Power system time domain model order reduction method based on Laguerre polynomial and approximate Gram matrix

Abstract

According to the power system time domain model order reduction method based on the Laguerre polynomial and the approximate Gram matrix, efficient order reduction can be performed on the state space model of the power system in combination with the Laguerre polynomial and the approximate Gram matrix, and the system response of the power system is accurately predicted by using the order reduction model. On one hand, in the order reduction process, two large Lyapunov equations do not need to be solved to calculate a controllable Gram matrix and a considerable Gram matrix, but a Laguerre orthogonal polynomial is utilized to perform low-rank approximation on the controllable Gram matrix and the considerable Gram matrix of the power system, so that the calculation complexity can be reduced, efficient order reduction can be realized, and the response prediction efficiency of the power system can be improved. And on the other hand, the reduced-order model has a global error boundary, so that the response prediction accuracy of the power system can be improved. Therefore, the method can give consideration to relatively good prediction accuracy and relatively low calculation complexity.

Description

Technical Field

[0001] The present application relates to the technical field of power systems, and in particular to a method for reducing the order of a time-domain model of a power system based on Laguerre polynomials and approximate Gram matrices. Background Art

[0002] As power systems continue to develop towards large-scale configuration, the system scale of power systems can reach thousands of buses and dynamic elements. The model equations used to describe the time domain response of power systems involve nonlinear ordinary differential equations, and the dimensions of these equations after linearization at the equilibrium point appear in large quantities. In order to better analyze the small signal stability of power systems, it is necessary to find a method that can reduce the model size, data storage requirements and computational complexity of power system time domain models. It can be seen that power system time domain model reduction is an important part of linear control system analysis and power system dynamic model design, and can be used to effectively reduce the computational complexity of controller design.

[0003] At present, the order reduction methods of power system time domain models mainly include moment matching method, mode truncation method and balanced truncation method. Among them, in the moment matching method, the system transfer function of the power system time domain model can be expanded around an operating point, and a reduced-order system can be constructed using Krylov subspace technology. Although this is an effective method, it does not have a global error bound. The mode truncation method retains the dominant mode of the system and truncates the modes of less importance, and cannot fully retain the input and output characteristics of the power system. The balanced truncation method is known for its ability to capture the input and output behavior and dominant mode of the power system. Compared with other order reduction methods, the balanced truncation method has theoretical advantages, including global error bounds and stability maintenance. However, when using the accurate balanced truncation method to reduce the order of the power system time domain model, two large Lyapunov equations need to be solved to obtain the controllable Gram matrix and the observable Gram matrix, which leads to significant computational complexity.

[0004] It can be seen that the existing technology cannot take into account both good response prediction accuracy and low computational complexity. Summary of the invention

[0005] The purpose of this application is to solve at least one of the above-mentioned technical deficiencies, especially the technical defect that the prior art cannot take into account both good response prediction accuracy and low computational complexity.

[0006] In one embodiment, the present application provides a method for reducing the order of a power system time domain model based on Laguerre polynomials and approximate Gram matrices, comprising:

[0007] Obtaining a state space model of the power system;

[0008] Using a Laguerre polynomial approximation algorithm, respectively calculating each first Laguerre coefficient corresponding to the state impulse response of the state-space model and each second Laguerre coefficient corresponding to the state impulse response of the dual model; wherein the dual model is dual to the state-space model;

[0009] Calculate a first low-rank decomposition factor matrix corresponding to a controllable Gram matrix according to each of the first Laguerre coefficients, and calculate a second low-rank decomposition factor matrix corresponding to an observable Gram matrix according to each of the second Laguerre coefficients;

[0010] Based on the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix, obtaining a reduced-order model corresponding to the state-space model;

[0011] In response to a system response prediction request, an input vector to be predicted of the power system is received, and a time domain response vector of the power system is predicted according to the input vector to be predicted and the reduced-order model.

