Lagguerre-Taylor mixed order reduction method for multi-time-delay power system

Abstract

The invention belongs to the technical field of electric power system time delay analysis, and relates to a Lagguerre-Taylor hybrid order reduction method for a multi-time-delay electric power system, which comprises the following steps of: S1, modeling the multi-time-delay electric power system and defining an order reduction problem; s2, a projection matrix is constructed, wherein a Laguerre function and Taylor are expanded and mixed; the method specifically comprises the steps that a projection matrix V1 is constructed based on scale Laguerre function expansion, a projection matrix V2 is constructed based on Taylor expansion, and a high-order block Krylov subspace orthogonal basis is generated based on a high-order block Arnoldi execution process; s3, constructing a Lagguerre-Taylor hybrid projection method of the multi-time-delay power system, and constructing an order reduction system; s4, analyzing the stability of the reduced-order system; according to the method, the reduced-order framework based on the projection matrix is constructed, the time-delay term algebraic structure of an original system is completely reserved, and the defect that the time-delay structure is damaged in an existing method is overcome.

Description

Technical Field

[0001] This invention belongs to the field of power system time delay analysis technology, and relates to a Laguerre-Taylor hybrid order reduction method for multi-time-delay power systems. Background Technology

[0002] As power systems evolve towards greater complexity, interconnectivity, and intelligence, their scale expands and operational characteristics become increasingly complex. Time delays have become a critical factor that cannot be ignored in power system modeling and analysis. For example, wide-area control systems based on PMUs (Phasor Measurement Units) rely on remote signal transmission, and data transmission and routing in the communication network introduce millisecond to second-level time delays. On the one hand, the introduction of time delays may disrupt the inherent stability of the system and even induce oscillatory instability (such as subsynchronous oscillations). Traditional stability criteria based on Lyapunov functionals or frequency domain methods require processing infinite-dimensional state spaces, resulting in exponentially increasing computational complexity, making them unsuitable for large-scale power networks. On the other hand, these time delays typically exhibit multi-source, distributed, and time-varying characteristics and may exist in state variables, control inputs, or output feedback, causing the power system model to evolve from differential algebraic equations to high-dimensional time-delay differential algebraic equations, significantly increasing the complexity of analysis and control.

[0003] To reduce the complexity of time-delay system analysis and control, existing model reduction techniques, by mapping high-dimensional time-delay models to low-dimensional spaces while preserving key dynamic characteristics, have become a core means of solving the aforementioned problems. However, existing methods face the following shortcomings in power system time-delay scenarios: 1) Delay structure destruction: The Krylov subspace reduction method transforms a time-delay system into a time-delay-free system through spectral discretization. Applying the Krylov subspace method to the equivalent linear system without time delay yields the reduced-order system. This method does not consider the distribution characteristics of time delays, leading to the loss of delay structure information in the reduced-order model. 2) Computational burden and applicability limitations: The equilibrium truncation method constructs transformations based on the Gramians matrix of the system to achieve equilibrium, then truncates the uncontrollable and unobservable states to obtain the reduced-order system, typically exhibiting high accuracy. However, this method requires solving two sets of delay Lyapunov equations to obtain the system's Gramians matrix, leading to significant computational complexity and making it difficult to handle large power systems with numerous time delays. 3) The contradiction between stability and accuracy: While rational function expansion methods such as Padé approximation can approximate time-delay elements, they cannot guarantee the stability of the reduced-order model. The principal pole method preserves the main modes of the system, ensuring the stability of the system to a certain extent. However, since this method is based on the frequency domain analysis of the system, it cannot completely preserve the time domain characteristics (such as input and output characteristics).

