Truncation number determination method, system and device suitable for HSS model and storage medium

Abstract

The invention discloses a truncation number determination method, system and device suitable for an HSS model and a storage medium, and the method comprises the steps: carrying out the time domain simulation based on a nonlinear time-varying periodic system model, obtaining the time domain waveform of a state variable during the instability of the system, obtaining the damping and oscillation frequency based on the time domain waveform of the state variable during the instability, and determining the truncation number of the HSS model. Determining a real part and an imaginary part of the instability mode; converting the linear time-varying periodic system after the periodic trajectory linearization into an infinite-order harmonic state space model, obtaining a characteristic root which is the same as a real part and an imaginary part of a to-be-analyzed instability mode by increasing a truncation order, and determining a reasonable truncation number of the harmonic state space model; and constructing a truncated harmonic state space model based on the reasonable truncation number, and analyzing the to-be-analyzed instability mode to obtain a state variable leading system instability. According to the method, the accuracy of the model is ensured, and intrinsic errors of a small-interference stability analysis result and adoption of a relatively large truncation number can be avoided.

Description

Technical Field

[0001] This invention relates to the technical field of power system stability control, and in particular to a method, system, device, and storage medium for determining the cutoff number applicable to HSS models. Background Technology

[0002] Traditional synchronous machine-dominated power systems exhibit nonlinear time-periodic (NLTP) characteristics. This characteristic is introduced by the inductance matrix in the motor equations and determined by the symmetrical power frequency sinusoidal AC grid. Therefore, the Park transform is widely used to make the system time-invariant. Furthermore, by linearizing the system at the steady-state operating point, classical linear time-invariant system stability analysis methods are used to analyze the system's small-disturbance stability. However, with the widespread application of power electronic equipment, dynamic coupling between sequences introduced by switching dynamics and negative-sequence control leads to complex time-varying characteristics in the system, rendering the time-invariant method based on the Park transform ineffective. Therefore, the Harmonic State-Space (HSS) modeling method based on Fourier series expansion and harmonic balance principles is used to model systems with complex time-varying characteristics as time-invariant. Based on the linearization of the trajectory of the NLTP system to obtain the Linear Time Periodic (LTP) system, the original time-varying characteristics in the time domain are divided into frequency components, thus making the system equivalently time-invariant under each frequency component. This modeling method overcomes the shortcomings of the Park transform, enabling the application of time-invariant classical stability analysis theory.

[0003] To address the above issues, existing methods have constructed a dq impedance model for a voltage source converter connected to a synchronous machine. This model analyzes the interaction between the two and the resulting subsynchronous oscillation problem, and proposes suppression strategies. Based on the Park transform, the time-varying inductance matrix in the motor equations and the symmetric structural parameters of the power frequency sinusoidal AC grid remain unchanged in the synchronously rotating dq coordinate system. However, this method faces the risk of failure when harmonic components are present in the grid and complex control is introduced. Other methods have constructed an HSS model for the MMC based on harmonic state-space modeling to characterize the coupling relationships between harmonics within the MMC, including submodule voltage ripple and bridge arm circulating current. The actual obtained HSS model is an infinite-order model; however, the truncation order is empirically set to 3 during analysis, which is unfounded and may lead to errors in stability analysis results. Since the theoretical harmonic state-space model (HSS model) is an infinite-order model, truncation is necessary in practical applications. The truncation order of an HSS model significantly impacts its accuracy. If the truncation order is too low, the model will be inaccurate, leading to intrinsic errors in small-disturbance stability analysis results. Conversely, if the truncation order is too high, while the modeling may be accurate, it can cause the curse of dimensionality, greatly increasing computational costs and potentially making analysis impossible. Therefore, a reasonable truncation order is crucial for HSS modeling. Summary of the Invention

[0004] In view of the aforementioned existing problems, this invention is proposed. Therefore, this invention provides a method, system, device, and storage medium for determining the truncation number in an HSS model, addressing the problems mentioned in the background art.

[0005] To solve the above-mentioned technical problems, the present invention provides the following technical solution:

[0006] In a first aspect, embodiments of the present invention provide a method for determining the cutoff number applicable to the HSS model, comprising: performing time-domain simulation based on a nonlinear time-varying periodic system model to obtain the time-domain waveform of the state variables when the system is unstable, and obtaining the damping and oscillation frequency based on the time-domain waveform of the state variables when the system is unstable, so as to determine the real part and imaginary part of the instability mode;

[0007] The linear time-varying periodic system after linearizing the periodic trajectory is transformed into an infinite-order harmonic state-space model. By increasing the truncation order, the same eigenvalues ​​as the real and imaginary parts of the instability mode to be analyzed are obtained, and the reasonable truncation number of the harmonic state-space model is determined.