[0012] In one embodiment, the state-space model includes a system matrix, an input matrix, and an output matrix;

[0013] The Laguerre polynomial approximation algorithm is used to calculate the first Laguerre coefficients corresponding to the state impulse response of the state space model and the second Laguerre coefficients corresponding to the state impulse response of the dual model, including:

[0014] The first Laguerre coefficients are calculated based on the following expressions:

[0015]

[0016] In the formula, is the coefficient of the first Laguerre expansion corresponding to the state impulse response of the state-space model; is the coefficient of the i+1th Laguerre expansion corresponding to the state impulse response of the state-space model; for dimensional identity matrix, is the number of differential variables in the linear differential algebraic equation of the power system, and the state space model is an equivalent model of the linear differential algebraic equation; is the system matrix, is the input matrix, is the predetermined number of Laguerre expansion terms;

[0017] The second Laguerre coefficients are calculated based on the following expressions:

[0018]

[0019] In the formula, is the coefficient of the first Laguerre expansion corresponding to the state impulse response of the dual model; is the coefficient of the i+1th Laguerre expansion corresponding to the state impulse response of the dual model; for The transposed matrix of for The transposed matrix of is the output matrix.

[0020] In one embodiment, the step of calculating a first low-rank decomposition factor matrix corresponding to a controllable Gram matrix according to each of the first Laguerre coefficients, and calculating a second low-rank decomposition factor matrix corresponding to an observable Gram matrix according to each of the second Laguerre coefficients, comprises:

[0021] Calculating the integral result of the product of the Laguerre polynomial within a preset time interval, and constructing an integral matrix according to the integral result;

[0022] Performing Cholesky decomposition on the integral matrix to obtain a first lower triangular matrix;

[0023] Obtaining the first low-rank decomposition factor matrix according to the first lower triangular matrix and each of the first Laguerre coefficients;

[0024] The second low-rank decomposition factor matrix is ​​obtained according to the first lower triangular matrix and each of the second Laguerre coefficients.

[0025] In one embodiment, the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix are obtained based on the following expressions:

[0026]

[0027] In the formula, is the first low-rank decomposition factor matrix; is a coefficient matrix composed of the first Laguerre coefficients; is the first lower triangular matrix; for dimensional identity matrix, is the number of inputs of the power system; is the second low-rank decomposition factor matrix; is a coefficient matrix composed of each of the second Laguerre coefficients; for dimensional identity matrix, is the number of outputs of the power system; is the Kronecker product operator.

[0028] In one embodiment, obtaining the reduced-order model corresponding to the state-space model based on the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix includes:

[0029] right The matrix is ​​decomposed by SVD, and the left truncation matrix is ​​obtained according to the SVD decomposition result and the preset reduction order. , diagonal truncation matrix and the right intercept matrix ;in, is the first low-rank decomposition factor matrix, for The transposed matrix of is the second low-rank decomposition factor matrix;

[0030] According to the left interception matrix , the diagonal intercept matrix and the right intercept matrix , calculate the left projection matrix and the right projection matrix respectively;

[0031] The reduced-order model is obtained according to the left projection matrix and the right projection matrix.

[0032] In one of the embodiments, the left projection matrix and the right projection matrix are obtained based on the following expressions:

[0033]

[0034] In the formula, is the left projection matrix; is the diagonal intercept matrix; for The transposed matrix of is the left intercept matrix; for The transposed matrix of is the second low-rank decomposition factor matrix; is the right projection matrix; is the first low-rank decomposition factor matrix; is the right truncation matrix.

[0035] In one embodiment, the state-space model includes the system matrix , input matrix , output matrix and the constant matrix ;

[0036] The step of obtaining the reduced-order model according to the left projection matrix and the right projection matrix comprises:

[0037] The reduced-order system matrices are obtained based on the following expressions: , the reduced-order input matrix and the reduced-order output matrix :

[0038]

[0039] In the formula, is the left projection matrix, is the right projection matrix;

[0040] The reduced-order model is constructed according to the reduced-order system matrix, the reduced-order input matrix, the reduced-order output matrix and the constant matrix.

