Harmonic state space model-based linear period time-varying system stability determination method and system, and medium

Abstract

The invention discloses a linear periodic time-varying system stability judgment method and system based on a harmonic state space model and a medium, and belongs to the field of linear system dynamics analysis, and the method comprises the steps: respectively constructing harmonic state space models MH and ML with a high truncation order H and a low truncation order L for the system; calculating a first characteristic value and a first characteristic vector of ML; for the MH, arranging state variables of the MH according to the sequence of # imgabs0 # so as to transform the MH; calculating the deviation of the system matrix of the ML under the first characteristic value relative to the system matrix of the transformed MH to obtain a deviation matrix; calculating a characteristic value deviation according to the deviation matrix and the first characteristic vector, and calculating the sum of the characteristic value deviation and the first characteristic value to obtain a second characteristic value of the MH; and determining the stability of the linear period time-varying system according to the second characteristic value. The calculation efficiency is improved while the characteristic value calculation accuracy is ensured, and the applicability of the harmonic state space model to a large-scale system is enhanced.

Description

Technical Field

[0001] The present invention belongs to the field of linear system dynamics analysis, and more specifically, relates to a method, a system and a medium for determining the stability of a linear periodically time-varying system based on a harmonic state space model. Background Art

[0002] In the real physical world, systems generally have non-linearity and time-variation, and non-linear time-varying models are usually obtained when modeling actual systems. When analyzing the small-signal stability of an actual system, it can be linearized in the neighborhood of the steady-state trajectory of the non-linear time-varying model to obtain a linear periodically time-varying model. The system matrix of the linear periodically time-varying model contains parameters that vary periodically with time. Therefore, the existing mature stability analysis and control theory of linear time-invariant systems are difficult to directly apply. In this regard, researchers have proposed a harmonic state space model, which transforms the linear periodically time-varying model into a linear time-invariant model based on Fourier series expansion and harmonic balance principle, and then applies the mature stability analysis and control theory.

[0003] The eigenvalues of the harmonic state space model can be used for the stability analysis of actual systems. However, the harmonic state space model has the following problems. The truncation order determines the number of frequency components considered by the harmonic state space model, and thus also determines the analysis accuracy and calculation efficiency of the model. When the truncation order takes a higher value, the model has high analysis accuracy, but slow calculation speed and low efficiency; when the truncation order takes a lower value, although the calculation efficiency is improved, the model has poor analysis accuracy and may even draw wrong stability analysis conclusions. Therefore, the existing research has proposed an optimal selection method for the truncation order, which can reduce the truncation order as much as possible on the premise of ensuring the analysis accuracy of the harmonic state space model. However, for some large-scale actual systems, such as a power electronic power system with high voltage direct current transmission, a higher truncation order must be taken to accurately analyze its small-signal stability. At this time, it is difficult to balance the analysis accuracy and calculation efficiency of the harmonic state space model, and it is not applicable to large-scale systems. This also hinders the subsequent research on the stability mechanism analysis and control methods by applying the linear time-invariant system theory. Summary of the Invention

[0004] Aiming at the defects and improvement requirements of the existing technology, the present invention provides a method, a system and a medium for determining the stability of a linear periodically time-varying system based on a harmonic state space model, aiming to solve the problem that the existing technology cannot simultaneously ensure the accuracy and efficiency of the eigenvalue calculation of the harmonic state space model, so as to improve the applicability of the harmonic state space model to large-scale actual systems.

[0005] To achieve the above object, according to one aspect of the present invention, a method for determining the stability of a linear periodic time-varying system based on a harmonic state space model is provided, including: constructing a harmonic state space model M with a first truncation order H for the linear periodic time-varying system H and a harmonic state space model M with a second truncation order L L , where H > L; calculating the first eigenvalues and first eigenvectors of model M L ; for model M H , arranging its state variables in the order of for transformation, including the -L to L Fourier coefficients of the state variables in the form of complex exponential modulation signals, including the -H to -(L + 1) and (L + 1) to H Fourier coefficients of the state variables in the form of complex exponential modulation signals; calculating the deviation of the system matrix of model M L under the first eigenvalues relative to the system matrix of the transformed model M H to obtain a deviation matrix; calculating eigenvalue deviations according to the deviation matrix and the first eigenvectors, and calculating the sum of the eigenvalue deviations and the first eigenvalues to obtain the second eigenvalues of model M H ; determining the stability of the linear periodic time-varying system according to the second eigenvalues.