[0004] Therefore, a method is needed to solve the above-mentioned technical problems by ensuring the stability, structural integrity, and computational efficiency of model reduction in power system multi-time-delay scenarios. Summary of the Invention

[0005] This invention aims to provide a hybrid expansion method integrating Taylor expansion and scaled Laguerre functions. The core features include: 1) Delay structure preservation: Constructing a model reduction method based on the projection matrix to preserve the delay structure of the original system. 2) Accuracy guarantee: Matching the system's frequency domain characteristics using the scaled Laguerre function, combined with Taylor expansion techniques to accurately capture the system's local modes, overcoming the distortion defects of traditional single expansion methods. 3) Computational efficiency optimization: Introducing high-order Krylov subspace techniques, combined with the high-order block Arnoldi algorithm, to efficiently generate the projection matrix. 4) Stability preservation: Constructing a reduced-order model based on hybrid basis functions, which can maintain the stability of the original system by introducing linear transformations.

[0006] The technical solution adopted by this invention to solve the technical problem is: a Laguerre-Taylor hybrid order reduction method for multi-time-delay power systems, comprising the following steps:

[0007] Step S1: Modeling and Defining the Order Reduction Problem of Multi-Time-Delay Power Systems;

[0008] Step S2, Projection Matrix Construction: This involves mixing the Laguerre function with Taylor expansion; specifically including:

[0009] Step S2-1: Construct the projection matrix V1 based on the scale Laguerre function expansion;

[0010] Step S2-2: Construct the projection matrix V2 based on Taylor expansion;

[0011] Step S2-3: Generate an orthogonal basis for the Krylov subspace of the higher-order block based on the Arnoldi execution process;

[0012] Step S3: Construct a reduced-order system using the Laguerre-Taylor hybrid projection method for multi-time-delay power systems;

[0013] Step S4: Stability analysis of the reduced-order system.

[0014] Preferably, in step S1, the multi-time-delay power system is modeled as follows:

[0015]

[0016] in, For system state variables, Let τ represent the derivative of x(t). k For the k-th time delay, E is the system constant matrix (E is usually a singular matrix). Let be the input and output matrices, where u(t) represents the input and y(t) represents the output, and the initial state is x(t)≡0 (t<0).

[0017] The model order reduction problem is:

[0018]

[0019] in, Represents the state vector of a reduced-order system. The derivative of the state vector Let V represent the reduced-order system matrix, V be the projection matrix to be determined, and u(t) represent the system input. Indicates system output;

[0020] The reduced-order model retains the number of time delay terms K and the time delay value τ of the original system. k The reduced-order system of the reduced-order model has the same stability criterion as the original system.

[0021] More preferably, in step S2-1, the projection matrix V1 is:

[0022]

[0023] in, Represents a higher-order Krylov subspace. Describe the inverse matrix of matrix B0, B i (i = 0, 1, L, s-1) represents the sequence of generating matrices in the projection matrix space V1, and B represents the input matrix of the original system.

[0024] More preferably, step S2-1 specifically includes the following steps:

[0025] make

[0026] Using the generating function of Laguerre polynomials Get the delayed term expansion coefficients

[0027] Using the bilinear transformation u=(s-α) / (s+α) and The coefficients of the expansion satisfy:

[0028]

[0029] Define matrix V1 as the higher-order Krylov subspace of the s-order block. A set of orthogonal bases.

[0030] More preferably, the projection matrix V2 in step S2-2 is:

[0031]

[0032] in, Represents a higher-order Krylov subspace. Represents the inverse matrix of matrix Θ0, Θ i (i = 0, 1, L, s-1) represents the sequence of generated matrices in the projection matrix space V2, and B represents the input matrix of the original system.

[0033] More preferably, the specific steps in step S2-2 include:

[0034] Given an expansion point s0, the moments of the transfer function at s0 are obtained by performing a Taylor expansion on the system's transfer function. In the formula M i =CN i B, matrix N i Construct it recursively;

[0035] Based on matrix sequence and Define the projection matrix V2 as the s-order block Krylov subspace as follows: A set of orthogonal bases.

