Method and apparatus for electromagnetic transient stability analysis based on frequency shift analysis

Abstract

The application provides an electromagnetic transient stability analysis method and device based on frequency shift analysis, and relates to the technical field of power systems.The method comprises the following steps: obtaining an electromagnetic transient dynamic equation with time-varying characteristics in a power system; performing frequency shift transformation on the electromagnetic transient dynamic equation to obtain a frequency shift frequency domain nonlinear dynamic equation; performing augmented projection on a complex envelope vector in the frequency shift frequency domain nonlinear dynamic equation based on a generalized frequency band truncation set to obtain an augmented state vector, and determining a time-invariant state matrix at a steady-state equilibrium point based on the augmented state vector; determining an augmented eigenvalue set based on the time-invariant state matrix, and performing steady-state characteristic reduction on the augmented eigenvalue set based on a physical eigenvalue mapping criterion to obtain a physical characteristic spectrum.The application can realize accurate analysis on electromagnetic transient stability of a power system.

Description

Technical Field

[0001] This invention relates to the field of power system technology, and in particular to an electromagnetic transient stability analysis method and apparatus based on frequency shift analysis. Background Technology

[0002] In modern power systems primarily based on power electronic converters, the AC grid is modeled using a stationary abc coordinate system, while the converter control system operates in a synchronously rotating dq coordinate system. The coupling between these two coordinate systems introduces an explicit periodic time-varying term into the system's Jacobian matrix, causing the system to exhibit linear time-varying (LTV) or linear time-varying periodic (LTP) characteristics. Directly extracting instantaneously frozen eigenvalues ​​from the time-varying state matrix can produce spurious oscillatory poles, rendering Lyapunov-based stability criteria ineffective.

[0003] In existing technologies, the above problems are generally solved based on the following methods.

[0004] (1) Dynamic Phasor (DP): Based on the generalized averaging method, the LTP system is transformed into an LTI (Linear Time Invariant) system. This method replaces the state vector with the sliding Fourier coefficients, but in multi-frequency coupling scenarios, the matrix dimension increases exponentially with the truncation order, resulting in a heavy computational burden. In addition, the separation mechanism between the real and virtual components will lose the coupling information between high-frequency components, leading to a loss of accuracy.

[0005] (2) Harmonic State Space (HSS): Based on the Toeplitz complex matrix structure, it reveals cross-frequency energy interactions. HSS can handle wideband impedance modeling, but the physical eigenvalues ​​in the augmented spectrum are severely mixed with the image or overtone eigenvalues, making it difficult to accurately identify physically meaningful stable modes from the high-dimensional eigenvalue spectrum.

[0006] Therefore, improving the analytical accuracy of electromagnetic transient stability is a technical problem that urgently needs to be solved. Summary of the Invention

[0007] This invention provides an electromagnetic transient stability analysis method and apparatus based on frequency shift analysis, which solves the problem of low accuracy in electromagnetic transient stability analysis in the prior art.

[0008] This invention provides an electromagnetic transient stability analysis method based on frequency shift analysis, comprising the following steps.

[0009] Obtain the electromagnetic transient dynamic equations with time-varying characteristics in the power system; The electromagnetic transient dynamic equations are subjected to frequency shift transformation to obtain frequency-shifted frequency domain nonlinear dynamic equations; Based on the generalized band cutoff set, the complex envelope vector in the frequency-shifting nonlinear dynamic equation is augmented and projected to obtain the augmented state vector. Based on the augmented state vector, the time-invariant state matrix at the steady-state equilibrium point is determined. Based on the time-invariant state matrix, an augmented eigenvalue set is determined, and the augmented eigenvalue set is restored to steady-state characteristics based on the physical eigenvalue mapping criterion to obtain the physical feature spectrum.

[0010] According to the electromagnetic transient stability analysis method based on frequency shift analysis provided by the present invention, the step of performing a frequency shift transformation on the electromagnetic transient dynamic equation to obtain a frequency-shifted frequency domain nonlinear dynamic equation includes: Perform a Hilbert transform on the state vector in the electromagnetic transient dynamic equation to obtain a complex domain analytic signal; Construct a diagonal frequency shift operator matrix; the diagonal frequency shift operator matrix is ​​used to characterize the multi-scale frequency shifting frequencies corresponding to different state vectors; the frequency shifting frequency corresponding to the electrical network state vector in the diagonal frequency shift operator matrix is ​​the fundamental frequency, and the frequency shifting frequency corresponding to the control system state vector is the zero frequency; Based on the complex domain analytic signal and the diagonal frequency shift operator matrix, the complex envelope vector is determined; Substituting the complex envelope vector into the electromagnetic transient dynamic equation yields the frequency-shifting nonlinear dynamic equation.

[0011] According to the electromagnetic transient stability analysis method based on frequency shift analysis provided by the present invention, the generalized frequency band cutoff set includes multiple harmonic components; The augmented state vector is obtained by performing an augmented projection on the complex envelope vector in the frequency-shifting nonlinear dynamic equation based on the generalized band cutoff set, including: Construct the orthogonal projection operator corresponding to the generalized frequency band truncation set; Based on the orthogonal projection operator, the complex envelope vector in the frequency-shifting nonlinear dynamic equation is decomposed to obtain the slowly varying complex envelope sub-vectors corresponding to each of the multiple harmonic components. The augmented state vector is determined based on all slowly varying complex envelope subvectors.

[0012] According to the electromagnetic transient stability analysis method based on frequency shift analysis provided by the present invention, determining the time-invariant state matrix at the steady-state equilibrium point based on the augmented state vector includes: Based on the augmented state vector, the nonlinear ordinary differential equation is determined; Based on the augmented state vector and the nonlinear terms in the nonlinear ordinary differential equation, determine the real-field Jacobian matrix at the steady-state equilibrium point in the nonlinear ordinary differential equation. Based on the real-domain Jacobian matrix and the augmented linear state matrix in the nonlinear ordinary differential equation, the time-invariant state matrix at the steady-state equilibrium point is determined.

[0013] According to the electromagnetic transient stability analysis method based on frequency shift analysis provided by the present invention, determining the real-domain Jacobian matrix at the steady-state equilibrium point in the nonlinear ordinary differential equation based on the augmented state vector and the nonlinear terms in the nonlinear ordinary differential equation includes: The augmented state vector is split into a real part vector and an imaginary part vector, and the nonlinear terms in the nonlinear ordinary differential equation are split into a real part mapping vector and an imaginary part mapping vector; At the steady-state equilibrium point, partial derivatives of the real part vector and the imaginary part vector are obtained based on the real part mapping vector to obtain the first real domain Jacobian sub-block and the second real domain Jacobian sub-block. At the steady-state equilibrium point, based on the imaginary part mapping vector, partial derivatives are taken with respect to the real part vector and the imaginary part vector respectively to obtain the third real domain Jacobian sub-block and the fourth real domain Jacobian sub-block. Based on the first real-domain Jacobian subblock, the second real-domain Jacobian subblock, the third real-domain Jacobian subblock, and the fourth real-domain Jacobian subblock, the real-domain Jacobian matrix at the steady-state equilibrium point in the nonlinear ordinary differential equation is determined.

[0014] According to the electromagnetic transient stability analysis method based on frequency shift analysis provided by the present invention, determining the time-invariant state matrix at the steady-state equilibrium point based on the real-domain Jacobian matrix and the augmented linear state matrix in the nonlinear ordinary differential equation includes: The augmented linear state matrix is ​​expanded into real part submatrices and imaginary part submatrices; Based on the real part submatrix, the imaginary part submatrix, and the real field Jacobian matrix, the time-invariant state matrix at the steady-state equilibrium point is determined.

[0015] According to the electromagnetic transient stability analysis method based on frequency shift analysis provided by the present invention, the physical eigenvalue mapping criteria include the baseband subspace orthogonal projection criterion, the Nyquist principal value branch mapping criterion, and the quotient space equivalence class folding criterion, wherein: The baseband subspace orthogonal projection criterion is used to select the first physical candidate eigenvalue from the augmented eigenvalue set based on the projection energy norm of each eigenvalue in the baseband subspace and the higher harmonic subspace. The Nyquist principal value branch mapping criterion is used to fold the imaginary part of each eigenvalue in the augmented eigenvalue set into the Nyquist principal value branch interval, wherein the Nyquist principal value branch interval has a period of the fundamental frequency. The quotient space equivalence class folding criterion is used to merge the eigenvalues ​​of the same physical mode in the augmented eigenvalue set.