[0006] Furthermore, the transformed model M H is:

[0007]

[0008] where s is the Laplace operator, is the system matrix of the transformed model M H , A 11 , A 12 , A 21 and A 22 are four submatrices in the system matrix of the transformed model M H , and A 11 contains the system matrix of model M L .

[0009] Furthermore, the deviation matrix is:

[0010]

[0011] where A Δ (s) is the deviation matrix, s takes the first eigenvalues, is the system matrix of model M L , is a Toeplitz matrix with a second truncation order L is a block diagonal matrix containing frequency information from -L to L times, and I is the identity matrix.

[0012] Furthermore, and are respectively:

[0013]

[0014] where A l is the l-th Fourier coefficient of the system matrix of the system linear periodic time-varying model, l = -L, -(L - 1), …, 0, …, L - 1, L, ω 0 is the fundamental frequency, and diag[·] represents a block diagonal matrix.

[0015] Furthermore, the eigenvalue deviation is:

[0016]

[0017] where Δλ Li is the eigenvalue deviation, A Δ (s) is the deviation matrix, s is the Laplace operator, λ Li is the first eigenvalue, and the first eigenvector includes l i and r i , l i is the left eigenvector, and r i is the right eigenvector.

[0018] Furthermore, constructing model M H and model M L specifically includes: establishing a linear periodic time-varying model for the linear periodic time-varying system; processing the system matrix and state variables in the linear periodic time-varying model based on Fourier series expansion; and arranging the processed model into a matrix form based on the harmonic balance principle to obtain model M H and model M L .

[0019] Furthermore, the method further includes: determining the influence degree of the (L + 1)-th to H-th frequency components on the second eigenvalue according to the eigenvalue deviation, and controlling the linear periodic time-varying system according to the influence degree.

[0020] According to another aspect of the present invention, there is provided a linear periodic time-varying system stability determination system based on a harmonic state space model, including: a processor; a memory storing computer-executable programs, and when the programs are executed by the processor, the processor executes the linear periodic time-varying system stability determination method as described above.

[0021] According to another aspect of the present invention, there is provided a computer-readable storage medium, on which a computer program is stored, and when the program is executed by a processor, it implements the method for determining the stability of a linear periodically time-varying system based on the harmonic state space model as described above.

[0022] Generally speaking, through the above technical solutions conceived by the present invention, the following beneficial effects can be achieved:

[0023] (1) A method for determining the stability of a linear periodically time-varying system based on the harmonic state space model is provided. First, calculate the eigenvalues of the harmonic state space model with a low truncation order, and reorder the state variables of the harmonic state space model with a high truncation order to facilitate subsequent calculation of the deviation of the eigenvalue relative to the eigenvalue of the harmonic state space model with a high truncation order, so as to obtain the eigenvalue of the harmonic state space model with a high truncation order, which improves the calculation efficiency and reduces the demand for computer storage for solving the eigenvalue of the harmonic state space model with a high truncation order. It can take into account the accuracy and calculation efficiency of the stability analysis of the linear periodically time-varying system, and improve the applicability of the harmonic state space model to large-scale actual systems;

[0024] (2) A specific method for calculating the eigenvalue deviation is provided. The deviation matrix is multiplied by the eigenvector of the harmonic state space model with a low truncation order to obtain it. The calculation method is simple and the result is accurate, which further improves the calculation efficiency of the eigenvalue of the harmonic state space model with a high truncation order;

[0025] (3) Quantify the influence of high-frequency components (the (L + 1)-th to H-th frequency components) in the steady-state trajectory of the linear periodically time-varying system on the eigenvalue according to the eigenvalue deviation, which lays a foundation for the subsequent analysis and stable control of the system. Description of the Drawings

[0026] Figure 1 It is a flowchart of the method for determining the stability of a linear periodically time-varying system based on the harmonic state space model provided by the embodiment of the present invention. Detailed Embodiments

[0027] In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0028] In the present invention, terms such as "first" and "second" in the present invention and the drawings (if any) are used to distinguish similar objects and do not necessarily need to describe a specific order or sequence.

[0029] Embodiment 1

[0030] A method for determining the stability of a linear periodic time-varying system based on a harmonic state space model, refer to Figure 1 , the method includes operations S1 - S6.

[0031] In operation S1, a harmonic state space model M with a first truncation order H and a harmonic state space model M with a second truncation order L are respectively constructed for the linear periodic time-varying system H and a harmonic state space model M with a second truncation order L L , where H > L.

