姜雪梅
刘仲尧
赵勇
成江东
潘凯岩
阎同东
吴昌川
高天亮
徐艳
刘海信
逄春
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Dongfang Electronics Co Ltd
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    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
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    • H—ELECTRICITY
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Abstract

The invention discloses a low-frequency oscillation analysis method based on Givens orthogonal similarity transformation, which comprises the following steps: reading an oscillation signal, and determining the number of sampling points and sampling intervals of an input signal; solving the differential equation coefficient of the low-frequency oscillation Prony algorithm autoregressive model, forming a Hessenberg matrix corresponding to a high-order equation according to the differential equation coefficient, and recording the Hessenberg matrix as a matrix H; solving the eigenvalue of the matrix H through a double QR algorithm based on Givens similarity transformation; solving attenuation factors and frequencies of all main modes of low-frequency oscillation according to the eigenvalue of the matrix H; and forming a Jacobian matrix according to the characteristic roots, and solving the amplitude and phase angle of each main mode of the low-frequency oscillation by using a least square method. The invention applies double QR transformation based on Givens orthogonal similarity transformation during analysis and calculation, thereby greatly reducing the calculated amount and improving the calculation efficiency of low-frequency oscillation analysis of the power system.

Description

Low-frequency oscillation analysis method based on Givens orthogonal similarity transformation
Technical Field
The invention relates to the technical field of power system automation, in particular to a low-frequency oscillation analysis method based on Givens orthogonal similarity transformation.
Background
With the increasingly expanded interconnected region of the power grid and the increasingly complex structure and model of the power system, the problems of stability and low-frequency oscillation of the power grid are increasingly serious. The low-frequency oscillation is a primary problem threatening the stable operation of the power system and limiting the transmission capability of the interconnected power grid, and even causing the collapse of the power grid in severe cases, so that the analysis and control of the low-frequency oscillation become one of the important subjects of the power system stability research. The low-frequency oscillation frequency range of the power system is generally 0.2-2.5 Hz, the Prony algorithm can directly extract the amplitude, the phase, the frequency and the attenuation factor of the oscillation signal, and the method is the most widely applied method for identifying the current low-frequency oscillation mode. The algorithm belongs to a time domain method for modal parameter identification, and essentially adopts the mathematics of fitting an equally-spaced sampling signal by using a group of multi-order linear combinations of exponential terms and directly calculating the information of amplitude, phase, frequency, damping factor and the like of the signal. When the oscillation mode is multi-step and the sampling rate increases, the amount of computation to identify the oscillation amplitude and phase out increases exponentially.
In the current low-frequency oscillation Prony algorithm of the power system, after solving a regular equation of an Autoregressive (AR) model to obtain a difference equation coefficient, the problem of solving a real coefficient characteristic polynomial to obtain a root is solved, which corresponds to a characteristic value of a Hessenberg matrix, and a QR algorithm is usually adopted to solve by using a householder orthogonal similarity transformation. The Hessenberg matrix formed in the Prony algorithm is a highly sparse matrix, the sparse matrix is usually changed into a full matrix through householder transformation, the calculation amount of solution is increased, and the number of sampling points in the Prony algorithm sampling is usually far greater than the actual order of the system. The QR algorithm based on householder transformation solves the characteristic value of the Hessenberg matrix with high dimensionality and high sparsity, greatly increases the calculation time consumption, and is difficult to meet the real-time analysis requirement of a power grid, so that a proper orthogonal transformation matrix is selected, and the method plays a significant role in improving the efficiency of low-frequency oscillation analysis of a power system.
Disclosure of Invention
The invention provides a low-frequency oscillation analysis method based on Givens orthogonal similarity transformation, which aims to: the calculation efficiency of the low-frequency oscillation Prony analysis of the power system is improved.
The technical scheme of the invention is as follows:
a low-frequency oscillation analysis method based on GivenS orthogonal similarity transformation comprises the following steps:
s1: reading an oscillation signal, and determining the number of sampling points and sampling intervals of an input signal;
s2: solving the differential equation coefficient of the low-frequency oscillation Prony algorithm autoregressive model, forming a Hessenberg matrix corresponding to a high-order equation according to the differential equation coefficient, and recording the Hessenberg matrix as a matrix H;
s3: solving the eigenvalue of the matrix H through a double QR algorithm based on Givens similarity transformation;
s4: solving attenuation factors and frequencies of all main modes of low-frequency oscillation according to the eigenvalue of the matrix H;
s5: and forming a Jacobian matrix according to the characteristic values, and solving the amplitude and phase angle of each main mode of the low-frequency oscillation by using a least square method.