[0016] According to the electromagnetic transient stability analysis method based on frequency shift analysis provided by the present invention, the step of selecting a first physical candidate eigenvalue from the augmented eigenvalue set based on the projected energy norms of each eigenvalue in the baseband subspace and the higher harmonic subspace respectively includes: Based on the left and right eigenvector matrices of each eigenvalue in the augmented eigenvalue set, the participation factor vector corresponding to each eigenvalue is determined. Based on the baseband subspace orthogonal projection operator and the participation factor vector corresponding to each eigenvalue, the baseband projection energy norm of each eigenvalue in the baseband subspace is determined; the baseband subspace orthogonal projection operator is used to project each eigenvalue to the baseband subspace. Based on the high-frequency extended subspace projection operator and the participation factor vector corresponding to each eigenvalue, the harmonic projection energy norm of each eigenvalue in the higher harmonic subspace is determined; the high-frequency extended subspace projection operator is used to project each eigenvalue to the higher harmonic subspace. The eigenvalues ​​whose baseband projection energy norm is greater than the harmonic projection energy norm are determined as the first physical candidate eigenvalues.

[0017] According to the electromagnetic transient stability analysis method based on frequency shift analysis provided by the present invention, the merging of eigenvalues ​​of the same physical mode in the augmented eigenvalue set includes: Determine the Euclidean norm for any two eigenvalues ​​in the augmented eigenvalue set; Any two eigenvalues ​​whose Euclidean norm is less than the truncation tolerance are equivalently merged.

[0018] The present invention also provides an electromagnetic transient stability analysis device based on frequency shift analysis, comprising the following modules.

[0019] The acquisition module is used to acquire the electromagnetic transient dynamic equations with time-varying characteristics in the power system; The frequency shift transformation module is used to perform a frequency shift transformation on the electromagnetic transient dynamic equation to obtain a frequency-shifted frequency domain nonlinear dynamic equation. An augmented projection module is used to augment the complex envelope vector in the frequency-shifting nonlinear dynamic equation based on a generalized band cutoff set, to obtain an augmented state vector, and to determine the time-invariant state matrix at the steady-state equilibrium point based on the augmented state vector. The steady-state analysis module is used to determine the augmented eigenvalue set based on the time-invariant state matrix, and to perform steady-state feature restoration on the augmented eigenvalue set based on the physical eigenvalue mapping criterion to obtain the physical feature spectrum.

[0020] The present invention also provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the electromagnetic transient stability analysis method based on frequency shift analysis as described above.

[0021] The present invention also provides a non-transitory computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the electromagnetic transient stability analysis method based on frequency shift analysis as described above.

[0022] The present invention also provides a computer program product, including a computer program that, when executed by a processor, implements the electromagnetic transient stability analysis method based on frequency shift analysis as described above.

[0023] The electromagnetic transient stability analysis method and apparatus based on frequency shift analysis provided by this invention performs frequency shift transformation on the time-varying electromagnetic transient dynamic equations of a power system to obtain a frequency-shifted frequency-domain nonlinear dynamic equation, eliminating linear time-varying characteristics while retaining full-frequency information. Based on a generalized band cutoff set, the complex envelope vector in the frequency-shifted frequency-domain nonlinear dynamic equation is augmented and projected to obtain an augmented state vector. Based on the augmented state vector, the time-invariant state matrix at the steady-state equilibrium point is determined, eliminating nonlinear time-varying characteristics while retaining multi-band coupling effects. Based on the time-invariant state matrix, an augmented eigenvalue set is determined, and the augmented eigenvalue set is restored to steady-state characteristics according to the physical eigenvalue mapping criterion to obtain the physical characteristic spectrum. This achieves accurate, efficient, and full-dimensional lossless characteristic spectrum restoration of linear time-varying power systems, thereby enabling accurate analysis of the electromagnetic transient stability of power systems. Furthermore, this invention does not rely on symmetry assumptions and is applicable to various operating conditions, including balanced, unbalanced, and multi-frequency control interactions, and is universally applicable to both grid-connected and grid-connected converters. Attached Figure Description

[0024] To more clearly illustrate the technical solutions in this invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0025] Figure 1 This is a flowchart illustrating the electromagnetic transient stability analysis method based on frequency shift analysis provided in an embodiment of the present invention.

[0026] Figure 2 This is a schematic diagram of LTV system transformation based on Floquet theory provided by existing technology.

[0027] Figure 3 This is a schematic diagram of the frequency shift converter provided in an embodiment of the present invention.

[0028] Figure 4 This is a flowchart illustrating the construction process of the augmented state vector provided in an embodiment of the present invention.

[0029] Figure 5 This is a schematic diagram of the eigenvalues ​​of the LTV EMT state-space model under the symmetric condition before feature mapping provided in the embodiments of the present invention.

[0030] Figure 6 This is a schematic diagram of the eigenvalues ​​of the LTV EMT state-space model under asymmetric conditions before feature mapping, provided in an embodiment of the present invention.

[0031] Figure 7 This is a schematic diagram of the SFA-SSM feature values ​​provided in the embodiments of the present invention.

[0032] Figure 8 This is a schematic diagram comparing the feature values ​​extracted by the SFA-SSM and dq-SSM methods under symmetrical working conditions provided in the embodiments of the present invention.

[0033] Figure 9 This is a schematic diagram of the feature values ​​extracted by the dq-SSM method under asymmetric conditions provided by existing technology.

[0034] Figure 10 This is a schematic diagram of the feature values ​​extracted by the SFA-SSM method under asymmetric operating conditions provided in the embodiments of the present invention.

[0035] Figure 11 This is a schematic diagram illustrating the calculation errors of different maximum order eigenvalues ​​provided in the embodiments of the present invention.

[0036] Figure 12 This is a schematic diagram of the electromagnetic transient stability analysis device based on frequency shift analysis provided in an embodiment of the present invention.

[0037] Figure 13 This is a schematic diagram of the structure of the electronic device provided in an embodiment of the present invention. Detailed Implementation

[0038] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.

[0039] To address the issue of low accuracy in existing electromagnetic transient stability analysis techniques, this invention provides an electromagnetic transient stability analysis method based on frequency shift analysis. Figure 1 This is a flowchart illustrating the electromagnetic transient stability analysis method based on frequency shift analysis provided in an embodiment of the present invention, as shown below. Figure 1 As shown, the method includes steps 110 to 140.

[0040] Step 110: Obtain the electromagnetic transient dynamic equations with time-varying characteristics in the power system.

[0041] Specifically, the power system includes a nonlinear electromagnetic transient (EMT) system, which comprises an electrical network and a converter control system. The EMT dynamic equation characterizes the interaction between the physical topology of the electrical network and the dynamics of the converter control system within the nonlinear EMT system. This EMT dynamic equation is shown in equation (1), which is: .

[0042] in, This represents a unified state vector in a power system, which includes electrical network state vectors and control system state vectors. The electrical network state vectors may include inductor currents and capacitor voltages, while the control system state vectors may include filter states and integral states. , Represents an n-dimensional real number; Let represent the linear characteristic matrix describing the topology and passive coupling relationships of an electrical network, and , express 3D real matrix; This represents an analytical operator that encapsulates all nonlinear feedback mechanisms, such as multipliers, limiters, pulse width modulation, and coordinate transformations. It represents the first derivative of the state vector with respect to time t.

[0043] Step 120: Perform a frequency shift transformation on the electromagnetic transient dynamic equation to obtain a frequency-shifted frequency domain nonlinear dynamic equation.

[0044] Specifically, after determining the electromagnetic transient dynamic equation, a small perturbation is applied to the nonlinear EMT system near a given steady-state trajectory point. This perturbation is then performed using Taylor expansion, retaining the linear terms, to obtain a small-signal state-space model (SSM). This small-signal state-space model is shown in equation (2), which is: .

[0045] in, Indicates a small perturbation. This represents the small perturbation input vector. Represents the system state matrix, and , express , Represents the steady-state trajectory point. Denotes the input matrix, and , This represents the external input vector.

[0046] In nonlinear EMT systems, the electrical network is modeled using the abc coordinate system, while the converter control system operates in the synchronous rotating dq coordinate system. Therefore, after the two coordinate systems are coupled, the system state matrix in equation (2) is a time-varying matrix. Under common asymmetric operating conditions or wideband multi-scale scenarios, nonlinear EMT systems exhibit extremely strong nonlinear time-varying characteristics, resulting in low analysis accuracy of traditional small-signal analysis methods based on a single steady-state operating point. Therefore, it is necessary to strictly convert the linear time-varying (LTV) system into a linear time-invariant (LTI) system operating at the steady-state equilibrium point. Figure 2 This is a schematic diagram of LTV system transformation based on Floquet theory provided by existing technology, such as... Figure 2 As shown, through Floquet theory, there exists mathematically a non-singular periodic Lyapunov transformation matrix P. F (t) can map the LTV system to an LTI system operating at the steady-state equilibrium point, but the periodic Lyapunov transformation matrix P F (t) is difficult to calculate. Therefore, in this embodiment of the invention, different frequency shift transformations are performed on different types of state vectors in the electromagnetic transient dynamic equation to eliminate the time dependence of the linear part. That is, in the frequency-shifted nonlinear dynamic equation obtained after the frequency shift transformation, the linear characteristic matrix eliminates the fast oscillation term and becomes a constant matrix.