[0008] Based on the reasonable truncation number, a truncated harmonic state-space model is constructed, and the instability mode to be analyzed is analyzed to obtain the state variables of the dominant system instability.

[0009] As a preferred embodiment of the truncation number determination method applicable to HSS models described in this invention, the step of obtaining the damping and oscillation frequency based on the time-domain waveform of the state variables during instability to determine the real and imaginary parts of the instability mode includes:

[0010] Based on the time-domain waveform of the state variables in the initial stage of system instability, a target time window is selected;

[0011] The time-domain waveform within the target time window is approximated by fitting a complex exponential decay model; where the term with a positive complex exponential decay factor is denoted as the unstable oscillation term.

[0012] For each unstable oscillation term, the corresponding damping and oscillation frequency are calculated; the damping value is regarded as the real part of the unstable mode, and the oscillation frequency is adjusted according to the grid frequency and used as the imaginary part of the unstable mode.

[0013] The beneficial effect of this preferred technical solution is that by representing the original waveform through a linear combination of complex exponential functions with different frequencies and attenuation rates, the error energy can be minimized.

[0014] As a preferred embodiment of the method for determining the cutoff number applicable to the HSS model described in this invention, the transformation of the linear time-varying periodic system after linearizing the periodic trajectory into an infinite-order harmonic state-space model includes:

[0015] A linear time-varying periodic system model is obtained by linearizing the system near its periodic trajectory. The linear time-varying periodic system model is represented in state-space form, which includes the Jacobian matrix.

[0016] By using Fourier series expansion and the harmonic balance method, a linear time-varying periodic system is transformed into an infinite-order harmonic state-space model; the harmonic state-space model is used to describe the dynamic behavior of the system under multiple frequency components.

[0017] As a preferred embodiment of the method for determining the cutoff number applicable to the HSS model described in this invention, the determination of a reasonable cutoff number for the harmonic state-space model by increasing the cutoff order to obtain eigenvalues ​​identical to the real and imaginary parts of the unstable mode to be analyzed includes:

[0018] The infinite harmonic state-space model is truncated to order m to construct a finite-dimensional harmonic state-space model, and the eigenvalues ​​of the truncated harmonic state-space model are calculated.

[0019] If the real and imaginary parts of any eigenvalue are consistent with the real and imaginary parts of the unstable mode to be analyzed, then the reasonable truncation number of the harmonic state-space model is determined to be m; if the real and imaginary parts of any eigenvalue are inconsistent with the real and imaginary parts of the unstable mode to be analyzed, then the value of the truncation order m is updated according to m = m + 1 until a matching eigenvalue is found, and the iteration is completed.

[0020] As a preferred embodiment of the truncation number determination method applicable to HSS models described in this invention, the truncation harmonic state-space model is constructed based on the reasonable truncation number, and the instability mode to be analyzed is analyzed to obtain the state variables of the dominant system instability, including:

[0021] Once a reasonable truncation order m is determined, eigenvalue decomposition is performed on the matrix after reasonable truncation of order m to obtain the left and right eigenvectors corresponding to the instability mode.

[0022] The participation factor of each state variable in the instability mode is calculated by multiplying the left and right eigenvectors; the state variable with the largest magnitude of the participation factor is taken as the dominant state variable in the system instability.

[0023] The beneficial effect of this preferred technical solution is that by identifying the unstable oscillation modes of the system and their dominant state variables, the accuracy of system stability assessment can be improved.

[0024] As a preferred embodiment of the method for determining the cutoff number applicable to the HSS model described in this invention, the harmonic state-space model is used to describe the dynamic behavior of the system under multiple frequency components, including:

[0025] The magnitudes of each frequency of the state variable x(t) are expressed as follows:

[0026] X hss =[…,X -2 X -1 [X0, X1, X2, ...] T

[0027] in, Let be the amplitude of the i-th frequency component after Fourier decomposition of x(t);

[0028] The magnitudes of each frequency of the system matrix A(t) are expressed as follows:

[0029]

[0030] in, Let be the amplitude of the i-th frequency component after Fourier decomposition of A(t);

[0031] The block diagonal matrix constructed from the system's fundamental angular frequency and identity matrix is ​​represented as follows:

[0032] N hss=diag[...-2jω0E-jω0E 0jω0E 2jω0E...]

[0033] Where E is the n-dimensional identity matrix, 0 is the n-dimensional zero matrix, diag[·] is the block diagonal matrix, ω0 is the fundamental angular frequency, and j is the imaginary unit.