[0041] In a second aspect, an embodiment of the present application provides a power system time domain model reduction device based on Laguerre polynomials and approximate Gram matrices, comprising:

[0042] A state space model acquisition module, used for acquiring a state space model of a power system;

[0043] A Laguerre coefficient acquisition module, used to calculate each first Laguerre coefficient corresponding to the state impulse response of the state space model and each second Laguerre coefficient corresponding to the state impulse response of the dual model by using a Laguerre polynomial approximation algorithm; wherein the dual model is dual to the state space model;

[0044] A low-rank decomposition module, used to calculate a first low-rank decomposition factor matrix corresponding to a controllable Gram matrix according to each of the first Laguerre coefficients, and to calculate a second low-rank decomposition factor matrix corresponding to an observable Gram matrix according to each of the second Laguerre coefficients;

[0045] A reduction module, used for obtaining a reduced-order model corresponding to the state-space model based on the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix;

[0046] The response prediction module is used to receive the input vector to be predicted of the power system in response to the system response prediction request, and predict the time domain response vector of the power system according to the input vector to be predicted and the reduced-order model.

[0047] In a third aspect, an embodiment of the present application provides a storage medium storing computer-readable instructions. When the computer-readable instructions are executed by one or more processors, the one or more processors execute the steps of the power system time domain model reduction method based on Laguerre polynomials and approximate Gram matrices described in any of the above embodiments.

[0048] In a fourth aspect, an embodiment of the present application provides a computer device, the computer device comprising: one or more processors, and a memory;

[0049] The memory stores computer-readable instructions, and when the computer-readable instructions are executed by the one or more processors, the steps of the power system time domain model reduction method based on Laguerre polynomials and approximate Gram matrices described in any of the above embodiments are executed.

[0050] In the power system time domain model reduction method, device, storage medium and computer equipment based on Laguerre polynomials and approximate Gram matrices provided in some embodiments of the present application, the state space model of the power system can be efficiently reduced in order by combining Laguerre polynomials and approximate Gram matrices, and the reduced-order model can be used to accurately predict the system response of the power system. On the one hand, the present application no longer needs to solve two large Lyapunov equations to calculate the controllable Gram matrix and the observable Gram matrix during the reduction process, but uses Laguerre orthogonal polynomials to perform low-rank approximation on the controllable Gram matrix and the observable Gram matrix of the power system, thereby reducing the computational complexity and achieving efficient reduction, thereby improving the efficiency of power system response prediction. On the other hand, the reduced-order model of the present application has a global error bound, thereby improving the accuracy of power system response prediction. It can be seen that the present application can take into account both good prediction accuracy and low computational complexity. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0052] Figure 1 A schematic diagram of a process flow of a power system time domain model reduction method based on Laguerre polynomials and approximate Gram matrices in one embodiment;

[0053] Figure 21 is a schematic flow chart of the Laguerre coefficient calculation step in one embodiment;

[0054] Figure 3 For one embodiment, when the input vector Schematic diagram of the output response of the LRBT reduced-order model and the BT reduced-order model when ;

[0055] Figure 4 For one embodiment, when the input vector Schematic diagram of the response error of the LRBT reduced-order model and the BT reduced-order model when ;

[0056] Figure 5 For one embodiment, when the input vector Schematic diagram of the output response of the LRBT reduced-order model and the BT reduced-order model when ;

[0057] Figure 6 For one embodiment, when the input vector Schematic diagram of the response error of the LRBT reduced-order model and the BT reduced-order model when ;

[0058] Figure 7 A comparison diagram of the prediction time consumption of the LRBT reduced-order model and the BT reduced-order model in one embodiment;

[0059] Figure 8 1 is a schematic diagram of a structure of a power system time domain model reduction device based on Laguerre polynomials and approximate Gram matrices in one embodiment;

[0060] Fig. 9 FIG. 1 is a diagram of the internal structure of a computer device in one embodiment. DETAILED DESCRIPTION

[0061] The following will be combined with the drawings in the embodiments of the present application to clearly and completely describe the technical solutions in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of this application.