[0036] More preferably, in step S3, the projection matrices [V1 V2] are merged and orthogonalized to obtain the final projection matrix V; the merging of the projection matrices is accomplished through QR decomposition or SVD decomposition, and the coefficient matrix of the reduced-order system is constructed using the projection matrix V.

[0037] More preferably, in step S4, the multi-time-delay power system is equivalently transformed into a linear system using Chebyshev discrete points, and then the matrix bundle (E) is solved. N A N The generalized eigenvalues ​​of the time-delay system are used to analyze its stability.

[0038] The beneficial effects of this invention are:

[0039] 1. This invention addresses the shortcomings of existing methods that destroy time-delay structures by constructing a reduction framework based on a projection matrix, which fully preserves the algebraic structure of the time-delay terms in the original system. Specifically, when generating the projection matrix, a higher-order block Arnoldi algorithm is used to directly apply the original time-delay differential algebraic equations, rather than discretizing the time-delay system into a time-delay-free equivalent system. This projection transformation only compresses the state-space dimension but retains the coupling relationship between the coefficient matrix of the time-delay terms and the time-delay parameters, ensuring that the reduced-order model still exhibits a multi-time-delay distribution consistent with the original system. This characteristic enables the reduced-order model to accurately reflect the coupling dynamics of actual time-delay scenarios such as signal transmission and multi-level routing in wide-area control systems, avoiding the modal distortion problem caused by structural simplification in traditional methods.

[0040] 2. This invention constructs a two-dimensional approximation mechanism using a hybrid basis function of scaled Laguerre functions and Taylor expansions, overcoming the shortcomings of existing methods in terms of single-distortion in the frequency or time domains. Technically, firstly, scaled Laguerre functions are used to perform orthogonal basis expansion on the time-delay element. By adjusting the scale factor to match the dominant frequency band of the system, the frequency response of the reduced-order model is ensured to be consistent with the original system. Simultaneously, Taylor expansions are introduced to approximate the transfer function, accurately capturing the local features of the system. The superposition of the two basis functions forms a composite projection space, achieving a high-precision approximation of the reduced-order system.

[0041] 3. This invention constructs a stability propagation path through a linear transformation of mixed basis functions, reducing the risk of introducing unstable poles that may occur with existing methods. Its core technology stems from solving a stable projection framework. This invention introduces a linear transformation to ensure that the Lyapunov functional of the reduced system remains isomorphic to the original system. This design ensures that the reduced model satisfies asymptotic stability and effectively prevents unstable pole phenomena caused by order reduction. Attached Figure Description

[0042] Figure 1 This is a schematic diagram of the Laguerre-Taylor hybrid order reduction method for multi-time-delay power systems according to the present invention, which constructs a projection matrix based on Laguerre expansion.

[0043] Figure 2 This is a schematic diagram of the projection matrix construction process based on Taylor expansion according to the present invention;

[0044] Figure 3 This is a schematic diagram of the order reduction process based on hybrid projection of the present invention;

[0045] Figure 4 This is a schematic diagram of the output response y1 and the corresponding absolute error of the original system and the 40th-order reduced system when the input u(t) = [sin(t); sin(t)] of the present invention;

[0046] Figure 5 This is a schematic diagram of the output response y2 and the corresponding absolute error of the original system and the 40th-order reduced system when the input u(t) = [sin(t); sin(t)] of the present invention;

[0047] Figure 6 These are the frequency domain singularity diagrams and gain diagrams of the original system and the 40th-order reduced system of this invention;

[0048] Figure 7 This is a schematic diagram of the eigenvalue distribution of the original system and the 40th-order reduced system of the present invention. Detailed Implementation

[0049] The related technologies of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0050] refer to Figures 1-7 The purpose of this implementation is to perform structure-preserving order reduction for multi-time-delay power systems, and to provide a hybrid projection model order reduction method based on the scale Laguerre function and Taylor expansion.