[0047] Step 130: Based on the generalized frequency band truncation set, perform augmented projection on the complex envelope vector in the frequency-shifting nonlinear dynamic equation to obtain the augmented state vector, and determine the time-invariant state matrix at the steady-state equilibrium point based on the augmented state vector.

[0048] Specifically, after determining the frequency-shifting nonlinear dynamic equation, the linear time-varying nature is eliminated, but the nonlinear part still contains time-varying characteristics. Therefore, in this embodiment of the invention, the complex envelope vector in the frequency-shifting nonlinear dynamic equation is augmented and projected using a generalized band cutoff set to obtain an augmented state vector superimposed with multiple frequency band components. This augmented projection can be understood as decomposing the multi-band coupled complex envelope vector according to the generalized band cutoff set. In this augmented state vector, based on the closure mapping property of the nonlinear operator, the frequency domain convolution after augmented projection is strictly constrained to pure algebraic interactions within a finite-dimensional closed subspace. That is, the nonlinear interactions between different frequency bands can be converted into pure algebraic operations between the frequency band components, no longer explicitly dependent on the time variable. Subsequently, based on the augmented state vector, the time-invariant state matrix at the steady-state equilibrium point, composed entirely of real numbers, is determined to eliminate nonlinear time-varying characteristics.

[0049] Step 140: Based on the time-invariant state matrix, determine the augmented eigenvalue set, and perform steady-state feature restoration on the augmented eigenvalue set based on the physical eigenvalue mapping criterion to obtain the physical feature spectrum.

[0050] Specifically, after determining the time-invariant state matrix, an augmented eigenvalue set is obtained by solving the time-invariant state matrix. This augmented eigenvalue set contains aliased physical eigenvalues, mirror eigenvalues ​​introduced by the frequency shift transform, and overtone eigenvalues ​​introduced by the augmented projection. Therefore, in this embodiment of the invention, the augmented eigenvalue set is restored to steady-state characteristics according to the physical eigenvalue mapping criterion, and the physical feature spectrum consistent with the order of the original nonlinear EMT system is accurately extracted from the high-dimensional aliased augmented eigenvalue set.

[0051] The electromagnetic transient stability analysis method based on frequency shift analysis provided in this invention transforms the time-varying electromagnetic transient dynamic equations of a power system using frequency shifting to obtain a frequency-shifted nonlinear dynamic equation in the frequency domain. This eliminates linear time-varying characteristics while retaining full-frequency information. Based on a generalized band cutoff set, the complex envelope vector in the frequency-shifted nonlinear dynamic equation is augmented and projected to obtain an augmented state vector. Based on this augmented state vector, the time-invariant state matrix at the steady-state equilibrium point is determined, eliminating nonlinear time-varying characteristics while retaining multi-band coupling effects. Based on the time-invariant state matrix, an augmented eigenvalue set is determined, and steady-state characteristics are restored to the augmented eigenvalue set according to the physical eigenvalue mapping criterion to obtain the physical characteristic spectrum. This achieves accurate, efficient, and lossless full-dimensional characteristic spectrum restoration of linear time-varying power systems, thereby enabling accurate analysis of the electromagnetic transient stability of power systems. Furthermore, this invention does not rely on symmetry assumptions and is applicable to various operating conditions, including balanced, unbalanced, and multi-frequency control interactions. It is universally applicable to both grid-connected and grid-connected converters.

[0052] In one embodiment, performing a frequency-shift transformation on the electromagnetic transient dynamic equation to obtain a frequency-shifted frequency domain nonlinear dynamic equation includes: Perform a Hilbert transform on the state vector in the electromagnetic transient dynamic equation to obtain a complex domain analytic signal; Construct a diagonal frequency shift operator matrix; the diagonal frequency shift operator matrix is ​​used to characterize the multi-scale frequency shifting frequencies corresponding to different state vectors; Based on the complex domain analytic signal and the diagonal frequency shift operator matrix, the complex envelope vector is determined; Substituting the complex envelope vector into the electromagnetic transient dynamic equation yields the frequency-shifting nonlinear dynamic equation.

[0053] Specifically, Figure 3 This is a schematic diagram of the frequency shift transform provided in an embodiment of the present invention, as shown below. Figure 3 As shown, the vertical axis represents the spectrum (i.e., Figure 3 In The horizontal axis represents the angular frequency (i.e., Figure 3 In After determining the electromagnetic transient dynamic equation, the Hilbert transform is performed on the real-domain state vector in the electromagnetic transient dynamic equation to obtain the complex-domain analytic signal, which is expressed as follows: , Represents an analytic signal in the complex field. The Hilbert transform operator is represented here. Next, a diagonal frequency shift operator matrix is ​​constructed, which includes multi-scale frequency shift frequencies corresponding to different state vectors to achieve adaptive translation of multi-scale frequencies. Then, a demodulation operator is constructed based on the diagonal frequency shift operator matrix. This demodulation operator is used to perform a rotational coordinate transformation on the complex domain analytic signal to obtain a complex envelope vector, which is represented as... ,in, Denotes the complex envelope vector. This represents the demodulation operator. Then, substituting this complex envelope vector into the electromagnetic transient dynamic equation yields the frequency-shifting nonlinear dynamic equation, which can be expressed as: ,in, That is, X(t) represents the complex envelope vector. This represents the linear part after the frequency shift transformation, and this linear part has eliminated the fast oscillation term, that is, this linear part has been converted into a real matrix. The term represents a nonlinear term, and since the nonlinear term includes a demodulation operator, it is still explicitly time-varying, meaning that the frequency-shifting nonlinear dynamic equation still has time-varying characteristics.

[0054] In one embodiment, the frequency shifting frequency corresponding to the electrical network state vector in the diagonal frequency shifting operator matrix is ​​the fundamental frequency, and the frequency shifting frequency corresponding to the control system state vector is zero frequency.

[0055] Specifically, the diagonal frequency shift operator matrix is ​​represented as follows: , This represents the diagonal frequency shift operator matrix, and diag(·) represents the diagonal matrix, such as... Figure 3 As shown, The fundamental angular frequency (or simply fundamental frequency) of the system is assigned to the row containing the electrical network state vector, while 0 represents zero frequency and is assigned to the row containing the control system state vector to preserve its baseband DC characteristics. After frequency shifting transformation through this diagonal frequency shifting operator matrix, the electrical network state vector (i.e., the fundamental frequency component) in this complex envelope vector is shifted to zero frequency, becoming a stationary DC component, while the control system state vector remains at zero frequency, preserving the baseband DC characteristics. This eliminates the explicit time-varying characteristics of the linear part in the frequency-shifting frequency domain nonlinear dynamic equation.

[0056] In one embodiment, the generalized band cutoff set includes multiple harmonic components; The augmented state vector is obtained by performing an augmented projection on the complex envelope vector in the frequency-shifting nonlinear dynamic equation based on the generalized band cutoff set, including: Construct the orthogonal projection operator corresponding to the generalized frequency band truncation set; Based on the orthogonal projection operator, the complex envelope vector in the frequency-shifting nonlinear dynamic equation is decomposed to obtain the slowly varying complex envelope sub-vectors corresponding to each of the multiple harmonic components. The augmented state vector is determined based on all slowly varying complex envelope subvectors.

[0057] Specifically, Figure 4 This is a flowchart illustrating the construction process of the augmented state vector provided in an embodiment of the present invention, such as... Figure 4 As shown in (a), the complex envelope vector is a broadband signal, including multiple frequency components, and the nonlinear terms in the frequency-shifting nonlinear dynamic equation still possess time-varying characteristics. To handle multi-band coupling information in a broadband environment and prevent accuracy degradation due to direct high-frequency truncation, this embodiment of the invention eliminates the time-varying characteristics of the nonlinear terms through augmented projection. Specifically, this includes: constructing a generalized band truncation set based on the multiple frequency components. This generalized band truncation set includes harmonic components of multiple frequency bands. This generalized band truncation set is used to capture the dominant dynamic characteristics of the nonlinear EMT system to determine the frequency coverage range of the augmented projection. This generalized band truncation set is represented as... , Let represent the harmonic components of the i-th frequency band, and , This represents the total number of lumped harmonic components in the generalized frequency band cutoff. Then, an orthogonal projection operator is constructed, such as... Figure 4 As shown in (b), the orthogonal projection operator decomposes the complex envelope vector in the frequency-shifting frequency domain nonlinear dynamic equation into multiple slowly varying complex envelope sub-vectors corresponding to each harmonic component. Each slowly varying complex envelope sub-vector is a slowly changing function of time t, and its frequency components are concentrated near the corresponding harmonic component. Then, all the slowly varying complex envelope sub-vectors are stacked sequentially to form a high-dimensional augmented state vector. Represented as ,in, Indicates the first The frequency-shifting sub-state vector (i.e., the slowly varying complex envelope vector) corresponding to the sub-harmonic components. , Let represent a g×m dimensional complex field matrix, where g represents the order of the nonlinear EMT system. The augmented state vector is as follows: Figure 4 As shown in (c) in the figure.