[0062] In one embodiment, Figure 1 As shown, the present application provides a method for reducing the order of a power system time domain model based on Laguerre polynomials and approximate Gram matrices, which specifically includes the following steps:

[0063] S102: Obtain a state space model of the power system.

[0064] The state space model is a dynamic time domain model used to predict the change of the response vector of the power system over time. In the present application, the state space model of the power system can be an ordinary differential system with an index of 1.

[0065] In this step, an equivalent state space model can be constructed for the linear differential algebraic equation of the whole system under the small disturbance stability analysis problem of the power system, so as to describe the relationship between the input and the response through the state space model.

[0066] Specifically, if the circuit system has p inputs and q outputs, the linear model of the power system can be expressed by the following n-order linear time-invariant differential algebraic equation: express:

[0067]

[0068] In the formula, is the input constant matrix, for The first-order differential of is the system matrix of the linear differential algebraic equation, is the generalized state vector, is the input matrix of the linear differential algebraic equation, is the input vector, is the response vector, is the output matrix of the linear differential algebraic equation, is the constant matrix of the linear differential algebraic equation.

[0069] . is the set of real numbers.

[0070] in, , , , and It has the following block structure:

[0071] , , , ,

[0072] In the formula, for dimensional identity matrix. is the number of differential variables in the linear differential algebraic equations of the power system, is the number of algebraic variables in the linear differential algebraic equations of the power system, . , , and Both submatrix of , and , , , . and Both submatrix of , and , . and Both submatrix of , and , . and Both The sub-matrix of is the state vector of the power system, is the operating vector of the power system. , .

[0073] when When non-singular, the system is a differential algebraic model with index 1, which can be rewritten as an equivalent ordinary differential system of the following form:

[0074]

[0075] In the formula, for The first-order differential of .

[0076] In actual power systems, is usually non-singular, so the running vector in the above formula can be eliminated , let the state variable , and then get the system Equivalent Order state space model, that is, ordinary differential system :

[0077]

[0078] in:

[0079]

[0080] In the formula, for The first-order differential of is the generalized state vector of the state-space model, is the system matrix of the state-space model, is the input matrix of the state-space model, is the output matrix of the state-space model, is the constant matrix of the state-space model, is the input vector, is the response vector.

[0081] S104: using a Laguerre polynomial approximation algorithm to respectively calculate each first Laguerre coefficient corresponding to the state impulse response of the state-space model and each second Laguerre coefficient corresponding to the state impulse response of the dual model; wherein the dual model is dual to the state-space model.

[0082] In this step, the Laguerre polynomial can be used to model the state space The state variables in are approximately expanded and the expansion coefficient is solved, and the expansion coefficient is the Laguerre coefficient described in the embodiment of the present application.

[0083] Specifically, the Laguerre polynomial is defined as:

[0084]

[0085] In the formula, is the first Laguerre expansion, is the (i+1)th Laguerre expansion, is a natural constant, t is time, and d is the differential sign.

[0086] The three-term recursive formula is:

[0087]

[0088] Laguerre polynomials also have the integral recursive property:

[0089]

[0090] In the formula, is the variable symbol.

[0091] Laguerre polynomial sequence Constructing Hilbert space A uniformly bounded orthogonal basis of , so for any function , which can be expanded into the following Laguerre polynomial series:

[0092]

[0093] In the formula, is the Laguerre coefficient.

[0094] In this step, let the state space model The input vector is a unit impulse function , then the corresponding state equation is:

[0095]

[0096] Apply the same method to both sides of the above state equation The integral on , that is:

[0097]

[0098] Assume that the initial condition is zero and use the integral property of the unit impulse function , then the above formula can be changed to:

[0099]

[0100] The state variables of the above formula, that is, the state impulse response of the power system, are approximately expanded using Laguerre polynomials:

[0101]

[0102] In the formula, l is the number of Laguerre expansion terms, and its specific value can be obtained according to actual conditions. In an example, The specific value can be increased or decreased according to the actual situation. is the i-th first Laguerre coefficient.