[0032] According to an embodiment of the present invention, in operation S1, constructing model M H and model M L specifically includes the following sub-operations S11 - S13.

[0033] In sub-operation S11, a linear periodic time-varying model is established for the linear periodic time-varying system. The linear periodic time-varying model of the actual system can be expressed as:

[0034]

[0035] where, x(t) = [x 1 (t), x 2 (t), …, x n (t)] T is the n-dimensional state variable of the linear periodic time-varying model, A(t) is the system matrix of the linear periodic time-varying model, A(t) = A(t + T), and T is the minimum period of the system.

[0036] In sub-operation S12, the system matrix and state variables in the linear periodic time-varying model are processed based on Fourier series expansion.

[0037] Based on Fourier series expansion, A(t) can be expressed as:

[0038]

[0039] where, A m is the m-th Fourier coefficient of A(t), and ω 0 is the fundamental frequency.

[0040] The state variable x(t) can be expressed in the form of a complex exponential modulation signal:

[0041]

[0042] where, X n is the n-th Fourier coefficient of x(t). Substituting equations (2) and (3) into equation (1), the processed model is obtained.

[0043] In sub-operation S13, based on the harmonic balance principle, the processed model is arranged in matrix form to obtain model M. H and model M L .

[0044] Model M H is:

[0045]

[0046] Model M L is:

[0047]

[0048] where A l is the l-th Fourier coefficient of the system matrix of the system linear periodic time-varying model, l = -L, -(L - 1), …, 0, …, L - 1, L, and diag[·] represents a block diagonal matrix. has a dimension of (2×H + 1)×n; has a dimension of (2×L + 1)×n.

[0049] Operation S2, calculate the first eigenvalue and the first eigenvector of model M L .

[0050] The calculation methods of the first eigenvalue and the first eigenvector are as follows:

[0051]

[0052] where λ Li is the first eigenvalue, and the first eigenvector includes l i and r i , l i is the left eigenvector, and r i is the right eigenvector.

[0053] Operation S3, for model M H , arrange its state variables in the order of to transform it. contains the -L to L-th Fourier coefficients of the state variables in the form of complex exponential modulation signals, contains the -H to -(L + 1) and (L + 1) to H-th Fourier coefficients of the state variables in the form of complex exponential modulation signals.

[0054] According to the embodiments of the present invention, the transformed model M H is:

[0055]

[0056] where s is the Laplace operator, is the transformed model M H of the system matrix, A 11 、A 12 、A 21 and A 22 are four sub - matrices in the system matrix of the transformed model M H where A 11 contains the system matrix of model M L

[0057] Operation S4, calculate the deviation of the system matrix of model M L at the first eigenvalue relative to the system matrix of the transformed model M H to obtain the deviation matrix.

[0058] According to an embodiment of the present invention, the deviation matrix is:

[0059]

[0060] where A Δ (s) is the deviation matrix, s takes the first eigenvalue, is the system matrix of model M L , is a Toeplitz matrix with the second truncation order L, is a block - diagonal matrix containing frequency information from - L to L, and I is the identity matrix.

[0061] Operation S5, calculate the eigenvalue deviation according to the deviation matrix and the first eigenvector, and calculate the sum of the eigenvalue deviation and the first eigenvalue to obtain the second eigenvalue of model M H .

[0062] According to an embodiment of the present invention, the eigenvalue deviation is:

[0063]

[0064] where Δλ Li is the eigenvalue deviation.

[0065] The second eigenvalue λ H of model M Hi = λ Li +Δλ Li . Through the above Operations S1 - S5, based on low - order matrix operations, the eigenvalues of the high - truncation - order harmonic state - space model are finally obtained, while ensuring the calculation accuracy and calculation efficiency.

[0066] Operation S6, determine the stability of the linear periodic time - varying system according to the second eigenvalue.

[0067] ​In this embodiment, the operation S6 can be implemented by using the existing method for determining the stability of a linear periodic time-varying system based on the eigenvalues of the harmonic state space model, which will not be elaborated here.

[0068] According to an embodiment of the present invention, the method further includes: quantifying the influence degree of the (L + 1)-th to H-th frequency components on the second eigenvalue according to the eigenvalue deviation, and performing a stability control design on the linear periodic time-varying system according to the influence degree.