As a further improvement of the method, the step S3 specifically includes the following steps:
s31: taking the matrix H as a current matrix A;
s32: finding out the position of the first zero element of the reciprocal of the secondary diagonal of the current matrix A to obtain the last diagonal block-irreducible quasi-triangular matrix C of the current matrix, wherein the matrix C is a lower right corner sub-matrix of the current matrix formed by all elements of the lower sides of the row partition lines and the right sides of the column partition lines by using the rows and the columns of the zero elements;
s33: judging the order of the diagonal block-irreducible quasi-triangular matrix C, if the order of the matrix C is 1 or 2, solving the eigenvalue of the matrix C, shrinking the current matrix A, returning to the step S32, otherwise, executing the step S34;
s34: performing double QR transformation based on Givens orthogonal similarity transformation on the current matrix A to obtain a quasi-triangular matrix B;
s35: and (4) taking the quasi-triangular matrix B as a new current matrix A, turning to the step S32, and performing loop iteration operation until the order of the current matrix A is shrunk to zero, namely all eigenvalues of the matrix H are solved.
As a further improvement of the method, in step S34, the step of performing Givens orthogonal similarity transformation-based double QR transformation on the current matrix a to obtain the quasi-triangular matrix B includes:
s341: determining an orthogonal transformation matrix G0And performing similarity transformation on the matrix A:
Figure BDA0003051267100000031
the orthogonal transformation matrix G0For melting
Figure BDA0003051267100000032
The first column is an upper triangle,
Figure BDA0003051267100000033
a dual QR transform matrix for matrix a to quasi-triangular matrix B:
Figure BDA0003051267100000034
Figure BDA0003051267100000035
B＝QTAQ
where Q is an orthogonal matrix, origin shifted by μ1And mu2Is the eigenvalue of the sub-matrix of 2 x 2 order in the lower right corner of A;
s342: continuing to form a plurality of new similarity transformation matrixes by using Givens transformation
G1，G2，…，Gi，…，Gn-2
By n-2 Givens transformations
Figure BDA0003051267100000041
Change matrix B1A quasi-triangular matrix B;
the relationship between the similarity transformation matrices is: qT＝Gn-2…G1G0。
As a further improvement of the method, the orthogonal transformation matrix G of step S3410The determination method comprises the following steps:
first, the lower right hand corner second order sub-matrix of matrix A
Figure BDA0003051267100000042
Has a characteristic value of mu1And mu2Let δ be μ1+μ2，ρ＝μ1μ2Then there is
Figure BDA0003051267100000043
Matrix array
Figure BDA0003051267100000044
Only the first three elements of (1) are non-zero, let this column be X0＝(p0，q0，r0，0，...，0)TX is obtained from the following formula0Non-zero elements of (a):
Figure BDA0003051267100000045
second, transform the matrix G using Givens01Through G01X0Elimination of element q0Is mixing X0Is changed to X'0；
The transformation matrix G01Is composed of
Figure BDA0003051267100000046
Wherein
Figure BDA0003051267100000051
The calculation result is X'0＝(p1，0，r0，0，...，0)T；
Thirdly, transforming the matrix G by using Givens02Through G02X′0Elimination of element r0(ii) a The transformation matrix G02Is composed of
Figure BDA0003051267100000052
Wherein
Figure BDA0003051267100000053
The transformation matrix G01And a transformation matrix G02The relationship between them is: g0＝G02G01。
As a further improvement of the method, the method for calculating the attenuation factor and the frequency of each main mode of low-frequency oscillation in step S4 includes:
Figure BDA0003051267100000054
wherein z isiThe characteristic root of the high-order equation corresponding to the characteristic value of the matrix H.
Compared with the prior art, the invention has the following beneficial effects: compared with householder transformation, the method applies double QR transformation based on Givens orthogonal similarity transformation in the Prony analysis and calculation, increases fewer non-zero injection elements in the transformation process, greatly reduces the calculated amount, improves the calculation efficiency of the Prony analysis of the low-frequency oscillation of the power system, and further improves the practicability of the low-frequency oscillation analysis of the power system.
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FIG. 1 is a flow chart of the present invention.
Detailed Description
The technical scheme of the invention is explained in detail in the following with the accompanying drawings:
as shown in fig. 1, a Givens orthogonal similarity transformation-based low-frequency oscillation analysis method includes the following steps:
s1: and reading the oscillation signal, and determining the number of sampling points and the sampling interval of the input signal.