[0034] As a preferred embodiment of the truncation number determination method applicable to HSS models described in this invention, the finite-dimensional harmonic state-space model is expressed as:

[0035]

[0036] Among them, X hss(m) Let A be the matrix of frequency amplitudes of x(t) after Fourier decomposition and truncated to order m. hss(m) Let A(t) be the frequency amplitude after Fourier decomposition, and N be the matrix after m-order truncation. hss(m) This is the matrix after m-order truncation of the diagonal matrix calculated using the n-dimensional identity matrix E and the n-dimensional zero matrix 0.

[0037] Secondly, the present invention provides a truncation number determination system suitable for HSS models, comprising:

[0038] The instability mode acquisition module is used to perform time-domain simulation based on a nonlinear time-varying periodic system model, acquire the time-domain waveform of the state variables when the system is unstable, and acquire the damping and oscillation frequency based on the time-domain waveform of the state variables when the system is unstable, so as to determine the real and imaginary parts of the instability mode.

[0039] The module for determining the reasonable truncation number of the model is used to transform the linear time-varying periodic system after linearizing the periodic trajectory into an infinite-order harmonic state-space model. By increasing the truncation order, the module obtains the same eigenvalues ​​as the real and imaginary parts of the unstable mode to be analyzed, and determines the reasonable truncation number of the harmonic state-space model.

[0040] The dominant system instability state variable discrimination module is used to construct a truncated harmonic state space model based on the reasonable truncation number, and to analyze the instability mode to be analyzed, so as to obtain the dominant system instability state variables.

[0041] Thirdly, the present invention provides an electronic device, comprising:

[0042] Memory and processor;

[0043] The memory is used to store computer-executable instructions, and the processor is used to execute the computer-executable instructions, which, when executed by the processor, implement the steps of the method for determining the truncation number applicable to the HSS model.

[0044] Fourthly, the present invention provides a computer-readable storage medium storing computer-executable instructions that, when executed by a processor, implement the steps of the method for determining the truncation number applicable to the HSS model.

[0045] Compared with the prior art, the beneficial effects of the present invention are as follows: The present invention extracts oscillation modes based on the Prony method and determines the reasonable truncation number of the infinite-order HSS model based on this; on the one hand, it ensures the accuracy of the model and avoids intrinsic errors in the stability analysis results of small disturbances; on the other hand, it avoids using a large truncation number, which is beneficial to the application of the infinite-order harmonic state model in large systems. By reasonably determining the order of truncation of the infinite-order HSS model, the state variables of the dominant system instability can be accurately identified. Attached Figure Description

[0046] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. Wherein:

[0047] Figure 1 This is a flowchart of a method for determining the cutoff number applicable to an HSS model according to an embodiment of the present invention;

[0048] Figure 2 This is a flowchart of a method for determining the truncation number applicable to an HSS model, as described in one embodiment of the present invention.

[0049] Figure 3 This is a diagram of a voltage source converter system connected to an infinite grid, illustrating a method for determining the cutoff number applicable to the HSS model according to an embodiment of the present invention.

[0050] Figure 4 The image shows the time-domain waveform of the state variables during system instability, according to an embodiment of the present invention, which describes a method for determining the cutoff number applicable to the HSS model.

[0051] Figure 5 This is a system eigenvalue plot showing the method for determining the cutoff number applicable to the HSS model as an embodiment of the present invention increases with the cutoff number m;

[0052] Figure 6 The results of the system instability mode participation factor analysis when the cutoff number m = 7 are provided in an embodiment of the present invention, which describes a method for determining the cutoff number applicable to the HSS model. Detailed Implementation

[0053] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the protection scope of the present invention.

[0054] Example 1, referring to Figures 1-2 This is one embodiment of the present invention, which provides a method for determining the cutoff number applicable to the HSS model, such as... Figure 1 As shown, it includes:

[0055] S100: Time-domain simulation is performed based on a nonlinear time-varying periodic system model to obtain the time-domain waveform of the state variables when the system is unstable. Damping and oscillation frequency are obtained based on the time-domain waveform of the state variables when the system is unstable in order to determine the real and imaginary parts of the instability mode.

[0056] S200: Transform the linear time-varying periodic system after linearizing the periodic trajectory into an infinite-order harmonic state-space model. By increasing the truncation order, obtain the same eigenvalues ​​as the real and imaginary parts of the instability mode to be analyzed, and determine the reasonable truncation number of the harmonic state-space model.

[0057] S300: Construct a truncated harmonic state-space model based on a reasonable truncation number, and analyze the instability mode to be analyzed to obtain the state variables of the dominant system instability.