[0103] Similarly, the state-space model The dual model is:

[0104]

[0105] In the formula, for The first-order differential of is the generalized state vector of the dual model. for The transposed matrix of for The transposed matrix of for The transposed matrix of . is the input vector of the dual model, is the response vector of the dual model.

[0106] Let the input vector of the dual model be the unit impulse function , to obtain the state variables of the dual model, that is, the state impulse response of the dual model. The state impulse response of the dual model is approximately expanded using Laguerre polynomials:

[0107]

[0108] In the formula, is the i-th second Laguerre coefficient.

[0109] S106: Calculate a first low-rank decomposition factor matrix corresponding to the controllable Gram matrix according to each first Laguerre coefficient, and calculate a second low-rank decomposition factor matrix corresponding to the observable Gram matrix according to each second Laguerre coefficient.

[0110] In this step, the state space model of the power system , in the time interval The controllable Gram matrix P and the observable Gram matrix Q are given above:

[0111]

[0112] Based on the specific form of controllable Gram matrix P and observable Gram matrix Q. Among them, It is a state space model The corresponding state impulse response when the input vector is a unit impulse function. are the state impulse responses of the dual model when the input vector is a unit impulse function. and , rewrite the controllable Gram matrix P and observable Gram matrix Q as:

[0113]

[0114] In the formula, for The transposed matrix of for The transposed matrix of .

[0115] Will The Laguerre polynomials and Substituting the Laguerre polynomial approximate expansion into the above formula, we can get:

[0116] ,

[0117] In the formula, for The transposed matrix of for The transposed matrix of .

[0118] It can be seen that this step can approximate the controllable Gram matrix according to each first Laguerre coefficient, and then obtain the first low-rank decomposition factor matrix corresponding to the controllable Gram matrix. Similarly, this step can approximate the observable Gram matrix according to each second Laguerre coefficient, and then obtain the second low-rank decomposition factor matrix corresponding to the observable Gram matrix.

[0119] S108: Based on the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix, obtain a reduced-order model corresponding to the state-space model.

[0120] In this step, after obtaining the low-rank decomposition factor matrix corresponding to the controllable Gram matrix and the observable Gram matrix, the state space model of the power system can be reduced based on the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix to obtain a reduced-order model.

[0121] S110: In response to a system response prediction request, receiving an input vector to be predicted of the power system, and predicting a time domain response vector of the power system according to the input vector to be predicted and a reduced-order model.

[0122] In this step, after obtaining the reduced-order model of the power system, the reduced-order model can be used to predict the response of the power system. In the prediction process, the input vector to be predicted of the power system can be received, and the input vector to be predicted can be used as the input vector in the reduced-order model. , in order to predict the time domain response vector of the power system through the reduced-order model and realize response prediction.

[0123] In this embodiment, the computational complexity of the response prediction mainly comes from the solution of the Laguerre coefficient. Since the solution of the Laguerre coefficient is a simple iterative calculation, the computational complexity can be greatly reduced and the memory usage can be reduced. In this way, the efficiency of the predicted response can be improved. At the same time, the reduced-order model of the present application has a global error bound, which can improve the accuracy of the power system response prediction. It can be seen that the present application can take into account both good prediction accuracy and low computational complexity.

[0124] In one embodiment, the state-space model includes the system matrix , input matrix And the output matrix About the system matrix , input matrix And the output matrix For the related description, please refer to the above embodiments, which will not be repeated here.

[0125] In this embodiment, each first Laguerre coefficient may be calculated based on the following expression:

[0126]

[0127] In the formula, is the coefficient of the first Laguerre expansion corresponding to the state impulse response of the state-space model; is the coefficient of the i+1th Laguerre expansion corresponding to the state impulse response of the state-space model; for dimensional identity matrix, is the number of differential variables in the linear differential algebraic equations of the power system. The state space model is an equivalent model of the linear differential algebraic equations. is the system matrix, is the input matrix, is the predetermined number of Laguerre expansion terms.