S2: and solving the coefficient of a differential equation of the low-frequency oscillation Prony algorithm autoregressive model, forming a corresponding high-order equation Hessenberg matrix according to the coefficient of the differential equation, and recording the matrix as a matrix H.
The concrete solving steps are as follows:
the Prony algorithm is directed to a signal identification method of sampling data at equal intervals, a model of the signal can be represented as a set of decaying sinusoidal components, and estimation values of sampling signals x (0), x (1), … and x (n-1) can be expressed as
Figure BDA0003051267100000061
Wherein, bi，ziAssume complex, i.e.:
Figure BDA0003051267100000062
in the formula, AiIs the amplitude; alpha is alphaiIs an attenuation factor; f. ofiIs the frequency; Δ t is the sampling interval.
To find the fitting parameters, the sum of squares of errors is minimized
Figure BDA0003051267100000063
A complex least squares problem can be solved, i.e., solved.
Namely, the linear difference equation of the constant coefficients is set as:
Figure BDA0003051267100000071
if z is to beiViewed as the root of a linear constant coefficient difference equation, then ziSatisfy the characteristic equation
Figure BDA0003051267100000072
Then
Figure BDA0003051267100000073
The fitting to x (n) is a homogeneous solution of a constant coefficient linear difference equation, and the estimation error is set as e (n), that is
Figure BDA0003051267100000074
Substituting (5) into (3) to obtain
Figure BDA0003051267100000075
Wherein
Figure BDA0003051267100000076
A is obtained by solving the formula (5)i、θi、αi、fiIs estimated. The coefficients a of the difference equation can be found from the output signal generated by an Autoregressive (AR) model excited by the formula (5) x (n) which can be regarded as the noise E (n), and solving the regular equation of the AR modeli. The least squares estimation for the (6) parameters is such that
Figure BDA0003051267100000077
And minimum.
According to formula (6) having (x), (n) with xnIs shown)
Figure BDA0003051267100000081
Consider equation (6) as a cost function with a variable as a
Figure BDA0003051267100000082
To minimize it, let
Figure BDA0003051267100000083
Then there is
Figure BDA0003051267100000084
Corresponding to a minimum error energy of
Figure BDA0003051267100000085
Definition of
Figure BDA0003051267100000086
Obtaining a normal equation form of the coefficient a in the Prony algorithm according to (10) and (11):
Figure BDA0003051267100000087
to obtain
Figure BDA0003051267100000091
For brevity, this is: r.a ═ R
Where the elements of the matrix R are calculated as follows:
Figure BDA0003051267100000092
wherein i, j is 1, 2, …, p
While
Figure BDA0003051267100000093
Where i is 1, 2, …, p.
The coefficient a of the difference equation is obtained from the aboveiThen substituting the formula to obtain z by polynomial root findingi
Figure BDA0003051267100000094
That is to say
Figure BDA0003051267100000095
As can be seen from the knowledge of linear algebra,
Figure BDA0003051267100000096
can be seen as an eigenpolynomial of some real matrix, and therefore the eigenroot z of the high-order equation is solvediThe problem of (i ═ 1, 2., p) becomes the problem of finding all eigenvalues of the matrix. It can be verified that the matrix corresponding to the characteristic polynomial is the Hessenberg matrix
Figure BDA0003051267100000101
S3: since the matrix H is an upper Hessenberg matrix, all eigenvalues can be directly obtained by the dual QR algorithm.
The general steps of the real matrix double QR algorithm, the double QR conversion from the matrix A to the matrix B, and the calculation process is summarized as
Figure BDA0003051267100000102
Figure BDA0003051267100000103
B＝QTAQ (20)
Where A and B are both real matrices, Q is an orthogonal matrix, and the origin is shifted by mu1And mu2Is the eigenvalue of the sub-matrix of order 2 × 2 in the lower right corner of a.
Obviously, the key to the transformation is how to solve the orthogonal matrix Q. Because the essence of QR decomposition is a real matrix
Figure BDA0003051267100000104
Is an upper triangle:
Figure BDA0003051267100000105
and QTCan take the product of n-1 orthogonal matrices
QT＝Gn-2…G1G0
Figure BDA0003051267100000106
Matrix G0，G1，…，Gn-2Have the effect of multiplying them to the left
Figure BDA0003051267100000107
Then, the first n-1 columns are sequentially arranged as upper triangles. Thus, the orthonormal transformation B ═ QTAQ can be decomposed into n-1 orthogonal similarity transforms
Figure BDA0003051267100000108
That is to say, QR decomposition
Figure BDA0003051267100000111
And similarity transformation B ═ QTAQ can be performed simultaneously without the need to find the matrix Q.