[0051] This implementation provides a hybrid projection reduction method based on scale Laguerre polynomials and Taylor expansions, including:

[0052] S1. Modeling and Order Reduction Problem Definition of Multi-Time-Delay Power Systems:

[0053] The model of a multi-time-delay power system can be represented by the following nth-order delay differential algebraic model:

[0054]

[0055] in, τ is a system state variable. k For the k-th time delay, E is the system constant matrix (E is usually a singular matrix). Let x(t) be the input and output matrix, and let x(t) ≡ 0 (t < 0) be the initial state.

[0056] Before performing model order reduction, the coefficient matrix of the system is usually preprocessed. By introducing an appropriate transformation, all the coefficient matrices of the system are rearranged so that matrix E has the following structure:

[0057]

[0058] For ease of explanation, the transformed coefficient matrix will still be represented by the same symbols.

[0059] The goal of order reduction is to construct a projection matrix. Satisfy V T V = I (order r << n), resulting in an r-order model:

[0060]

[0061] Wherein, the coefficient matrix At the same time, the reduced-order model is required to satisfy two conditions: first, preserve the delay structure, that is, retain the number of time delay terms K and the time delay value τ of the original system. k Second, maintain stability: the reduced-order system has the same stability criteria as the original system.

[0062] S2. Construction of the projection matrix: A hybrid method combining Laguerre functions and Taylor expansion.

[0063] S2.1 Constructing the projection matrix V1 based on the scale Laguerre function expansion:

[0064] Laguerre orthogonal functions are frequently used by engineers to solve system control problems, and their important properties facilitate the development of efficient model reduction methods. First, the three-term recurrence formula for the Laguerre polynomial is given:

[0065]

[0066] Based on Laguerre polynomials, scaled Laguerre orthogonal basis functions can be defined. Where L i (·) represents the i-th order Laguerre polynomial, and α > 0 is the scaling factor used to adjust the decay rate of the basis functions.

[0067] Using the Laplace transform, the Laguerre function in the frequency domain can be expressed as:

[0068]

[0069] sequence Forming Hardy space An orthonormal basis that satisfies orthogonality Where δ ij For the Kronecker function, space yes Laplace transform of . Arbitrary function Expand as in These are the Laguerre coefficients of F(s) in the frequency domain.

[0070] Through the Laplace transform, the system has a transfer function in the frequency domain:

[0071]

[0072] make Now consider the matrix function H. B (s) Laguerre function with respect to frequency domain scaling The expansion of .

[0073] First, consider the expansion of the delay term in the frequency domain:

[0074]

[0075] Where {T i k}yes The Laguerre coefficients. By employing the bilinear transformation u = (s-α) / (s+α), the above expression can be transformed into:

[0076]

[0077] Using the generating function of Laguerre polynomials It can be proven

[0078] Now consider the matrix function H. B The Laguerre function expansion of (s) at scale. Clearly, we have:

[0079]

[0080] Combine H B The expansion of (s) And the expansion of the delay term further yields:

[0081]

[0082] From both sides of the equation u i The coefficients of (i = 0, 1, ..., k-1) are equal, which gives H. B The coefficient X of (s) i Satisfies the linear equation:

[0083]

[0084] And matrix elements

[0085]

[0086] Note that the coefficient matrix of the above equation is a Toeplitz-like matrix, but not a true block Toeplitz matrix, because the single block element B i It is not a Toeplitz matrix. If matrix B0 is invertible, then X... i It can be calculated iteratively, that is:

[0087]

[0088] Within the framework of the projection method, matrix X can be used. i Constructing the projection matrix Make:

[0089] span{V1} = span{X0,X1,…,X…} s-1}

[0090] Note the special structure of the solution, using a set of matrices and We can define the higher-order Krylov subspace of an s-order block as follows: Matrix V1 can be constructed as an orthonormal basis for this higher-order block Krylov subspace, i.e.:

[0091]

[0092] The construction process of projection matrix V1 is as follows: Figure 1 .