[0128] In this embodiment, each second Laguerre coefficient can be calculated based on the following expression:

[0129]

[0130] In the formula, is the coefficient of the first Laguerre expansion corresponding to the state impulse response of the dual model; is the coefficient of the i+1th Laguerre expansion corresponding to the state impulse response of the dual model; for The transposed matrix of for The transposed matrix of is the output matrix.

[0131] Specifically, the state impulse response of the state space model is expanded by Laguerre polynomial approximation as follows: Then, we can substitute it into the integral equation In, we get:

[0132]

[0133] Using the integral recursive property of Laguerre polynomials, the above formula can be transformed into:

[0134]

[0135] The left and right sides of the equation Eliminate and you can get the first Laguerre coefficient for:

[0136]

[0137] In the formula, for The inverse matrix of .

[0138] Through the specific forms of the state space model and the dual model, it can be determined that the second Laguerre coefficient is actually the sum of the first Laguerre coefficient. Replace with , Replace with As a result, the second Laguerre coefficient for:

[0139]

[0140] Based on the first Laguerre coefficient and the second Laguerre coefficient The expression can be determined that the first Laguerre coefficient and the second Laguerre coefficient The solution of can be obtained by using the state equation and the recursive properties of Laguerre polynomials. In one example, the calculation steps of Laguerre coefficients can be as follows: Figure 2 shown.

[0141] In one embodiment, calculating a first low-rank decomposition factor matrix corresponding to the controllable Gram matrix according to each first Laguerre coefficient, and calculating a second low-rank decomposition factor matrix corresponding to the observable Gram matrix according to each second Laguerre coefficient, includes:

[0142] Step A1: Calculate the integral result of the product of Laguerre polynomials within a preset time interval, and construct an integral matrix according to the integral result;

[0143] Step A3: Perform Cholesky decomposition on the integral matrix to obtain the first lower triangular matrix;

[0144] Step A5: Obtain a first low-rank decomposition factor matrix according to the first lower triangular matrix and each first Laguerre coefficient;

[0145] Step A7: Obtain a second low-rank decomposition factor matrix according to the first lower triangular matrix and each second Laguerre coefficient.

[0146] In this embodiment, in the process of performing low-rank approximate decomposition on the controllable Gram matrix and the observable Gram matrix, firstly, a matrix with the integral of the product of Laguerre polynomials as elements and its Cholesky decomposition over a given time interval may be calculated.

[0147] Specifically, in a given time interval On the Laguerre polynomial and The integral of the product is:

[0148]

[0149] In the formula, . Defined by is the integral matrix of the elements , subscript Represents the row number of the element in the integral matrix, subscript Represents the number of columns of the element in the integral matrix. If the integral matrix H is positive definite, the Cholesky decomposition of the integral matrix H can be performed directly:

[0150]

[0151] In the formula, is the first lower triangular matrix.

[0152] If the integral matrix H is not positive definite, the damping factor can be introduced Correct the integral matrix. When correcting, let , so that the resulting matrix Positive definite, then the result matrix Perform Cholesky decomposition to obtain the first lower triangular matrix. is the l-dimensional identity matrix.

[0153] After obtaining the first lower triangular matrix, the Laguerre approximate expansion result and the first lower triangular matrix can be used to calculate the first low-rank decomposition factor matrix corresponding to the controllable Gram matrix and the second low-rank decomposition factor matrix corresponding to the observable Gram matrix respectively.

[0154] Furthermore, if we express it in matrix form and , we can get:

[0155]

[0156] in, for dimensional identity matrix, is the number of inputs to the power system; is the second low-rank decomposition factor matrix. for dimensional identity matrix, is the number of outputs of the power system.

[0157] Let the matrix , , combined with the integral matrix The specific form of the above formula can be further rewritten as:

[0158]

[0159] Where P is the controllable Gram matrix, for The transposed matrix of , Q is the observable Gram matrix, for The transposed matrix of . is the Kronecker product operator.

[0160] Since the integral matrix H has a Cholesky decomposition ,and , , it is not difficult to deduce:

[0161] ,

[0162]

[0163] in, , .