In this embodiment, the eigenvalue of the matrix H is solved by a dual QR algorithm based on Givens similarity transformation, and the specific steps are as follows:
s31: taking the matrix H as the current matrix A.
S32: and finding out the position of the zero element which is the first zero element of the last diagonal of the current matrix A to obtain the final diagonal block-irreducible quasi-triangular matrix C of the current matrix, wherein the matrix C is a sub-matrix at the lower right corner of the current matrix formed by all elements of the row and column parting lines where the zero element is located, the lower sides of the row parting lines and the right sides of the column parting lines.
S33: and judging the order of the diagonal block-irreducible quasi-triangular matrix C, if the order of the matrix C is 1 or 2, solving the eigenvalue of the matrix C, shrinking the current matrix A (the shrunk matrix is the upper left corner sub-matrix corresponding to the lower right corner sub-matrix in the step S32), returning to the step S32, and otherwise, executing the step S34.
S34: performing double QR transformation based on Givens orthogonal similarity transformation on the current matrix A to obtain a quasi-triangular matrix B, which comprises the following specific steps:
s341: determining an orthogonal transformation matrix G0And performing similarity transformation on the matrix A:
Figure BDA0003051267100000112
Figure BDA0003051267100000113
the orthogonal transformation matrix G0For melting
Figure BDA0003051267100000114
The first column is an upper triangle.
The orthogonal transformation matrix G0The determination method comprises the following steps:
first, the lower right hand corner second order sub-matrix of matrix A
Figure BDA0003051267100000115
Has a characteristic value of mu1And mu2Let δ be μ1+μ2，ρ＝μ1μ2Then there is
Figure BDA0003051267100000116
Since A is a quasi-triangular matrix, the matrix
Figure BDA0003051267100000121
Only the first three elements of (1) are non-zero, let this column be X0＝(p0，q0，r00, 0.., 0) T, X is obtained from the following formula0Non-zero elements of (a):
Figure BDA0003051267100000122
second, transform the matrix G using Givens01Through G01X0Elimination of element q0Is mixing X0Is changed to X'0；
The transformation matrix G01Is composed of
Figure BDA0003051267100000123
Wherein
Figure BDA0003051267100000124
In-2Is an n-2 order identity matrix, matrix G01Wherein the elements not shown represent zero elements, and the calculation result is X'0＝(p1，0，r0，0，...，0)T；
Thirdly, transforming the matrix G by using Givens02Through G02X′0Elimination of element r0；
The transformation matrix G02Is composed of
Figure BDA0003051267100000125
Wherein
Figure BDA0003051267100000126
The transformation matrix G01And a transformation matrix G02The relationship between them is: g0＝G02G01By G0The result of the similarity transformation performed on the alignment triangle array A is
Figure BDA0003051267100000131
Where elements not indicated in the matrix represent zero elements, x represents non-zero elements, and … represents elements similar to the preceding matrix elements (zero or non-zero elements).
The right side position of the secondary diagonal is 0 element after Givens is multiplied by right, the position is 0, the multiplication operation of the position element is reduced in the transformation process, and the calculation iteration times are very large under the condition that the matrix dimension is very large, so that the calculation amount is greatly reduced.
It can be seen that the current matrix B1Still not a Hessenberg matrix (quasi-triangular matrix), quantization matrix B1The quasi-triangular matrix B is calculated by using n-2 Givens transformations, as detailed in step S342.
S342: continuing to form a new similarity transformation matrix G by using Givens transformation1，G2，…，Gi，…，Gn-2By n-2 Givens transformations
Figure BDA0003051267100000132
Figure BDA0003051267100000133
Change matrix B1A quasi-triangular matrix B; the relationship between the similarity transformation matrices is: qT＝Gn-2…G1G0。
The above-mentioned elementary orthogonal matrix Gi(i-1, 2.., n-2) has the same meaning as G0Similar simple equations.