[0058] It should be noted that since the theoretical harmonic state-space model (HSS) is an infinite-order model, truncation is necessary in practical applications. The truncation order of the HSS model significantly affects its accuracy. If the truncation order is too low, the model will be inaccurate, leading to intrinsic errors in small-disturbance stability analysis results. If the truncation order is too high, although the modeling will be accurate, it will lead to the curse of dimensionality, greatly increasing the computational cost of the analysis, or even making analysis impossible. Therefore, a reasonable truncation order is crucial for HSS modeling. Addressing the shortcomings of existing technologies and the need for improvement, this invention provides a method for determining the truncation order suitable for HSS models. Its purpose is to reasonably determine the truncation order for an infinite-order HSS model and accurately identify the state variables that cause instability in the dominant system.

[0059] like Figure 2 As shown in this embodiment, the general expression for the nonlinear time-varying periodic system in step S100 is:

[0060]

[0061] in, Let n be the system's state variable, n be the system dimension, and f be the system's dynamic equation.

[0062] It should be noted that all state variables of the NLTP system are consistent with the system dimension n.

[0063] In an optional embodiment, when the present application performs time-domain simulation of the determined system using a general expression, the time-domain waveform of the state variables when the system becomes unstable is calculated by ode45.

[0064] In another optional embodiment, other numerical calculation methods can be selected to complete the calculation of the time-domain waveform, depending on the actual calculation accuracy requirements.

[0065] In this embodiment of the application, step S100, which involves obtaining the damping and oscillation frequency based on the time-domain waveform of the state variables during instability to determine the real and imaginary parts of the instability mode, includes:

[0066] Based on the time-domain waveform of the state variables in the initial stage of system instability, a target time window is selected;

[0067] The time-domain waveform within the target time window is approximated by fitting a complex exponential decay model; where the term with a positive complex exponential decay factor is denoted as the unstable oscillation term.

[0068] For each unstable oscillation term, the corresponding damping and oscillation frequency are calculated; the damping value is regarded as the real part of the unstable mode, and the oscillation frequency is adjusted according to the grid frequency and used as the imaginary part of the unstable mode.

[0069] Specifically, Prony analysis is performed on the time-domain waveform of the state variables during instability to obtain the damping and oscillation frequency. The time-domain waveform of the state variables in the initial stage of instability is selected as the time window for Prony analysis. A linear combination of complex exponentially decaying frequency components of appropriate order is chosen to approximate the time-domain waveform within the selected time window, minimizing the error energy. The term with a positive complex exponential decay factor is the instability oscillation term, from which the damping and frequency of the oscillation can be further obtained. The damping obtained from the Prony analysis is the real part of the instability mode, and the oscillation frequency is 1 / (2πf) of the imaginary part of the instability mode, where f is the grid frequency.

[0070] It should be noted that, based on the acquired time-domain waveform, a time window is selected for the waveform oscillation and divergence phase for Prony analysis. The length of the time window is determined specifically according to the oscillation waveform and analysis requirements.

[0071] In this embodiment of the application, step S200, which transforms the linear time-varying periodic system after linearizing the periodic trajectory into an infinite-order harmonic state-space model, includes:

[0072] A linear time-varying periodic system model is obtained by linearizing the system near its periodic trajectory. The linear time-varying periodic system model is represented in state-space form, which includes the Jacobian matrix.

[0073] By using Fourier series expansion and the harmonic balance method, a linear time-varying periodic system is transformed into an infinite-order harmonic state-space model; the harmonic state-space model is used to describe the dynamic behavior of the system under multiple frequency components.

[0074] Specifically, the LTP system obtained after linearization near the periodic trajectory is transformed into an infinite-order HSS model, and the truncation order m = 1 is initialized. In this embodiment, linearization is specifically performed using Taylor series expansion and ignoring second-order and higher-order terms. The general expression of the incremental form state-space model of the linearized LTP system is as follows:

[0075]

[0076] in, Let be the Jacobian matrix of the system near the periodic trajectory.

[0077] Furthermore, using Fourier series expansion and the harmonic balance principle, the state-space model of the LTP system is transformed into an infinite-order HSS model. The expression of the infinite-order HSS model is:

[0078]

[0079] In this embodiment of the application, the harmonic state-space model used in step S200 to describe the dynamic behavior of the system under multiple frequency components includes:

[0080] The magnitudes of each frequency of the state variable x(t) are expressed as follows:

[0081] X hss =[…,X -2 X -1 [X0, X1, X2, ...] T

[0082] in, Let be the amplitude of the i-th frequency component after Fourier decomposition of x(t);

[0083] The magnitudes of each frequency of the system matrix A(t) are expressed as follows:

[0084]

[0085] in, Let be the amplitude of the i-th frequency component after Fourier decomposition of A(t);

[0086] The block diagonal matrix constructed from the system's fundamental angular frequency and identity matrix is ​​represented as follows:

[0087] N hss =diag[...-2jω0E-jω0E 0jω0E 2jω0E...]