[0093] S2.2 Constructing the projection matrix V2 based on Taylor expansion

[0094] Given any real number expansion point The Taylor expansion of the system transfer function is:

[0095]

[0096] The expression for the series coefficients is:

[0097]

[0098] Assuming matrix Θ0 is invertible, the system's transfer function can be approximated as:

[0099]

[0100] Where the expansion coefficients M i =CN i B is the moment of the system, and matrix N is... i The following recursive method can be used to construct it:

[0101]

[0102] Note that the expression for the system's moments is also related to the higher-order Krylov subspace, based on the matrix sequence. and The Krylov subspace of an s-order block can be defined as follows: This construction can be used to form a projection matrix through basis extraction. Right now:

[0103]

[0104] The construction process of projection matrix V2 is as follows: Figure 2 .

[0105] S2.3. Generate an orthogonal basis for the Krylov subspace of a higher-order block based on the execution process of the higher-order block Arnoldi:

[0106] With higher-order Krylov subspaces For example, where the matrix The steps for generating orthogonal bases using the higher-order block Arnoldi process are described below.

[0107] Input parameters:

[0108] 1) Matrix sequence

[0109] 2) Initial block matrix

[0110] 3) Krylov sequence order k.

[0111] Output result:

[0112] The orthogonal basis matrix Q of a higher-order Krylov subspace k ; Low-dimensional orthogonal basis used for projection order reduction.

[0113] Execution process

[0114] Step 1: Initialize the orthogonal basis and intermediate variables:

[0115] 1) QR decomposition initialization:

[0116] Perform QR decomposition on the input matrix U to obtain the initial orthogonal basis matrix. Here, the column vectors of Q1 are the orthogonalized basis vectors of U.

[0117] 2) Pre-setting intermediate variables:

[0118] Initialize l-1 intermediate matrices Set all elements to zero:

[0119]

[0120] Step 2: Generate Krylov subspace orthogonal basis column by column

[0121] Process each column of basis vectors in a loop (total number of columns is m×k):

[0122] 1) Calculation of the current column vector:

[0123] For the j-th column basis vector q j ∈Q k A vector v is generated through a linear combination of multiple matrices:

[0124]

[0125] 2) Temporary storage of intermediate variables:

[0126] Temporarily store the relevant intermediate variables of the current column in {z i}:

[0127]

[0128] Step 3: Orthogonalization

[0129] Orthogonalize the candidate vector v to eliminate its linear dependence on existing basis vectors:

[0130]

[0131] v = vq i t ij ,

[0132]

[0133] Step 4: Normalization and Termination Condition Judgment

[0134] 1) Norm calculation and termination test:

[0135] Calculate the 2-norm of the corrected vector v:

[0136] t j+1，j =||v||2

[0137] If t j+1，j =0 indicates that the candidate vector is linearly dependent on the existing basis space, and the algorithm is terminated to avoid redundant calculations.

[0138] 2) Basis vector normalization:

[0139] If t j+1，j If the value is not equal to 0, generate new orthogonal basis vectors and update intermediate variables:

[0140] q j+1 =v / t j+1,j ,

[0141]

[0142] Step 5: Iterate until completion.

[0143] Repeat steps 2-4 until m×k orthogonal basis vectors are generated, and finally output the matrix:

[0144] Q k =[q1 q2…q mk ]

[0145] S3. Using the Laguerre-Taylor hybrid projection method to construct multi-time-delay power systems, a reduced-order system is obtained:

[0146] Step 1: Construct the projection matrix V1 based on the scale Laguerre function expansion:

[0147] Technical features:

[0148] 1) Scaling Laguerre function expansion of time-delay elements.