[0058] Based on the closure mapping characteristics of nonlinear operators, the frequency domain convolution operation after truncation and projection is strictly constrained to pure algebraic interactions within a finite-dimensional closed subspace. That is, the interaction between slowly varying complex envelope vectors of different frequency bands can be described by algebraic operations, no longer explicitly dependent on time variables, thus eliminating the time-varying characteristics in the nonlinear terms.

[0059] In one embodiment, determining the time-invariant state matrix at the steady-state equilibrium point based on the augmented state vector includes: Based on the augmented state vector, the nonlinear ordinary differential equation is determined; Based on the augmented state vector and the nonlinear terms in the nonlinear ordinary differential equation, determine the real-field Jacobian matrix at the steady-state equilibrium point in the nonlinear ordinary differential equation. Based on the real-domain Jacobian matrix and the augmented linear state matrix in the nonlinear ordinary differential equation, the time-invariant state matrix at the steady-state equilibrium point is determined.

[0060] Specifically, after determining the augmented state vector, based on the closure mapping characteristics of the nonlinear operator, it is ensured that there exists an algebraic mapping completely independent of the explicit time variables. Based on this algebraic mapping and the augmented state vector, the nonlinear dynamic equation in the complex domain corresponding to the nonlinear EMT system is folded into a completely autonomous nonlinear ordinary differential equation, as shown in equation (3). Equation (3) is: .

[0061] in, Indicates the linear part, Let represent the augmented linear state matrix, which is a diagonal matrix of frequency shift blocks generated independently from different frequency bands, and the augmented linear state matrix is ​​expressed as follows: , Represents an n-order identity matrix. Represents a block diagonal matrix. Represents nonlinear terms (i.e., algebraic mappings). This represents the first derivative of the augmented state vector with respect to the time variable t.

[0062] Subsequently, the nonlinear term is linearized at the steady-state equilibrium point to obtain a real-domain Jacobian matrix. This real-domain Jacobian matrix is ​​used to describe the small-perturbation dynamic characteristics of the nonlinear EMT system at the steady-state equilibrium point. Then, the augmented linear state matrix is ​​expanded into a real-domain form and combined with the real-domain Jacobian matrix to obtain a time-invariant state matrix composed entirely of constants, thus avoiding the eigenvalue pseudo-oscillation problem caused by time-varying matrices in existing technologies.

[0063] In one embodiment, determining the real-domain Jacobian matrix at the steady-state equilibrium point in the nonlinear ordinary differential equation based on the augmented state vector and the nonlinear terms in the nonlinear ordinary differential equation includes: The augmented state vector is split into a real part vector and an imaginary part vector, and the nonlinear terms in the nonlinear ordinary differential equation are split into a real part mapping vector and an imaginary part mapping vector; At the steady-state equilibrium point, partial derivatives of the real part vector and the imaginary part vector are obtained based on the real part mapping vector to obtain the first real domain Jacobian sub-block and the second real domain Jacobian sub-block. At the steady-state equilibrium point, based on the imaginary part mapping vector, partial derivatives are taken with respect to the real part vector and the imaginary part vector respectively to obtain the third real domain Jacobian sub-block and the fourth real domain Jacobian sub-block. Based on the first real-domain Jacobian subblock, the second real-domain Jacobian subblock, the third real-domain Jacobian subblock, and the fourth real-domain Jacobian subblock, the real-domain Jacobian matrix at the steady-state equilibrium point in the nonlinear ordinary differential equation is determined.

[0064] Specifically, since the nonlinear term is a non-holomorphic function in the complex domain, including conjugates and amplitude limiting, it cannot be directly linearized using the Cauchy-Riemann derivative. Therefore, in this embodiment of the invention, the augmented state vector is decomposed into a real part vector and an imaginary part vector. The decomposed augmented state vector is represented as follows: , Represents a vector with the real part. Let represent the imaginary part vector. Simultaneously, the nonlinear term is decomposed into a real part mapping vector and an imaginary part mapping vector; the decomposed nonlinear term is represented as . ,in, Represents the real part mapping vector, This represents the imaginary part of the mapping vector.

[0065] Next, the steady-state constant complex envelope equilibrium point (i.e., the steady-state equilibrium point) in the nonlinear ordinary differential equation is determined. This steady-state equilibrium point contains the steady-state amplitude and phase information of all harmonic components. Then, at this steady-state equilibrium point, the first real-domain Jacobian sub-block is obtained by taking the partial derivative of the real part mapping vector with respect to the real part vector. The second real-domain Jacobian sub-block is obtained by taking the partial derivative of the real part mapping vector with respect to the imaginary part vector. The third real-domain Jacobian sub-block is obtained by taking the partial derivative of the imaginary part mapping vector with respect to the real part vector. The fourth real-domain Jacobian sub-block is obtained by taking the partial derivative of the imaginary part mapping vector with respect to the imaginary part vector. Based on the first, second, third, and fourth real-domain Jacobian sub-blocks, the real-domain Jacobian matrix at the steady-state equilibrium point is determined. This real-domain Jacobian matrix is ​​shown in equation (4), which is: .

[0066] in, Represents a real-field Jacobian matrix. Represents the first real-field Jacobian subblock, and , Indicates the second real field Jacobian subblock, and , Represents the third real field Jacobian subblock, and , Represents the fourth real field Jacobian subblock, and .

[0067] In this embodiment of the invention, by splitting the augmented state vector and nonlinear terms into real and imaginary parts, non-differentiable holomorphic functions in the complex domain are transformed into differentiable real-valued functions in the real domain, thereby achieving precise linearization of nonlinear terms. At the same time, the four real-domain Jacobian sub-blocks obtained after the splitting retain all coupling relationships between the real and imaginary parts without losing any cross-coupling information.

[0068] In one embodiment, determining the time-invariant state matrix at the steady-state equilibrium point based on the real-domain Jacobian matrix and the augmented linear state matrix in the nonlinear ordinary differential equation includes: The augmented linear state matrix is ​​expanded into real part submatrices and imaginary part submatrices; Based on the real part submatrix, the imaginary part submatrix, and the real field Jacobian matrix, the time-invariant state matrix at the steady-state equilibrium point is determined.

[0069] Specifically, the augmented linear state matrix is ​​expanded into real-field form, and the expanded augmented linear state matrix is ​​represented as follows: ,in, This represents the real part submatrix of the expanded augmented linear state matrix. Let represent the imaginary submatrix in the expanded augmented linear state matrix. Then, the expanded augmented linear state matrix and the real-field Jacobian matrix are combined to obtain the time-invariant state matrix, as shown in equation (5). Equation (5) is: .

[0070] in, This represents a time-invariant state matrix, and all elements in this time-invariant state matrix are constants. Based on this, the time-varying characteristics in the nonlinear term are completely eliminated.

[0071] In one embodiment, the physical eigenvalue mapping criterion includes the baseband subspace orthogonal projection criterion, the Nyquist principal value branch mapping criterion, and the quotient space equivalence class folding criterion, wherein: The baseband subspace orthogonal projection criterion is used to select the first physical candidate eigenvalue from the augmented eigenvalue set based on the projection energy norm of each eigenvalue in the baseband subspace and the higher harmonic subspace. The Nyquist principal value branch mapping criterion is used to fold the imaginary part of each eigenvalue in the augmented eigenvalue set into the Nyquist principal value branch interval, wherein the Nyquist principal value branch interval has a period of the fundamental frequency. The quotient space equivalence class folding criterion is used to merge the eigenvalues ​​of the same physical mode in the augmented eigenvalue set.