[0088] Where E is the n-dimensional identity matrix, 0 is the n-dimensional zero matrix, diag[·] is the block diagonal matrix, ω0 is the fundamental angular frequency, and j is the imaginary unit.

[0089] In this embodiment of the application, step S200 involves increasing the truncation order to obtain eigenvalues ​​that are identical to the real and imaginary parts of the unstable mode to be analyzed, and determining a reasonable truncation number for the harmonic state-space model, including:

[0090] The infinite-order harmonic state-space model is truncated to the m-th order to construct a finite-dimensional harmonic state-space model, and the eigenvalues ​​of the truncated harmonic state-space model are calculated.

[0091] If the real and imaginary parts of any eigenvalue are consistent with the real and imaginary parts of the unstable mode to be analyzed, then the reasonable truncation number of the harmonic state-space model is determined to be m; if the real and imaginary parts of any eigenvalue are inconsistent with the real and imaginary parts of the unstable mode to be analyzed, then the value of the truncation order m is updated according to m = m + 1 until a matching eigenvalue is found, and the iteration is completed.

[0092] In this embodiment of the application, the finite-dimensional harmonic state-space model in step S200 is represented as follows:

[0093]

[0094] Among them, X hss(m) Let A be the matrix of frequency amplitudes of x(t) after Fourier decomposition and truncated to order m. hss(m) Let A(t) be the frequency amplitude after Fourier decomposition, and N be the matrix after m-order truncation. hss(m) This is the matrix after m-order truncation of the diagonal matrix calculated using the n-dimensional identity matrix E and the n-dimensional zero matrix 0.

[0095] For example, taking m=2 as an example, after m-order truncation, it can be expressed as:

[0096]

[0097] N hss(2) =diag[-2jω0E-jω0E O jω0E 2jω0E]

[0098] In this embodiment of the application, step S300 involves constructing a truncated harmonic state-space model based on a reasonable truncation number and analyzing the instability mode to be analyzed, obtaining the state variables of the dominant system instability, including:

[0099] Once a reasonable truncation order m is determined, eigenvalue decomposition is performed on the matrix after reasonable truncation of order m to obtain the left and right eigenvectors corresponding to the instability mode.

[0100] The participation factor of each state variable in the instability mode is calculated by multiplying the left and right eigenvectors; the state variable with the largest participation factor magnitude is taken as the dominant state variable in the system instability.

[0101] It should be noted that this invention extracts oscillation modes based on the Prony method and determines a reasonable truncation number for the infinite-order HSS model based on this. On the one hand, this ensures the accuracy of the model and avoids intrinsic errors in the stability analysis results of small disturbances. On the other hand, it avoids using a large truncation number, which is beneficial for the application of the infinite-order harmonic state model in large systems. By reasonably determining the order of truncation of the infinite-order HSS model, the state variables of the dominant system instability can be accurately identified.

[0102] Example 2 is an embodiment of the present invention. This embodiment differs from the first embodiment in that it provides a truncation number determination system suitable for the HSS model, comprising:

[0103] The instability mode acquisition module is used to perform time-domain simulation based on a nonlinear time-varying periodic system model, acquire the time-domain waveform of the state variables when the system is unstable, and acquire the damping and oscillation frequency based on the time-domain waveform of the state variables when the system is unstable, so as to determine the real and imaginary parts of the instability mode.

[0104] The module for determining the reasonable truncation number of the model is used to transform the linear time-varying periodic system after linearizing the periodic trajectory into an infinite-order harmonic state-space model. By increasing the truncation order, the module obtains the same eigenvalues ​​as the real and imaginary parts of the unstable mode to be analyzed, and determines the reasonable truncation number of the harmonic state-space model.

[0105] The dominant system instability state variable discrimination module is used to construct a truncated harmonic state space model based on a reasonable truncation number, and to analyze the instability mode to be analyzed, thereby obtaining the state variables of the dominant system instability.

[0106] Furthermore, the instability mode acquisition module includes a time-domain simulation unit and a Prony analysis unit;

[0107] The time-domain simulation unit is used to perform time-domain simulation on the NLTP system model, obtain the time-domain waveforms of the state variables when the system becomes unstable, and trigger the Prony analysis unit.