[0149] Generate Laguerre polynomials to facilitate the calculation of coefficients {T} i k}, by choosing an appropriate scaling factor α, the transfer function H(s) is scaled within the Laguerre basis functions. Expand upwards to generate a matrix sequence. Constructing higher-order Krylov subspaces

[0150] 2) Higher-order Krylov subspace projection:

[0151] Generate higher-order Krylov subspaces using the higher-order block Arnoldi process A set of orthogonal bases, namely the projection matrix V1, is used to preserve a certain number of Laguerre expansion coefficients of the original system's transfer function.

[0152] Step 2: Construct the projection matrix V2 based on Taylor expansion

[0153] Technical features:

[0154] 1) Taylor series expansion of time-delay components:

[0155] Select a suitable expansion point s0 and calculate the expansion coefficients. Constructing higher-order Krylov subspaces

[0156] 2) The higher-order block Arnoldi process generates the projection matrix V2:

[0157] Generate higher-order Krylov subspaces using the higher-order block Arnoldi process A set of orthogonal bases, namely the projection matrix V2, is used to preserve a certain number of moments of the original system.

[0158] Step 3: Merge the projection matrices V = [V1 V2] and orthogonalize them:

[0159] Technical features:

[0160] 1) Matrix concatenation:

[0161] Concatenate V1 and V2 column-wise, and obtain the orthogonal projection matrix through QR decomposition or SVD decomposition. Where r = r1 + r2.

[0162] 2) Feature Preservation:

[0163] The column space of the projection matrix V simultaneously covers the frequency domain characteristics (Laguerre function) and the dominant mode (Taylor expansion) of the time delay term.

[0164] Step 4: Construct a reduced-order system:

[0165] By using QR decomposition or SVD decomposition, and combining two projection matrices V1 and V2, the projection matrix is ​​generated. The system satisfies span{V}=span{[V1,V2]}. The coefficient matrix of the reduced-order system is constructed using the projection matrix.

[0166] The construction process of the hybrid projection method for multi-time-delay power systems is as follows: Figure 3 .

[0167] Technical features

[0168] 1) Time-delay structure:

[0169] This projection transformation only reduces the dimension of the state space, but retains the coupling relationship between the coefficient matrix of the time delay term and the time delay, ensuring that the reduced-order model retains the delay structure of the original system.

[0170] S4. Stability analysis of the reduced-order system:

[0171] The characteristic equation of a time-delay system is a transcendental equation, and its solution involves exponential functions, leading to infinitely many roots. Here, we utilize Chebyshev discrete points to equivalently transform it into a linear system, and then solve for the matrix bundle (E... N A N We analyze the stability of the generalized eigenvalues ​​of ).

[0172]

[0173] in, The matrix D represents the row of the k-th lag variable. N =[D k,k The elements of ] are:

[0174]

[0175] And when i = 0 or N, c i =2, and 1 in all other cases.

[0176] If the system is asymptotically stable, meaning all eigenvalues ​​of the matrix bundle have negative real parts, then for any symmetric positive definite matrix Q, there exists a unique symmetric positive definite matrix P satisfying the matrix equation. The converse also holds. Given the projection matrix V, construct... Therefore, it is true:

[0177]

[0178] Since congruent transformations do not change the positive definiteness of a matrix, the matrix... It remains a symmetric positive definite matrix. Therefore, there exists a symmetric positive definite matrix. Make the matrix Established, among which If the coefficient matrix is ​​the equivalent linear system of the reduced-order system, then the reduced-order system is stable.

[0179] Therefore, the stability of the reduced-order system can be theoretically guaranteed. However, in practical analysis, due to the singularity of E, to ensure the stability of the reduced-order system, a transformation T needs to be introduced to make the matrix... Negative definite, where The process of constructing the transformation T is described below. Solving a generalized Lyapunov equation... We obtain a symmetric positive definite solution P, and perform Cholesky decomposition on matrix P: P = LL T The transformation T = L is obtained. T .

[0180] Finally, the steps for stability analysis of reduced-order systems are given.

[0181] Step 1: Transform the reduced-order system into a linear system through spectral discretization, obtaining the matrix. and

[0182] Technical features: Generates a discretized model for calculating the rightmost eigenvalue of a power system containing time delays.