[0072] Specifically, after determining the time-invariant state matrix, the full eigenvalue spectrum is solved based on the time-invariant state matrix to obtain the augmented eigenvalue set. Due to the introduction of the demodulation operator, the augmented eigenvalue set contains aliased physical eigenvalues, mirror-enhanced eigenvalues ​​introduced by the frequency shift transform, and overtone eigenvalues ​​introduced by the augmented projection. Therefore, in this embodiment of the invention, three screening criteria are set, namely the baseband subspace orthogonal projection criterion, the Nyquist principal value branch mapping criterion, and the quotient space equivalence class folding criterion, to accurately extract the precise small-signal spectrum (i.e., the physical eigenvalue spectrum) that strictly matches the dimension of the original physical system from the high-dimensional aliased augmented eigenvalue set.

[0073] In the augmented eigenvalue set, the projection energy of the true physical eigenvalues ​​is concentrated in the baseband subspace, while the projection energy of the adjoint eigenvalues ​​introduced by the frequency shift transform and augmented projection is distributed in the higher harmonic subspace. Therefore, the orthogonal projection criterion of the baseband subspace determines whether each eigenvalue is a true physical eigenvalue or an adjoint eigenvalue by the projection energy norm of each eigenvalue in the baseband subspace and the higher harmonic subspace, thereby eliminating false eigenvalues ​​in the augmented eigenvalue set.

[0074] Through frequency shift transformation, the imaginary parts of each eigenvalue in this augmented eigenvalue set will be transformed in the frequency shift coordinate system by... The frequency shift occurs by a multiple of the frequency shift. Therefore, in the Nyquist principal value branch mapping criterion, the imaginary part of each eigenvalue is restored from the frequency-shifted coordinate system to the Nyquist principal value branch interval corresponding to the true frequency, eliminating the imaginary part offset caused by the frequency shift transformation. This Nyquist principal value branch interval has a period of the fundamental frequency and is represented as follows: Eigenvalues ​​after eliminating imaginary part offset Represented as ,in, It can be the d-th eigenvalue in the augmented eigenvalue set, the first physical candidate eigenvalue after being screened by the baseband subspace orthogonal projection criterion, or the representative element after being folded by the quotient space equivalence class folding criterion. Denotes the real part of the d-th eigenvalue. This represents the imaginary part of the d-th eigenvalue. This represents the frequency shift corresponding to the imaginary part.

[0075] The same physical mode may manifest as multiple sidelobe eigenvalues ​​in the augmented space, and these sidelobe eigenvalues ​​will coincide on the complex plane after principal value mapping. In the quotient space equivalence class folding criterion, the overlapping eigenvalues ​​are merged into the same equivalence class, and representative elements are selected, thereby restoring the true dimension of the physical feature spectrum and achieving dimensionality reduction of the augmented feature set.

[0076] It should be noted that the execution order of the three screening criteria is not limited. For example, they can be executed sequentially as follows: the baseband subspace orthogonal projection criterion, the Nyquist principal value branch mapping criterion, and the quotient space equivalence class folding criterion. That is, the baseband subspace orthogonal projection criterion is executed to screen and obtain the first physical candidate eigenvalues ​​after removing spurious eigenvalues. Then, the Nyquist principal value branch mapping criterion is applied to the first physical candidate eigenvalues ​​to perform imaginary part frequency shift restoration, obtaining the second physical candidate eigenvalues. Finally, the second physical candidate eigenvalues ​​after frequency shift restoration are equivalently merged to obtain the true physical eigenvalues. Alternatively, the Nyquist principal value branch mapping criterion, the quotient space equivalence class folding criterion, and the baseband subspace orthogonal projection criterion can also be executed sequentially.

[0077] In one embodiment, the step of selecting a first physical candidate eigenvalue from the augmented eigenvalue set based on the projected energy norms of each eigenvalue in the baseband subspace and the higher harmonic subspace includes: Based on the left and right eigenvector matrices of each eigenvalue in the augmented eigenvalue set, the participation factor vector corresponding to each eigenvalue is determined. Based on the baseband subspace orthogonal projection operator and the participation factor vector corresponding to each eigenvalue, the baseband projection energy norm of each eigenvalue in the baseband subspace is determined; the baseband subspace orthogonal projection operator is used to project each eigenvalue to the baseband subspace. Based on the high-frequency extended subspace projection operator and the participation factor vector corresponding to each eigenvalue, the harmonic projection energy norm of each eigenvalue in the higher harmonic subspace is determined; the high-frequency extended subspace projection operator is used to project each eigenvalue to the higher harmonic subspace. The eigenvalues ​​whose baseband projection energy norm is greater than the harmonic projection energy norm are determined as the first physical candidate eigenvalues.

[0078] Specifically, the time-invariant state matrix includes a left eigenvector matrix U and a right eigenvector matrix V that satisfy the bioorthogonality normalization condition, i.e., UV=I. For the d-th eigenvalue in this augmented eigenvalue set, the participation factor vector corresponding to the d-th eigenvalue is determined based on the left and right eigenvector matrices of the d-th eigenvalue. The participation factor vector corresponding to the d-th eigenvalue is expressed as... ,and , Let d be an N-dimensional complex number, and let q be the element in the participating factor vector corresponding to the d-th eigenvalue. Then, based on all elements in the participating factor vector corresponding to the eigenvalue, the 1-norm between the baseband subspace orthogonal projection operator and the participating factor vector corresponding to each eigenvalue is determined. This 1-norm is the baseband projection energy norm of the eigenvalue in the baseband subspace, as shown in equation (6). Equation (6) is: .

[0079] in, Let d represent the baseband projection energy norm of the d-th eigenvalue in the baseband subspace. This represents the baseband subspace orthogonal projection operator. This represents the set of state indices corresponding to the baseband subspace.

[0080] Simultaneously, the 1-norm between the high-frequency extended subspace projection operator and the participating factor vectors corresponding to each eigenvalue is determined. This 1-norm is the harmonic projection energy norm of the eigenvalue in the higher-order harmonic subspace, expressed as: .

[0081] Next, the baseband projected energy norm is compared with the harmonic projected energy norm. If the baseband projected energy norm is greater than the harmonic projected energy norm, it indicates that the projected energy of this eigenvalue is concentrated in the baseband subspace. Therefore, this eigenvalue is an intrinsic physical root and is determined as a first physical candidate eigenvalue. If the baseband projected energy norm is less than or equal to the harmonic projected energy norm, it indicates that the projected energy of this eigenvalue is concentrated in the higher harmonic subspace. Therefore, this eigenvalue is a false eigenvalue and is discarded. Repeating the above operation for each eigenvalue allows us to determine the type of each eigenvalue, thus obtaining all first physical candidate eigenvalues ​​after discarding false eigenvalues.

[0082] In one embodiment, merging the eigenvalues ​​of the same physical mode in the augmented eigenvalue set includes: Determine the Euclidean norm for any two eigenvalues ​​in the augmented eigenvalue set; Any two eigenvalues ​​whose Euclidean norm is less than the truncation tolerance are equivalently merged.

[0083] Specifically, for any two eigenvalues ​​in the augmented eigenvalue set, the Euclidean norm between the two eigenvalues ​​is determined. This Euclidean norm is used to characterize the geometric distance between the two eigenvalues ​​in the complex plane. Then, this Euclidean norm is compared with the truncation tolerance. If the Euclidean norm is less than the truncation tolerance, there is an equivalence relationship between the two eigenvalues, and these two eigenvalues ​​can be equivalently merged to obtain an equivalence class. This equivalence relationship is shown in equation (7), which is: .

[0084] in, This represents the a-th eigenvalue. This represents the b-th eigenvalue. Denotes the Euclidean norm. Indicates equivalence. This indicates the truncation tolerance.

[0085] It should be noted that the eigenvalues ​​for equivalent merging can be eigenvalues ​​in the augmented eigenvalue set, the first physical candidate eigenvalues ​​after screening by the baseband subspace orthogonal projection criterion, or eigenvalues ​​after principal value mapping.

[0086] Through the above equivalent merging, all equivalence classes can be obtained. Selecting the feature with the largest baseband projected energy norm from each equivalence class as the representative yields the dominant physical feature spectrum after equivalent merging. If the folding criterion for this quotient space equivalence class is the last screening criterion executed, then this dominant physical feature spectrum is the true physical feature spectrum, and its dimension is restored to n, consistent with the original physical system.

[0087] In this embodiment of the invention, a 16th-order grid-forming converter (GFM) is used as an example for experimentation. This 16th-order grid-forming converter adopts a grid-connected system of a GFM converter controlled by a virtual synchronous generator (VSG). The control scheme mainly includes an external pressure loop, an internal current loop, and a power synchronization loop based on the VSG. The converter subsystem is established as a 16th-order state-space model. The electrical network is modeled using a stationary abc coordinate system, and the control system is modeled using a synchronous rotating dq coordinate system. The system state variables corresponding to this grid-forming converter system are shown in Table 1.