[0108] The Prony analysis unit is used to perform Prony analysis on the time-domain waveform of the state variable during instability, obtain the damping and oscillation frequency, and determine the real and imaginary parts of the instability mode.

[0109] Furthermore, the module for determining the reasonable truncation number of the model includes an initialization unit, a truncation unit, and a control unit;

[0110] An initialization unit is used to transform the state-space model of the LTP system into an infinite-order HSS model and initialize the truncation order m = 1.

[0111] The truncation unit is used to truncate the infinite-order HSS model to order m and trigger the control unit.

[0112] The control unit is used to calculate the eigenvalues ​​of the truncated infinite-order HSS model and determine whether there are eigenvalues ​​that are the same as the real and imaginary parts of the unstable mode to be analyzed. If so, the reasonable truncation number of the HSS model is determined to be m; otherwise, the truncation order m is updated according to m = m + 1 and the truncation unit is triggered.

[0113] The dominant system instability state variable discrimination module is used to construct a truncated HSS model using the truncation number m determined by the model reasonable truncation number determination module, and to analyze the instability mode to be analyzed obtained by the instability mode acquisition module using modal participation factor analysis to obtain the state variables of the dominant system instability.

[0114] Specifically, each module of the truncation number determination system for the HSS model in this embodiment implements the steps of the truncation number determination method for the HSS model in Embodiment 1 during execution, for example:

[0115] In one implementation, a truncation number determination system applicable to the HSS model may perform the following steps:

[0116] Based on the time-domain waveform of the state variables in the initial stage of system instability, a target time window is selected;

[0117] The time-domain waveform within the target time window is approximated by fitting a complex exponential decay model; where the term with a positive complex exponential decay factor is denoted as the unstable oscillation term.

[0118] For each unstable oscillation term, the corresponding damping and oscillation frequency are calculated; the damping value is regarded as the real part of the unstable mode, and the oscillation frequency is adjusted according to the grid frequency and used as the imaginary part of the unstable mode.

[0119] A linear time-varying periodic system model is obtained by linearizing the system near its periodic trajectory. The linear time-varying periodic system model is represented in state-space form, which includes the Jacobian matrix.

[0120] By using Fourier series expansion and the harmonic balance method, a linear time-varying periodic system is transformed into an infinite-order harmonic state-space model; the harmonic state-space model is used to describe the dynamic behavior of the system under multiple frequency components.

[0121] The magnitudes of each frequency of the state variable x(t) are expressed as follows:

[0122] X hss =[…,X -2 X -1 [X0, X1, X2, ...] T

[0123] in, Let be the amplitude of the i-th frequency component after Fourier decomposition of x(t);

[0124] The magnitudes of each frequency of the system matrix A(t) are expressed as follows:

[0125]

[0126] in, Let be the amplitude of the i-th frequency component after Fourier decomposition of A(t);

[0127] The block diagonal matrix constructed from the system's fundamental angular frequency and identity matrix is ​​represented as follows:

[0128] N hss =diag[...-2jω0E-jω0E 0jω0E 2jω0E...]

[0129] Where E is the n-dimensional identity matrix, 0 is the n-dimensional zero matrix, diag[·] is the block diagonal matrix, ω0 is the fundamental angular frequency, and j is the imaginary unit.

[0130] The infinite-order harmonic state-space model is truncated to the m-th order to construct a finite-dimensional harmonic state-space model, and the eigenvalues ​​of the truncated harmonic state-space model are calculated.

[0131] The finite-dimensional harmonic state-space model is represented as:

[0132]

[0133] Among them, X hss(m) Let A be the matrix of frequency amplitudes of x(t) after Fourier decomposition and truncated to order m. hss(m) Let A(t) be the frequency amplitude after Fourier decomposition, and N be the matrix after m-order truncation. hss(m) This is the matrix after m-order truncation of the diagonal matrix calculated using the n-dimensional identity matrix E and the n-dimensional zero matrix 0.

[0134] If the real and imaginary parts of any eigenvalue are consistent with the real and imaginary parts of the unstable mode to be analyzed, then the reasonable truncation number of the harmonic state-space model is determined to be m; if the real and imaginary parts of any eigenvalue are inconsistent with the real and imaginary parts of the unstable mode to be analyzed, then the value of the truncation order m is updated according to m = m + 1 until a matching eigenvalue is found, and the iteration is completed.

[0135] Once a reasonable truncation order m is determined, eigenvalue decomposition is performed on the matrix after reasonable truncation of order m to obtain the left and right eigenvectors corresponding to the instability mode.

[0136] The participation factor of each state variable in the instability mode is calculated by multiplying the left and right eigenvectors; the state variable with the largest participation factor magnitude is taken as the dominant state variable in the system instability.