[0183] Step 2: Calculate the generalized Lyapunov equation Solution For matrix Perform Cholesky decomposition The transformation T = L is obtained T .

[0184] Technical feature: The introduction of transformation T allows the reduced-order system to retain the dominant modes of the original system.

[0185] Step 3: Calculate the matrix Solving matrix bundles The generalized eigenvalues.

[0186] Technical characteristics: When the original system is stable, the resulting reduced-order system is also stable, that is, the eigenvalues ​​of the reduced-order system all have negative real parts.

[0187] Example

[0188] This embodiment is based on a time-delay dynamic model constructed for testing on a typical New England power grid architecture (IEEE 39-bus system). The system includes 10 generators, 39 bus nodes, and 46 transmission lines, with basic data sourced from [source missing]. To characterize the delay effects in actual operation, the model introduces 140 time-delay terms by extending the traditional differential-algebraic equations, forming a time-delay differential-algebraic system. Its state variables (such as generator dynamic parameters) and algebraic variables (such as node voltage and power) reach 46 and 141 respectively, used to simulate power system stability problems under complex operating conditions such as communication delays and control response lags.

[0189] The S1 procedure is executed to rearrange the elements of the system's coefficient matrix, obtaining the coefficient matrix E,A required for order reduction. k ,τ k (k = 1, 2, L, 140), B, C; Set the scale factor α = 48, parameter s = 10, and execute procedure S2.1 (see flowchart). Figure 1 The process involves constructing an orthogonal basis for the higher-order block Krylov subspace using the Arnoldi procedure in S2.3, thereby obtaining the projection matrix V1 based on the scale Laguerre function; taking the expansion point s0 = 40 and the parameter s = 10, and executing S2.2 (see process details). Figure 2 This also involves the higher-order block Arnoldi process in S2.3, obtaining the Taylor-expansion-based projection matrix V2; combining the two projection matrices, S3 is executed (see process). Figure 3 ), thus obtaining a 40th-order reduced-order system.

[0190] Given a time interval [0, T] = [0, 2 × 10 -4 For this test system, when the input is u(t) = [sin(t); sin(t)], Figure 4 and 5 The two outputs of the system in the time domain and their corresponding absolute errors are described, which shows that the proposed method can approximate the transient response of the original system well in the time domain.

[0191] Given a frequency domain interval ω∈[0,6000] rad / s, the maximum singular value images and gains of the original system and the reduced-order system are as follows: Figure 6 As shown, the reduced-order model has almost identical dynamic response to the original system, and performs better in the high-frequency band, which demonstrates the high accuracy of the proposed method in the frequency domain.

[0192] The spectral discretization process of S4 is performed to calculate the eigenvalues ​​of the original system, all of which have negative real parts, indicating the stability of the system. Then, the stability analysis steps in S4 are performed on the reduced-order system to calculate its eigenvalues. Figure 7 The distribution of eigenvalues ​​of the original system and the reduced system is described, which shows that the obtained reduced system can maintain the stability of the original system.

[0193] In summary, this invention addresses the shortcomings of existing methods in destroying the time-delay structure by constructing a reduced-order framework based on the projection matrix, which fully preserves the algebraic structure of the time-delay term in the original system. Furthermore, it overcomes the limitations of existing methods in terms of single-distortion in the frequency or time domain by constructing a two-dimensional approximation mechanism using a hybrid basis function of the scaling Laguerre function and Taylor expansion. Finally, it reduces the risk of introducing unstable poles by constructing a stability transfer channel through linear transformation of the hybrid basis function.

[0194] It should be emphasized that the above are merely preferred embodiments of the present invention and are not intended to limit the present invention in any way. Any simple modifications, equivalent changes and alterations made to the above embodiments based on the technical essence of the present invention shall still fall within the scope of the technical solution of the present invention.