[0088] Table 1 System state variables for grid-type converter systems

[0089] First, before using the electromagnetic transient stability analysis method based on frequency shift analysis provided in this embodiment of the invention, the eigenvalue behavior of the LTV EMT state-space model under different operating conditions is analyzed. Figure 5 This is a schematic diagram of the eigenvalues ​​of the LTV EMT state-space model under the symmetric condition before feature mapping provided in an embodiment of the present invention, as shown below. Figure 5 As shown, under symmetrical operating conditions, the real part of the system's eigenvalues ​​(i.e., Figure 5 (a) and the imaginary part (i.e.) Figure 5 (b) remains completely unchanged at the steady-state equilibrium point because the symmetric system can be equivalently transformed into an LTI system through similarity transformation. Figure 6 This is a schematic diagram of the eigenvalues ​​of the LTV EMT state-space model under asymmetric operating conditions before feature mapping, provided in an embodiment of the present invention. Figure 6 In this context, (a) represents the real part of the eigenvalue. Figure 6 In the diagram, (b) represents the imaginary part of the eigenvalue. Figure 6 In the diagram, (c) represents the magnified real part of the eigenvalue. Figure 6 In this context, (d) represents the imaginary part of the magnified eigenvalue, such as... Figure 6 As shown, under asymmetrical operating conditions, imbalance is introduced by adjusting the resistance of phase A, and the system characteristic values ​​exhibit time-varying oscillation characteristics.

[0090] Subsequently, the electromagnetic transient stability analysis method based on frequency shift analysis (i.e., SFA-SSM) provided in this embodiment of the invention is used to extract eigenvalues. Under symmetric operating conditions, a generalized frequency band cutoff set K = {0,1,2,3} is taken, where the maximum order K = 3. SFA-SSM can generate a 64-dimensional augmented eigenspectrum. Figure 7 This is a schematic diagram of the SFA-SSM eigenvalues ​​provided in an embodiment of the present invention. The 64-dimensional augmented eigenspectrum is restored to its original form after physical eigenvalue mapping. Figure 7 The 16-dimensional physical characteristic spectrum is shown. Simultaneously, the dq-SSM method is used to extract eigenvalues, and the eigenvalues ​​extracted by the SFA-SSM method are compared with those extracted by the dq-SSM method. The dq-SSM method is based on an electromagnetic transient state-space model established in the dq coordinate system (i.e., a synchronously rotating coordinate system). It converts the sinusoidal alternating current in the abc coordinate system into a direct current in the dq coordinate system to perform eigenvalue analysis at the equilibrium point. Figure 8 This is a comparative diagram of the feature values ​​extracted by the SFA-SSM and dq-SSM methods under symmetrical working conditions provided in this embodiment of the invention. Figure 8 In this context, (a) represents the real part of the SFA-SSM. Figure 8 In this context, (b) represents the imaginary part of the SFA-SSM. Figure 8 In this context, (c) represents the real part of the dq-SSM. Figure 8In the equation (d), the imaginary part of the dq-SSM is represented. Both models obtain constant eigenvalues ​​at the equilibrium point, but the total number and specific values ​​differ. The comparison results are shown in Table 2. Table 2 shows that the SFA-SSM method can extract 16 complete physical eigenvalues ​​(including λ8, 9, λ...). 10 (e.g., modes omitted by the dq-SSM method), and the imaginary part of the high-frequency eigenvalues ​​in the dq-SSM method is systematically shifted due to the frequency bias of the single-frequency rotating reference frame, which cannot accurately reflect the true oscillation frequency of the system.

[0091] Table 2. Comparison of eigenvalues ​​extracted by SFA-SSM and dq-SSM methods under symmetrical conditions.

[0092] Figure 9 This is a schematic diagram of the feature values ​​extracted by the dq-SSM method under asymmetric conditions provided by existing technology. Figure 9 In this context, (a) represents the real part of the dq-SSM. Figure 9 In this context, (b) represents the imaginary part of the dq-SSM. Figure 9 In this context, (c) represents the magnified real part of the dq-SSM. Figure 9 In this context, (d) represents the magnified imaginary part of the dq-SSM. For example... Figure 9 As shown, the intrinsic values ​​extracted based on the dq-SSM method exhibit time-varying characteristics under asymmetric conditions. This is because the traditional dq transform is only theoretically effective under ideal three-phase symmetry. In the asymmetric case, conventional dq modeling cannot preserve the LTI structure, and the neglect of zero-order and negative-order components leads to inherent errors in modal analysis. The traditional eigenvalue freezing method fails, and the dq transform cannot establish the LTI structure; therefore, alternative methods such as Floquet theory must be employed.

[0093] Figure 10 This is a schematic diagram of the feature values ​​extracted by the SFA-SSM method under asymmetric operating conditions provided in this embodiment of the invention. Figure 10 In this context, (a) represents the real part of the SFA-SSM. Figure 10 In this context, (b) represents the imaginary part of the SFA-SSM. For example... Figure 10 As shown, the physical feature spectrum extracted based on SFA-SSM maintains its time-invariant property. The influence of system asymmetry only shifts the steady-state equilibrium point to a new location without causing time-varying phenomena in the physical feature spectrum. Therefore, the method proposed in this embodiment of the invention exhibits robust performance under various operating scenarios, especially when the power grid conditions are unbalanced and the converter operates with an asymmetric control objective, where its performance is particularly excellent.

[0094] Under asymmetric conditions, the accuracy of the SFA method was verified using equivalent Floquet eigenvalues ​​as a benchmark. The comparison results are shown in Table 3. Table 3 shows that the SFA-SSM method can achieve full-dimensional lossless spectral reconstruction under asymmetric conditions, which is in perfect agreement with the theoretical benchmark results of Floquet.

[0095] Table 3. Comparison of equivalent Floquet eigenvalues ​​and SFA-SSM eigenvalues ​​under asymmetric conditions.

[0096] Figure 11 This is a schematic diagram illustrating the calculation errors of different maximum order eigenvalues ​​provided in an embodiment of the present invention. Figure 11 (a) in the text represents an asymmetrical operating condition. Figure 11 (b) in the diagram represents a symmetrical operating condition. For example... Figure 11 As shown, as the maximum order K gradually increases from 1 to 4, the eigenvalue calculation error of SFA-SSM monotonically converges. When K=3, it already fully covers the dominant frequency band of asymmetric excitation. Further increasing K only brings marginal accuracy improvement while significantly increasing the computational cost. The computational efficiency comparison is shown in Table 4. As can be seen from Table 4, the SFA-SSM method improves the computational speed by approximately 4.4 times while maintaining equivalent accuracy to Floquet theory, demonstrating significant engineering application value.

[0097] Table 4 Comparison of computational efficiency between Floquet theory and SFA-SSM method

[0098] The electromagnetic transient stability analysis device based on frequency shift analysis provided by the present invention will be described below. The electromagnetic transient stability analysis device based on frequency shift analysis described below can be referred to in correspondence with the electromagnetic transient stability analysis method based on frequency shift analysis described above.

[0099] This invention also provides an electromagnetic transient stability analysis device based on frequency shift analysis. Figure 12 This is a schematic diagram of the electromagnetic transient stability analysis device based on frequency shift analysis provided in an embodiment of the present invention, as shown below. Figure 12 As shown, the electromagnetic transient stability analysis device 1200 based on frequency shift analysis includes: an acquisition module 1210, a frequency shift transformation module 1220, an augmented projection module 1230, and a steady-state analysis module 1240.

[0100] The acquisition module 1210 is used to acquire the electromagnetic transient dynamic equations with time-varying characteristics in the power system.

[0101] The frequency shift transformation module 1220 is used to perform frequency shift transformation on the electromagnetic transient dynamic equation to obtain the frequency-shifted frequency domain nonlinear dynamic equation.

[0102] The augmented projection module 1230 is used to augment the complex envelope vector in the frequency-shifting nonlinear dynamic equation based on the generalized frequency band truncation set, to obtain the augmented state vector, and to determine the time-invariant state matrix at the steady-state equilibrium point based on the augmented state vector.

[0103] The steady-state analysis module 1240 is used to determine the augmented eigenvalue set based on the time-invariant state matrix, and to perform steady-state feature restoration on the augmented eigenvalue set based on the physical eigenvalue mapping criterion to obtain the physical feature spectrum.