[0164] Then the low-rank decomposition form of the controllable Gram matrix P and the low-rank decomposition form of the observable Gram matrix Q can be:

[0165]

[0166] The first low-rank decomposition factor matrix R and the second low-rank decomposition factor matrix L satisfy:

[0167]

[0168] In the formula, is the first low-rank decomposition factor matrix; is a coefficient matrix composed of the first Laguerre coefficients; is the first lower triangular matrix; for dimensional identity matrix, is the number of inputs to the power system; is the second low-rank decomposition factor matrix; is a coefficient matrix composed of the second Laguerre coefficients; for dimensional identity matrix, is the number of outputs of the power system; is the Kronecker product operator.

[0169] In one embodiment, based on the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix, a reduced-order model corresponding to the state-space model is obtained, including:

[0170] Step B1: The matrix is ​​decomposed by SVD, and the left truncation matrix is ​​obtained according to the SVD decomposition result and the preset reduction order. , diagonal truncation matrix and the right intercept matrix ;in, is the first low-rank decomposition factor matrix, for The transposed matrix of is the second low-rank decomposition factor matrix;

[0171] Step B3: According to the left intercept matrix , diagonal truncation matrix and the right intercept matrix , calculate the left projection matrix and the right projection matrix respectively;

[0172] Step B5: Obtain a reduced-order model according to the left projection matrix and the right projection matrix.

[0173] In this embodiment, the state space model can be reduced in order based on the approximate balanced truncation algorithm (LRBT algorithm for short), and the reduced-order model can be obtained. In this way, when the reduced-order model is used to predict the response of the power system, the prediction accuracy can be further improved.

[0174] Specifically, after obtaining the first low-rank decomposition factor matrix and the second low-rank factorization matrix Afterwards, you can The matrix is ​​processed by SVD (Singular Value Decomposition) to obtain the left singular vector matrix , diagonal matrix and the right singular vector matrix , .in, for The transposed matrix of .

[0175] If the reduced order is r, the left singular vector matrix can be , diagonal matrix and the right singular vector matrix Intercept to obtain the left intercept matrix , diagonal truncation matrix and the right intercept matrix In one example, the left truncation matrix is the left singular vector matrix The first r columns of . Right intercept matrix is the first r columns of the right singular vector matrix, that is . Diagonal intercept matrix is a diagonal matrix The first r-order principal minor form of .

[0176] It can be understood that the reduction order r can be predetermined according to practical factors such as computational complexity requirements and computational accuracy requirements, and this document does not impose any specific restrictions on this.

[0177] Get the left intercept matrix , diagonal truncation matrix and the right intercept matrix After that, we can use the left intercept matrix , diagonal truncation matrix and the right intercept matrix , respectively calculate the left projection matrix and the right projection matrix , and based on this, the state space model is reduced to obtain a reduced-order model.

[0178] In one embodiment, the left projection matrix and the right projection matrix are obtained based on the following expressions:

[0179]

[0180] In the formula, is the left projection matrix; is the diagonal intercept matrix; for The transposed matrix of is the left intercept matrix; for The transposed matrix of is the second low-rank decomposition factor matrix; is the right projection matrix; is the first low-rank decomposition factor matrix; is the right intercept matrix.

[0181] In one embodiment, the state-space model includes the system matrix , input matrix , output matrix and the constant matrix About the system matrix , input matrix , output matrix and the constant matrix For relevant instructions, please refer to the description of the above embodiment, which will not be repeated here.

[0182] According to the left projection matrix and the right projection matrix, the reduced-order model is obtained, including:

[0183] Step B51: Obtain the reduced-order system matrix based on the following expressions: , the reduced-order input matrix and the reduced-order output matrix :

[0184]

[0185] In the formula, is the left projection matrix, is the right projection matrix;

[0186] Step B53: Construct a reduced-order model according to the reduced-order system matrix, the reduced-order input matrix, the reduced-order output matrix and the constant matrix.