Claims

1. A Laguerre-Taylor hybrid order reduction method for multi-time-delay power systems, characterized in that, Includes the following steps: Step S1: Modeling and Defining the Order Reduction Problem of Multi-Time-Delay Power Systems; Step S2, Projection Matrix Construction: Laguerre function and Taylor expansion are combined; Specifically, this includes: Step S2-1: Construct the projection matrix V1 based on the scale Laguerre function expansion; Step S2-2: Construct the projection matrix V2 based on Taylor expansion; Step S2-3: Generate an orthogonal basis for the Krylov subspace of the higher-order block based on the Arnoldi execution process; Step S3: Construct a reduced-order system using the Laguerre-Taylor hybrid projection method for multi-time-delay power systems; Step S4: Stability analysis of the reduced-order system.

2. The Laguerre-Taylor hybrid order reduction method for multi-time-delay power systems according to claim 1, characterized in that, In step S1, the multi-time-delay power system is modeled as follows: in, For system state variables, Let τ represent the derivative of x(t). k For the k-th time delay, The system constant matrix, Let y(t) be the input and output matrices, where u(t) represents the input and y(t) represents the output. The model order reduction problem is: in, Represents the state vector of a reduced-order system. The derivative of the state vector Let V represent the reduced-order system matrix, and V be the projection matrix to be determined for the reduced order. Indicates system output; The reduced-order model retains the number of time delay terms K and the time delay value τ of the original system. k The reduced-order system of the reduced-order model has the same stability criterion as the original system.

3. The Laguerre-Taylor hybrid order reduction method for multi-time-delay power systems according to claim 2, characterized in that, In step S2-1, the projection matrix V1 is: in, Represents a higher-order Krylov subspace. Describe the inverse matrix of matrix B0, B i (i = 0, 1, L, s-1) represents the sequence of generating matrices in the projection matrix space V1, and B represents the input matrix of the original system.

4. The Laguerre-Taylor hybrid order reduction method for multi-time-delay power systems according to claim 3, characterized in that, Step S2-1 specifically includes the following steps: make Using the generating function of Laguerre polynomials Get the delayed term expansion coefficients Using the bilinear transformation u=(s-α) / (s+α) and The coefficients of the expansion satisfy: Define matrix V1 as the higher-order Krylov subspace of the s-order block. A set of orthogonal bases.

5. The Laguerre-Taylor hybrid order reduction method for multi-time-delay power systems according to claim 2, characterized in that, The projection matrix V2 in step S2-2 is: in, Represents a higher-order Krylov subspace. Represents the inverse matrix of matrix Θ0, Θ i (i = 0, 1, L, s-1) represents the sequence of generated matrices in the projection matrix space V2, and B represents the input matrix of the original system.

6. The Laguerre-Taylor hybrid order reduction method for multi-time-delay power systems according to claim 5, characterized in that, The specific steps in step S2-2 include: Given an expansion point s0, the moments of the transfer function at s0 are obtained by performing a Taylor expansion on the system's transfer function. In the formula M i =CN i B, matrix N i Construct it recursively; Based on matrix sequence and Define the projection matrix V2 as the s-order block Krylov subspace as follows: A set of orthogonal bases.

7. The Laguerre-Taylor hybrid order reduction method for multi-time-delay power systems according to claim 2, characterized in that, In step S3, the projection matrices [V1 V2] are merged and orthogonalized to obtain the final projection matrix V. The merging of the projection matrices is accomplished through QR decomposition or SVD decomposition, and the coefficient matrix of the reduced-order system is constructed using the projection matrix V.

8. The Laguerre-Taylor hybrid order reduction method for multi-time-delay power systems according to claim 2, characterized in that, In step S4, the multi-time-delay power system is equivalently transformed into a linear system using Chebyshev discrete points. Then, the stability of the time-delay system is analyzed by solving the generalized eigenvalues ​​of the matrix bundle.

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