[0104] The electromagnetic transient stability analysis device based on frequency shift analysis provided in this invention transforms the time-varying electromagnetic transient dynamic equations of a power system using frequency shift analysis to obtain a frequency-shifted frequency-domain nonlinear dynamic equation. This eliminates linear time-varying characteristics while retaining full-frequency information. Based on a generalized band cutoff set, the complex envelope vector in the frequency-shifted frequency-domain nonlinear dynamic equation is augmented and projected to obtain an augmented state vector. Based on this augmented state vector, the time-invariant state matrix at the steady-state equilibrium point is determined, eliminating nonlinear time-varying characteristics while retaining multi-band coupling effects. Based on the time-invariant state matrix, an augmented eigenvalue set is determined, and steady-state characteristics are restored to the augmented eigenvalue set according to the physical eigenvalue mapping criterion to obtain the physical characteristic spectrum. This achieves accurate, efficient, and lossless full-dimensional characteristic spectrum restoration of linear time-varying power systems, thereby enabling precise analysis of the electromagnetic transient stability of power systems. Furthermore, this invention does not rely on symmetry assumptions and is applicable to various operating conditions, including balanced, unbalanced, and multi-frequency control interactions. It is universally applicable to both grid-connected and grid-connected converters.

[0105] Optionally, the frequency shift conversion module 1220 is specifically used for: Perform a Hilbert transform on the state vector in the electromagnetic transient dynamic equation to obtain a complex domain analytic signal; Construct a diagonal frequency shift operator matrix; the diagonal frequency shift operator matrix is ​​used to characterize the multi-scale frequency shifting frequencies corresponding to different state vectors; the frequency shifting frequency corresponding to the electrical network state vector in the diagonal frequency shift operator matrix is ​​the fundamental frequency, and the frequency shifting frequency corresponding to the control system state vector is the zero frequency; Based on the complex domain analytic signal and the diagonal frequency shift operator matrix, the complex envelope vector is determined; Substituting the complex envelope vector into the electromagnetic transient dynamic equation yields the frequency-shifting nonlinear dynamic equation.

[0106] Optionally, the generalized band cutoff set includes multiple harmonic components.

[0107] Optionally, the augmentation projection module 1230 is specifically used for: Construct the orthogonal projection operator corresponding to the generalized frequency band truncation set; Based on the orthogonal projection operator, the complex envelope vector in the frequency-shifting nonlinear dynamic equation is decomposed to obtain the slowly varying complex envelope sub-vectors corresponding to each of the multiple harmonic components. The augmented state vector is determined based on all slowly varying complex envelope subvectors.

[0108] Optionally, the augmentation projection module 1230 is specifically used for: Based on the augmented state vector, the nonlinear ordinary differential equation is determined; Based on the augmented state vector and the nonlinear terms in the nonlinear ordinary differential equation, determine the real-field Jacobian matrix at the steady-state equilibrium point in the nonlinear ordinary differential equation. Based on the real-domain Jacobian matrix and the augmented linear state matrix in the nonlinear ordinary differential equation, the time-invariant state matrix at the steady-state equilibrium point is determined.

[0109] Optionally, the augmentation projection module 1230 is specifically used for: The augmented state vector is split into a real part vector and an imaginary part vector, and the nonlinear terms in the nonlinear ordinary differential equation are split into a real part mapping vector and an imaginary part mapping vector; At the steady-state equilibrium point, partial derivatives of the real part vector and the imaginary part vector are obtained based on the real part mapping vector to obtain the first real domain Jacobian sub-block and the second real domain Jacobian sub-block. At the steady-state equilibrium point, based on the imaginary part mapping vector, partial derivatives are taken with respect to the real part vector and the imaginary part vector respectively to obtain the third real domain Jacobian sub-block and the fourth real domain Jacobian sub-block. Based on the first real-domain Jacobian subblock, the second real-domain Jacobian subblock, the third real-domain Jacobian subblock, and the fourth real-domain Jacobian subblock, the real-domain Jacobian matrix at the steady-state equilibrium point in the nonlinear ordinary differential equation is determined.

[0110] Optionally, the augmentation projection module 1230 is specifically used for: The augmented linear state matrix is ​​expanded into real part submatrices and imaginary part submatrices; Based on the real part submatrix, the imaginary part submatrix, and the real field Jacobian matrix, the time-invariant state matrix at the steady-state equilibrium point is determined.

[0111] Optionally, the physical eigenvalue mapping criteria include the baseband subspace orthogonal projection criterion, the Nyquist principal value branch mapping criterion, and the quotient space equivalence class folding criterion, wherein: The baseband subspace orthogonal projection criterion is used to select the first physical candidate eigenvalue from the augmented eigenvalue set based on the projection energy norm of each eigenvalue in the baseband subspace and the higher harmonic subspace. The Nyquist principal value branch mapping criterion is used to fold the imaginary part of each eigenvalue in the augmented eigenvalue set into the Nyquist principal value branch interval, wherein the Nyquist principal value branch interval has a period of the fundamental frequency. The quotient space equivalence class folding criterion is used to merge the eigenvalues ​​of the same physical mode in the augmented eigenvalue set.

[0112] Optionally, the steady-state analysis module 1240 is specifically used for: Based on the left and right eigenvector matrices of each eigenvalue in the augmented eigenvalue set, the participation factor vector corresponding to each eigenvalue is determined. Based on the baseband subspace orthogonal projection operator and the participation factor vector corresponding to each eigenvalue, the baseband projection energy norm of each eigenvalue in the baseband subspace is determined; the baseband subspace orthogonal projection operator is used to project each eigenvalue to the baseband subspace. Based on the high-frequency extended subspace projection operator and the participation factor vector corresponding to each eigenvalue, the harmonic projection energy norm of each eigenvalue in the higher harmonic subspace is determined; the high-frequency extended subspace projection operator is used to project each eigenvalue to the higher harmonic subspace. The eigenvalues ​​whose baseband projection energy norm is greater than the harmonic projection energy norm are determined as the first physical candidate eigenvalues.

[0113] Optionally, the steady-state analysis module 1240 is specifically used for: Determine the Euclidean norm for any two eigenvalues ​​in the augmented eigenvalue set; Any two eigenvalues ​​whose Euclidean norm is less than the truncation tolerance are equivalently merged.

[0114] Figure 13 This is a schematic diagram of the structure of the electronic device provided in the embodiment of the present invention, such as... Figure 13As shown, the electronic device may include: a processor 1310, a communication interface 1320, a memory 1330, and a communication bus 1340, wherein the processor 1310, the communication interface 1320, and the memory 1330 communicate with each other through the communication bus 1340. The processor 1310 can call logic instructions in the memory 1330 to execute an electromagnetic transient stability analysis method based on frequency shift analysis. The method includes: obtaining the electromagnetic transient dynamic equation with time-varying characteristics in the power system; performing a frequency shift transformation on the electromagnetic transient dynamic equation to obtain a frequency-shifted nonlinear dynamic equation; performing an augmented projection on the complex envelope vector in the frequency-shifted nonlinear dynamic equation based on a generalized band cutoff set to obtain an augmented state vector, and determining the time-invariant state matrix at the steady-state equilibrium point based on the augmented state vector; determining the augmented eigenvalue set based on the time-invariant state matrix, and performing steady-state feature restoration on the augmented eigenvalue set based on the physical eigenvalue mapping criterion to obtain the physical feature spectrum.

[0115] Furthermore, the logical instructions in the aforementioned memory 1330 can be implemented as software functional units and, when sold or used as independent products, can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, or the part that contributes to the prior art, or a part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0116] On the other hand, the present invention also provides a computer program product, which includes a computer program that can be stored on a non-transitory computer-readable storage medium. When the computer program is executed by a processor, the computer can execute the electromagnetic transient stability analysis method based on frequency shift analysis provided by the above methods. The method includes: obtaining the electromagnetic transient dynamic equation with time-varying characteristics in the power system; performing a frequency shift transformation on the electromagnetic transient dynamic equation to obtain a frequency-shifted frequency domain nonlinear dynamic equation; performing an augmented projection on the complex envelope vector in the frequency-shifted frequency domain nonlinear dynamic equation based on a generalized band cutoff set to obtain an augmented state vector, and determining the time-invariant state matrix at the steady-state equilibrium point based on the augmented state vector; determining the augmented eigenvalue set based on the time-invariant state matrix, and performing steady-state feature restoration on the augmented eigenvalue set based on the physical eigenvalue mapping criterion to obtain the physical feature spectrum.

[0117] In another aspect, the present invention also provides a non-transient computer-readable storage medium storing a computer program thereon, which, when executed by a processor, implements the electromagnetic transient stability analysis method based on frequency shift analysis provided by the above methods. This method includes: obtaining the electromagnetic transient dynamic equations with time-varying characteristics in a power system; performing a frequency shift transformation on the electromagnetic transient dynamic equations to obtain a frequency-shifted frequency domain nonlinear dynamic equation; performing an augmented projection on the complex envelope vector in the frequency-shifted frequency domain nonlinear dynamic equation based on a generalized band cutoff set to obtain an augmented state vector, and determining the time-invariant state matrix at the steady-state equilibrium point based on the augmented state vector; determining the augmented eigenvalue set based on the time-invariant state matrix, and performing steady-state feature restoration on the augmented eigenvalue set based on a physical eigenvalue mapping criterion to obtain a physical feature spectrum.