For matrix B1If B is ordered1Is first column of (1) is X1Then matrix G1Should make the linear transformation G1X1Elimination of X1Element q of (1)1And r1In the visible, see G1Equation of (c) and block diagonal form and G0And the like. Successively transformed according to the respective similarity transformation matrices, in general, BiAnd B1Similarly, there are only three non-zero elements below the minor diagonal (two in column i, one in column i + 1)
Figure BDA0003051267100000141
If the ith column is Xi＝(×，×，...，pi，qi，ri，0，...，0)TParameter p thereofi、qi、ri(i-0, 1.., n-2) from the corresponding matrix BiThe ith column of (1):
Figure BDA0003051267100000142
when i is n-2, there are
Figure BDA0003051267100000143
Transforming matrix G with GivensiThe purpose of which is to convert by pair XiIs linearly transformed GiXiPost-elimination of element q thereiniAnd riThe conversion steps are divided into two steps:
first, transform the matrix G using Givensi1Through Gi1XiElimination of element q0Is mixing XiIs changed to X'i；
The transformation matrix Gi1Is composed of
Figure BDA0003051267100000144
Wherein
Figure BDA0003051267100000145
The result of the calculation is Xi1＝(pi1，0，ri，0，...，0)T。
Second, transform the matrix G using Givensi2Through Gi2X′iElimination of element ri；
The transformation matrix Gi2Is composed of
Figure BDA0003051267100000151
Wherein
Figure BDA0003051267100000152
The transformation matrix Gi1And a transformation matrix Gi2The relationship between them is: gi＝Gi2Gi1。
S35: and (4) taking the quasi-triangular matrix B as a new current matrix A, turning to the step S32, and performing loop iteration operation until the order of the current matrix A is shrunk to zero, namely all eigenvalues of the matrix H are solved.
S4: and solving the attenuation factor and the frequency of each main mode of the low-frequency oscillation according to the eigenvalue of the matrix H, wherein the calculation formula is as follows:
Figure BDA0003051267100000153
wherein z isiThe characteristic root of the high-order equation corresponding to the characteristic value of the matrix H.
S5: and forming a Jacobian matrix according to the characteristic roots, and solving the amplitude and phase angle of each main mode of the low-frequency oscillation by using a least square method.
In summary, in the Givens orthogonal similarity transformation-based power system low-frequency oscillation Prony algorithm, fitting of a sampling value is a homogeneous solution of a constant coefficient linear differential equation, and under the condition that the coefficients of the differential equation are known, root solving can be carried out according to the characteristic equation of the differential equation, namely, the characteristic value of a Hessenberg matrix formed by the coefficients of the differential equation is solved. In actual sampling, the number of sampling points is usually much larger than the actual order of the system, and under the condition of unclear system models, in order to make the sample matrix contain the comprehensive characteristics of signals as much as possible, the dimension of the sample matrix takes the maximum value, and the initial order is generally selected to be half of the number of sampling points, namely N/2, so that the dimension of the Hessenberg matrix is larger.
Method for repeatedly utilizing similarity transformation GAG in double QR algorithmTUsually, the left multiplication is to realize the elimination operation of non-zero elements under the secondary diagonal through Givens orthogonal transformation, when the elements above the diagonal of the matrix a are full arrays or matrices with higher packing density, the calculation amount is relatively small by using householder transformation, but a Hessenberg matrix formed by difference equation coefficients in a Prony algorithm has the characteristic of high sparseness, original corresponding rows and columns can be changed into full arrays after the householder transformation, on one hand, the Givens transformation increases relatively few non-zero injection elements in the transformation process, one element is less in each calculation, and after the calculation times increase explosively, the correspondingly reduced calculation amount is very considerable. On the other hand, programming is simple, and a 3 × 3 matrix of householder transforms is not required to be calculated.
Through practical verification of a low-frequency oscillation signal of a certain power grid, the sampling frequency is 20hz, the number of sampling points is 200, the order is 100, a Hessenberg matrix is 100 multiplied by 100, the characteristic value multiplication and addition operation is 6189681 times by adopting a double QR algorithm based on Householder transformation, the characteristic value multiplication and addition operation frequency is 4933812 by adopting the method, the operation efficiency is improved by 21%, the floating point multiplication and addition operation frequency is reduced, the calculation efficiency is improved, and the practicability of low-frequency oscillation analysis of the power system is further improved.

Claims (5)

1. A low-frequency oscillation analysis method based on Givens orthogonal similarity transformation is characterized in that: the method comprises the following steps:
s1: reading an oscillation signal, and determining the number of sampling points and sampling intervals of an input signal;
s2: solving the differential equation coefficient of the low-frequency oscillation Prony algorithm autoregressive model, forming a Hessenberg matrix corresponding to a high-order equation according to the differential equation coefficient, and recording the Hessenberg matrix as a matrix H;
s3: solving the eigenvalue of the matrix H through a double QR algorithm based on Givens similarity transformation;
s4: solving attenuation factors and frequencies of all main modes of low-frequency oscillation according to the eigenvalue of the matrix H;
s5: and forming a Jacobian matrix according to the characteristic values, and solving the amplitude and phase angle of each main mode of the low-frequency oscillation by using a least square method.