[0137] This embodiment also provides an electronic device applicable to a method for determining the cutoff number in an HSS model, including:

[0138] The memory and processor are used to store computer-executable instructions and execute the computer-executable instructions to implement the truncation number determination method applicable to the HSS model as proposed in the above embodiments.

[0139] This embodiment also provides a storage medium storing a computer program that, when executed by a processor, implements the truncation number determination method applicable to the HSS model as proposed in the above embodiments.

[0140] The storage medium proposed in this embodiment and the method for determining the truncation number applicable to the HSS model proposed in the above embodiments belong to the same inventive concept. Technical details not described in detail in this embodiment can be found in the above embodiments, and this embodiment has the same beneficial effects as the above embodiments.

[0141] Based on the above description of the implementation methods, those skilled in the art can clearly understand that the present invention can be implemented using software and necessary general-purpose hardware, and of course, it can also be implemented using hardware, but in many cases the former is a better implementation method. Based on this understanding, the technical solution of the present invention, or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a computer-readable storage medium, such as a computer floppy disk, read-only memory (ROM), random access memory (RAM), flash memory, hard disk, or optical disk, etc., including several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods of the various embodiments of the present invention.

[0142] Example 3, referring to Figures 3-6 This is one embodiment of the present invention, which verifies the beneficial effects of the present invention through scientific experiments.

[0143] like Figure 3 As shown, this embodiment takes a voltage source converter connected to an infinite power grid as an example; when a single-phase ground fault occurs in the system, it causes system oscillation, and the oscillation waveform is as follows. Figure 4 As shown; Prony analysis of the oscillation waveform reveals two oscillation modes:

[0144] mode1=0.663±j63.785π, mode2=0.663±j36.214×2π;

[0145] It can be seen that the damping of the two instability modes is basically the same, and the oscillation frequencies differ by 100Hz. The eigenvalues ​​of the truncated harmonic state-space model are as follows: Figure 5 As shown, when the truncation order m < 7, there is a large error between the eigenvalue analysis results and the Prony analysis results, and the harmonic state-space model at this truncation number cannot accurately analyze the oscillation mode; the oscillation mode is accurately revealed only when the truncation number m = 7; when the truncation number m > 7, although the oscillation mode can still be accurately revealed, the redundant eigenvalues ​​increase the computational burden, which is not conducive to its application in large-scale systems. Figure 6 As shown, eigenvalue decomposition is performed on the matrix after reasonable truncation of order m (m=7) to obtain the left and right eigenvectors corresponding to the instability mode. The participation factors of each state variable for this instability mode are calculated by the dot product of the left and right eigenvectors. Participation factor analysis shows that the participation factor related to phase-locked control has the largest magnitude, therefore it is the dominant state variable for system instability.

[0146] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for determining the cutoff number applicable to the HSS model, characterized in that, include: Time-domain simulation is performed based on a nonlinear time-varying periodic system model to obtain the time-domain waveform of the state variables when the system is unstable. Damping and oscillation frequency are obtained based on the time-domain waveform of the state variables when the system is unstable in order to determine the real and imaginary parts of the instability mode. The linear time-varying periodic system after linearizing the periodic trajectory is transformed into an infinite-order harmonic state-space model. By increasing the truncation order, the same eigenvalues ​​as the real and imaginary parts of the instability mode to be analyzed are obtained, and the reasonable truncation number of the harmonic state-space model is determined. Based on the reasonable truncation number, a truncated harmonic state-space model is constructed, and the instability mode to be analyzed is analyzed to obtain the state variables of the dominant system instability.

2. The method for determining the cutoff number applicable to the HSS model as described in claim 1, characterized in that, The step of obtaining the damping and oscillation frequency based on the time-domain waveform of the state variables during instability, in order to determine the real and imaginary parts of the instability mode, includes: Based on the time-domain waveform of the state variables in the initial stage of system instability, a target time window is selected; The time-domain waveform within the target time window is approximated by fitting a complex exponential decay model; where the term with a positive complex exponential decay factor is denoted as the unstable oscillation term. For each unstable oscillation term, the corresponding damping and oscillation frequency are calculated; the damping value is regarded as the real part of the unstable mode, and the oscillation frequency is adjusted according to the grid frequency and used as the imaginary part of the unstable mode.

3. The method for determining the cutoff number applicable to the HSS model as described in claim 2, characterized in that, Transforming a linear time-varying periodic system after linearizing its periodic trajectory into an infinite-order harmonic state-space model includes: A linear time-varying periodic system model is obtained by linearizing the system near its periodic trajectory. The linear time-varying periodic system model is represented in state-space form, which includes the Jacobian matrix. By using Fourier series expansion and the harmonic balance method, a linear time-varying periodic system is transformed into an infinite-order harmonic state-space model; the harmonic state-space model is used to describe the dynamic behavior of the system under multiple frequency components.