[0118] The device embodiments described above are merely illustrative. The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this embodiment according to actual needs. Those skilled in the art can understand and implement this without any creative effort.

[0119] Through the above description of the embodiments, those skilled in the art can clearly understand that each embodiment can be implemented by means of software plus necessary general-purpose hardware platforms, and of course, it can also be implemented by hardware. Based on this understanding, the above technical solutions, in essence 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 ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods described in the various embodiments or some parts of the embodiments.

[0120] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for electromagnetic transient stability analysis based on frequency shift analysis, characterized in that, include: Obtain the electromagnetic transient dynamic equations with time-varying characteristics in the power system; The electromagnetic transient dynamic equations are subjected to frequency shift transformation to obtain frequency-shifted frequency domain nonlinear dynamic equations; Based on the generalized band cutoff set, the complex envelope vector in the frequency-shifting nonlinear dynamic equation is augmented and projected to obtain the augmented state vector. Based on the augmented state vector, the time-invariant state matrix at the steady-state equilibrium point is determined. Based on the time-invariant state matrix, an augmented eigenvalue set is determined, and the augmented eigenvalue set is restored to steady-state characteristics based on the physical eigenvalue mapping criterion to obtain the physical feature spectrum.

2. The electromagnetic transient stability analysis method based on frequency shift analysis according to claim 1, characterized in that, The step of performing a frequency-shift transformation on the electromagnetic transient dynamic equation to obtain a frequency-shifted frequency domain nonlinear dynamic equation includes: Perform a Hilbert transform on the state vector in the electromagnetic transient dynamic equation to obtain a complex domain analytic signal; Construct a diagonal frequency shift operator matrix; the diagonal frequency shift operator matrix is ​​used to characterize the multi-scale frequency shifting frequencies corresponding to different state vectors; the frequency shifting frequency corresponding to the electrical network state vector in the diagonal frequency shift operator matrix is ​​the fundamental frequency, and the frequency shifting frequency corresponding to the control system state vector is the zero frequency; Based on the complex domain analytic signal and the diagonal frequency shift operator matrix, the complex envelope vector is determined; Substituting the complex envelope vector into the electromagnetic transient dynamic equation yields the frequency-shifting nonlinear dynamic equation.

3. The electromagnetic transient stability analysis method based on frequency shift analysis according to claim 1, characterized in that, The generalized band cutoff set includes multiple harmonic components; The augmented state vector is obtained by performing an augmented projection on the complex envelope vector in the frequency-shifting nonlinear dynamic equation based on the generalized band cutoff set, including: Construct the orthogonal projection operator corresponding to the generalized frequency band truncation set; Based on the orthogonal projection operator, the complex envelope vector in the frequency-shifting nonlinear dynamic equation is decomposed to obtain the slowly varying complex envelope sub-vectors corresponding to each of the multiple harmonic components. The augmented state vector is determined based on all slowly varying complex envelope subvectors.

4. The electromagnetic transient stability analysis method based on frequency shift analysis according to claim 1, characterized in that, Determining the time-invariant state matrix at the steady-state equilibrium point based on the augmented state vector includes: Based on the augmented state vector, the nonlinear ordinary differential equation is determined; Based on the augmented state vector and the nonlinear terms in the nonlinear ordinary differential equation, determine the real-field Jacobian matrix at the steady-state equilibrium point in the nonlinear ordinary differential equation. Based on the real-domain Jacobian matrix and the augmented linear state matrix in the nonlinear ordinary differential equation, the time-invariant state matrix at the steady-state equilibrium point is determined.

5. The electromagnetic transient stability analysis method based on frequency shift analysis according to claim 4, characterized in that, The step of determining the real-domain Jacobian matrix at the steady-state equilibrium point in the nonlinear ordinary differential equation based on the augmented state vector and the nonlinear terms in the nonlinear ordinary differential equation includes: The augmented state vector is split into a real part vector and an imaginary part vector, and the nonlinear terms in the nonlinear ordinary differential equation are split into a real part mapping vector and an imaginary part mapping vector; At the steady-state equilibrium point, partial derivatives of the real part vector and the imaginary part vector are obtained based on the real part mapping vector to obtain the first real domain Jacobian sub-block and the second real domain Jacobian sub-block. At the steady-state equilibrium point, based on the imaginary part mapping vector, partial derivatives are taken with respect to the real part vector and the imaginary part vector respectively to obtain the third real domain Jacobian sub-block and the fourth real domain Jacobian sub-block. Based on the first real-domain Jacobian subblock, the second real-domain Jacobian subblock, the third real-domain Jacobian subblock, and the fourth real-domain Jacobian subblock, the real-domain Jacobian matrix at the steady-state equilibrium point in the nonlinear ordinary differential equation is determined.

6. The electromagnetic transient stability analysis method based on frequency shift analysis according to claim 4, characterized in that, Determining the time-invariant state matrix at the steady-state equilibrium point based on the real-domain Jacobian matrix and the augmented linear state matrix in the nonlinear ordinary differential equation includes: The augmented linear state matrix is ​​expanded into real part submatrices and imaginary part submatrices; Based on the real part submatrix, the imaginary part submatrix, and the real field Jacobian matrix, the time-invariant state matrix at the steady-state equilibrium point is determined.

7. The electromagnetic transient stability analysis method based on frequency shift analysis according to claim 1, characterized in that, The physical eigenvalue mapping criteria include the baseband subspace orthogonal projection criterion, the Nyquist principal value branch mapping criterion, and the quotient space equivalence class folding criterion, wherein: The baseband subspace orthogonal projection criterion is used to select the first physical candidate eigenvalue from the augmented eigenvalue set based on the projection energy norm of each eigenvalue in the baseband subspace and the higher harmonic subspace. The Nyquist principal value branch mapping criterion is used to fold the imaginary part of each eigenvalue in the augmented eigenvalue set into the Nyquist principal value branch interval, wherein the Nyquist principal value branch interval has a period of the fundamental frequency. The quotient space equivalence class folding criterion is used to merge the eigenvalues ​​of the same physical mode in the augmented eigenvalue set.

8. The electromagnetic transient stability analysis method based on frequency shift analysis according to claim 7, characterized in that, The first physical candidate eigenvalue is obtained by filtering from the augmented eigenvalue set based on the projected energy norms of each eigenvalue in the baseband subspace and the higher harmonic subspace, respectively, including: Based on the left and right eigenvector matrices of each eigenvalue in the augmented eigenvalue set, the participation factor vector corresponding to each eigenvalue is determined. Based on the baseband subspace orthogonal projection operator and the participation factor vector corresponding to each eigenvalue, the baseband projection energy norm of each eigenvalue in the baseband subspace is determined; the baseband subspace orthogonal projection operator is used to project each eigenvalue to the baseband subspace. Based on the high-frequency extended subspace projection operator and the participation factor vector corresponding to each eigenvalue, the harmonic projection energy norm of each eigenvalue in the higher harmonic subspace is determined; the high-frequency extended subspace projection operator is used to project each eigenvalue to the higher harmonic subspace. The eigenvalues ​​whose baseband projection energy norm is greater than the harmonic projection energy norm are determined as the first physical candidate eigenvalues.

9. The electromagnetic transient stability analysis method based on frequency shift analysis according to claim 7, characterized in that, The merging of eigenvalues ​​of the same physical mode in the augmented eigenvalue set includes: Determine the Euclidean norm for any two eigenvalues ​​in the augmented eigenvalue set; Any two eigenvalues ​​whose Euclidean norm is less than the truncation tolerance are equivalently merged.

10. An electromagnetic transient stability analysis device based on frequency shift analysis, characterized in that, include: The acquisition module is used to acquire the electromagnetic transient dynamic equations with time-varying characteristics in the power system; The frequency shift transformation module is used to perform a frequency shift transformation on the electromagnetic transient dynamic equation to obtain a frequency-shifted frequency domain nonlinear dynamic equation. An augmented projection module is used to augment the complex envelope vector in the frequency-shifting nonlinear dynamic equation based on a generalized band cutoff set, to obtain an augmented state vector, and to determine the time-invariant state matrix at the steady-state equilibrium point based on the augmented state vector. The steady-state analysis module is used to determine the augmented eigenvalue set based on the time-invariant state matrix, and to perform steady-state feature restoration on the augmented eigenvalue set based on the physical eigenvalue mapping criterion to obtain the physical feature spectrum.

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