2. A Givens orthogonal similarity transformation-based low-frequency oscillation analysis method as claimed in claim 1, wherein: step S3 specifically includes the following steps:
s31: taking the matrix H as a current matrix A;
s32: finding out the position of the first zero element of the reciprocal of the secondary diagonal of the current matrix A to obtain the last diagonal block-irreducible quasi-triangular matrix C of the current matrix, wherein the matrix C is a lower right corner sub-matrix of the current matrix formed by all elements of the lower sides of the row partition lines and the right sides of the column partition lines by using the rows and the columns of the zero elements;
s33: judging the order of the diagonal block-irreducible quasi-triangular matrix C, if the order of the matrix C is 1 or 2, solving the eigenvalue of the matrix C, shrinking the current matrix A, returning to the step S32, otherwise, executing the step S34;
s34: performing double QR transformation based on Givens orthogonal similarity transformation on the current matrix A to obtain a quasi-triangular matrix B;
s35: and (4) taking the quasi-triangular matrix B as a new current matrix A, turning to the step S32, and performing loop iteration operation until the order of the current matrix A is shrunk to zero, namely all eigenvalues of the matrix H are solved.
3. A Givens orthogonal similarity transformation-based low-frequency oscillation analysis method as claimed in claim 2, wherein: step S34, the step of obtaining the quasi-triangular matrix B by performing Givens orthogonal similarity transformation-based double QR transformation on the current matrix a includes:
s341: determining an orthogonal transformation matrix G0And performing similarity transformation on the matrix A:
Figure FDA0003051267090000021
the orthogonal transformation matrix G0For melting
Figure FDA0003051267090000022
The first column isThe upper part of the triangle is provided with a triangle,
Figure FDA0003051267090000023
a dual QR transform matrix for matrix a to quasi-triangular matrix B:
Figure FDA0003051267090000024
Figure FDA0003051267090000025
B＝QTAQ
where Q is an orthogonal matrix, origin shifted by μ1And mu2Is the eigenvalue of the sub-matrix of 2 x 2 order in the lower right corner of A;
s342: continuing to form a plurality of new similarity transformation matrixes G by utilizing Givens transformation1，G2，…，Gi，…，Gn-2
By n-2 Givens transformations
Figure FDA0003051267090000026
Figure FDA0003051267090000027
Change matrix B1A quasi-triangular matrix B;
the relationship between the similarity transformation matrices is: qT＝Gn-2…G1G0。
4. A Givens orthogonal similarity transformation-based low-frequency oscillation analysis method as claimed in claim 3, wherein: step S341 the orthogonal transformation matrix G0The determination method comprises the following steps:
first, the lower right hand corner second order sub-matrix of matrix A
Figure FDA0003051267090000031
Has a characteristic value of mu1And mu2Let δ be μ1+μ2，ρ＝μ1μ2Then there is
Figure FDA0003051267090000032
Matrix array
Figure FDA0003051267090000033
Only the first three elements of (1) are non-zero, let this column be X0＝(p0，q0，r0，0，...，0)TX is obtained from the following formula0Non-zero elements of (a):
Figure FDA0003051267090000034
second, transform the matrix G using Givens01Through G01X0Elimination of element q0Is mixing X0Is changed to X'0；
The transformation matrix G01Is composed of
Figure FDA0003051267090000035
Wherein
Figure FDA0003051267090000036
The calculation result is X'0＝(p1，0，r0，0，...，0)T；
Thirdly, transforming the matrix G by using Givens02Through G02X′0Elimination of element r0；
The transformation matrix G02Is composed of
Figure FDA0003051267090000037
Wherein
Figure FDA0003051267090000038
The transformation matrix G01And a transformation matrix G02The relationship between them is: g0＝G02G01。
5. A Givens orthogonal similarity transformation-based low-frequency oscillation analysis method as claimed in any one of claims 1 to 4, wherein: step S4 is a method for calculating the attenuation factor and the frequency of each main mode of the low-frequency oscillation, including:
Figure FDA0003051267090000041
wherein z isiThe characteristic root of the high-order equation corresponding to the characteristic value of the matrix H.
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