4. The method for determining the cutoff number applicable to the HSS model as described in claim 3, characterized in that, By increasing the truncation order, eigenvalues ​​identical to the real and imaginary parts of the unstable mode to be analyzed are obtained, and the reasonable truncation number for the harmonic state-space model is determined, including: The infinite harmonic state-space model is truncated to order m to construct a finite-dimensional harmonic state-space model, and the eigenvalues ​​of the truncated harmonic state-space model are calculated. If the real and imaginary parts of any eigenvalue are consistent with the real and imaginary parts of the unstable mode to be analyzed, then the reasonable truncation number of the harmonic state-space model is determined to be m; if the real and imaginary parts of any eigenvalue are inconsistent with the real and imaginary parts of the unstable mode to be analyzed, then the value of the truncation order m is updated according to m = m + 1 until a matching eigenvalue is found, and the iteration is completed.

5. The method for determining the cutoff number applicable to the HSS model as described in claim 4, characterized in that, Based on the aforementioned reasonable truncation number, a truncated harmonic state-space model is constructed, and the instability mode to be analyzed is determined. The state variables that dominate the system instability include: Once a reasonable truncation order m is determined, eigenvalue decomposition is performed on the matrix after reasonable truncation of order m to obtain the left and right eigenvectors corresponding to the instability mode. The participation factor of each state variable in the instability mode is calculated by multiplying the left and right eigenvectors; the state variable with the largest magnitude of the participation factor is taken as the dominant state variable in the system instability.

6. The method for determining the cutoff number applicable to the HSS model as described in claim 3 or 5, characterized in that, Harmonic state-space models are used to describe the dynamic behavior of a system across multiple frequency components, including: The magnitudes of each frequency of the state variable x(t) are expressed as follows: X hss ?[...,X -2 ,X -1 ,X0,X1,X2,. T in, Let be the amplitude of the i-th frequency component after Fourier decomposition of x(t); The magnitudes of each frequency of the system matrix A(t) are expressed as follows: in, Let be the amplitude of the i-th frequency component after Fourier decomposition of A(t); The block diagonal matrix constructed from the system's fundamental angular frequency and identity matrix is ​​represented as follows: N hss =diag[...-2jω0E-jω0E 0jω0E 2jω0E...] Where E is the n-dimensional identity matrix, 0 is the n-dimensional zero matrix, diag[·] is the block diagonal matrix, ω0 is the fundamental angular frequency, and j is the imaginary unit.

7. The method for determining the cutoff number applicable to the HSS model as described in claim 4, characterized in that: The finite-dimensional harmonic state-space model is expressed as: Among them, X hss(m) Let A be the matrix of frequency amplitudes of x(t) after Fourier decomposition and truncated to order m. hss(m) Let A(t) be the frequency amplitude after Fourier decomposition, and N be the matrix after m-order truncation. hss(m) This is the matrix after m-order truncation of the diagonal matrix calculated using the n-dimensional identity matrix E and the n-dimensional zero matrix 0.

8. A truncation number determination system suitable for HSS models, applied to the method described in any one of claims 1-7, characterized in that, include: The instability mode acquisition module is used to perform time-domain simulation based on a nonlinear time-varying periodic system model, acquire the time-domain waveform of the state variables when the system is unstable, and acquire the damping and oscillation frequency based on the time-domain waveform of the state variables when the system is unstable, so as to determine the real and imaginary parts of the instability mode. The module for determining the reasonable truncation number of the model is used to transform the linear time-varying periodic system after linearizing the periodic trajectory into an infinite-order harmonic state-space model. By increasing the truncation order, the module obtains the same eigenvalues ​​as the real and imaginary parts of the unstable mode to be analyzed, and determines the reasonable truncation number of the harmonic state-space model. The dominant system instability state variable discrimination module is used to construct a truncated harmonic state space model based on the reasonable truncation number, and to analyze the instability mode to be analyzed, so as to obtain the dominant system instability state variables.

9. An electronic device, comprising: Memory and processor; The memory is used to store computer-executable instructions, and the processor is used to execute the computer-executable instructions, which, when executed by the processor, implement the steps of the truncation number determination method applicable to the HSS model as described in any one of claims 1 to 7.

10. A computer-readable storage medium storing computer-executable instructions that, when executed by a processor, implement the steps of the method for determining the truncation number applicable to the HSS model as described in any one of claims 1 to 7.

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