[0069] Embodiment 2

[0070] A stability determination system for a linear periodic time-varying system based on a harmonic state space model includes: a processor; a memory storing computer-executable programs, which when executed by the processor cause the processor to execute the above-mentioned stability determination method for a linear periodic time-varying system based on a harmonic state space model. The related technical solutions are the same as those in Embodiment 1 and will not be elaborated here.

[0071] Embodiment 3

[0072] A computer-readable storage medium stores a computer program, which when executed by a processor implements the above-mentioned stability determination method for a linear periodic time-varying system based on a harmonic state space model. The related technical solutions are the same as those in Embodiment 1 and will not be elaborated here.

[0073] Those skilled in the art can easily understand that the above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principles of the present invention shall be included in the protection scope of the present invention.

Claims

1. A method for determining the stability of a linear periodic time-varying system based on a harmonic state space model, characterized in that: include: A harmonic state space model M with a first truncation order H is constructed for the linear periodic time-varying system. H and the harmonic state space model M with the second truncation order L L , H>L; Calculation Model M L The first eigenvalue and the first eigenvector of ; For model M H , its state variables are to transform them, Contains the -L to Lth order Fourier coefficients of the state variables in the form of complex exponential modulation signals, Contains -H to -(L+1) and (L+1) to H order Fourier coefficients of state variables in the form of complex exponential modulation signals; Calculate the first eigenvalue under the model M L The system matrix is ​​relative to the transformed model M H The deviation of the system matrix is ​​obtained to obtain the deviation matrix; Calculate the eigenvalue deviation according to the deviation matrix and the first eigenvector, calculate the sum of the eigenvalue deviation and the first eigenvalue, and obtain the model M H The second eigenvalue of The stability of the linear periodic time-varying system is determined according to the second eigenvalue.

2. The method for determining the stability of a linear periodic time-varying system based on a harmonic state space model according to claim 1, characterized in that: The transformed model M H for: Where s is the Laplace operator, is the transformed model M H The system matrix, A 11 , A 12 , A 21 and A 22 is the transformed model M H The four sub-matrices in the system matrix, A 11 Contains model M L The system matrix.

3. The method for determining the stability of a linear periodic time-varying system based on a harmonic state space model according to claim 2, characterized in that: The deviation matrix is: Among them, A Δ (s) is the deviation matrix, s takes the first eigenvalue, For model M L The system matrix, is a Toeplitz matrix with second truncation order L, is a block diagonal matrix containing -L to L frequency information, and I is the identity matrix.

4. The method for determining the stability of a linear periodic time-varying system based on a harmonic state space model according to claim 3, characterized in that: and They are: Among them, A l is the l-th order Fourier coefficient of the system matrix of the linear periodic time-varying model of the system, l = -L, -(L-1),…,0,…,L-1,L, ω0 is the fundamental frequency, and diag[·] represents a block diagonal matrix.

5. The method for determining the stability of a linear periodic time-varying system based on a harmonic state space model according to claim 1, characterized in that: The characteristic value deviation is: Among them, Δλ Li is the eigenvalue deviation, A Δ (s) is the deviation matrix, s is the Laplace operator, λ Li is the first eigenvalue, and the first eigenvector includes l i and r i , l i is the left eigenvector, r i is the right eigenvector.

6. The method for determining the stability of a linear periodic time-varying system based on a harmonic state space model according to claim 1, characterized in that: Build Model M H and Model M L Specifically include: Establishing a linear periodic time-varying model for the linear periodic time-varying system; Processing the system matrix and state variables in the linear periodic time-varying model based on Fourier series expansion; Based on the principle of harmonic balance, the processed model is organized into a matrix form to obtain the model M H and Model M L .

7. The method for determining the stability of a linear periodic time-varying system based on a harmonic state space model according to claim 1, characterized in that: The method further comprises: The influence of the L+1th to Hth frequency components on the second eigenvalue is quantified according to the eigenvalue deviation, and stable control design is performed on the linear periodic time-varying system according to the influence degree.

8. A linear periodic time-varying system stability determination system based on a harmonic state space model, characterized in that: include: processor; A memory storing a computer executable program, wherein when the program is executed by the processor, the processor executes the method for determining the stability of a linear periodic time-varying system based on a harmonic state space model as described in any one of claims 1 to 7.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the program is executed by a processor, the method for determining the stability of a linear periodic time-varying system based on a harmonic state space model as described in any one of claims 1 to 7 is implemented.

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