[0187] In this embodiment, the reduced-order model can be constructed according to the left projection matrix and the right projection matrix Among them, the reduced-order model Can be:

[0188]

[0189] In the formula, for The first-order differential of is the generalized state vector of the reduced-order model, is the input vector, is the response vector of the power system output by the reduced-order model.

[0190] The following are the performance verification results of the power system time domain model reduction method based on Laguerre polynomials and approximate Gram matrices provided in an embodiment of the present application.

[0191] The LRBT reduced-order model is generated by the method provided in the present application, wherein the method provided in the present application can convert the original 13251-dimensional differential algebraic model into a state space model containing 1664 state variables and reduce the order to obtain the LRBT reduced-order model. In addition, the existing balanced truncation method is used to generate the BT reduced-order model. The LRBT reduced-order model and the BT reduced-order model are respectively used to test the model reduction benchmark example BIPS / 97 for small disturbance stability analysis of power systems.

[0192] For the preset time interval , when the input vector When, this application takes the Laguerre expansion coefficient ,make , and a 10th-order LRBT reduced-order model is obtained. Using the existing balanced truncation method, we get a BT reduced-order model. In the input vector In the case of , the response of the power system is predicted by the LRBT reduced-order model and the BT model respectively. The time domain response vectors output by the two models are as follows: Figure 3 As shown in the figure, the absolute error between the time domain response vectors of the two models and the actual response vector of the power system can be expressed as Figure 4 shown.

[0193] For the preset time interval , when the input vector When, this application takes the Laguerre expansion coefficient ,make , and a 20th-order LRBT reduced-order model is obtained. Using the existing balanced truncation method, we get a BT reduced-order model. In the input vector In the case of , the response of the power system is predicted by the LRBT reduced-order model and the BT model respectively. The time domain response vectors output by the two models are as follows: Figure 5 As shown in the figure, the absolute error between the time domain response vectors of the two models and the actual response vector of the power system can be expressed as Figure 6 shown.

[0194] In this test, the time consumption of using the method provided by this application to predict the response of the power system and using the existing balanced truncation method to predict the response of the power system can be compared as follows: Figure 7 As shown. Figure 3-7 It can be seen that both the present application and the existing balanced truncation method can well approximate the transient response of the power system. In addition, in the time domain, the method provided by the present application has a better approximation effect than the classical balanced truncation method, and can obtain a reduced-order system with a smaller order and higher accuracy. In terms of running time, compared with the classical balanced truncation method, the method provided by the present application has an improvement of about 98% in computing speed and has higher computational efficiency. It can be seen that the present application can take into account both high prediction accuracy and low computational complexity.

[0195] The following is a description of the power system time domain model reduction device based on Laguerre polynomials and approximate Gram matrices provided in an embodiment of the present application. The power system time domain model reduction device based on Laguerre polynomials and approximate Gram matrices described below and the power system time domain model reduction method based on Laguerre polynomials and approximate Gram matrices described above can be referenced to each other.

[0196] In one embodiment, Figure 8As shown, the present application provides a power system time domain model reduction device 200 based on Laguerre polynomials and approximate Gram matrices, comprising:

[0197] A state space model acquisition module 202 is used to acquire a state space model of the power system;

[0198] A Laguerre coefficient acquisition module 204, used to calculate each first Laguerre coefficient corresponding to the state impulse response of the state-space model and each second Laguerre coefficient corresponding to the state impulse response of the dual model by using a Laguerre polynomial approximation algorithm; wherein the dual model is dual to the state-space model;

[0199] A low-rank decomposition module 206, configured to calculate a first low-rank decomposition factor matrix corresponding to a controllable Gram matrix according to each of the first Laguerre coefficients, and to calculate a second low-rank decomposition factor matrix corresponding to an observable Gram matrix according to each of the second Laguerre coefficients;

[0200] A reduction module 208, configured to obtain a reduced-order model corresponding to the state-space model based on the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix;

[0201] The response prediction module 210 is used to receive the input vector to be predicted of the power system in response to the system response prediction request, and predict the time domain response vector of the power system according to the input vector to be predicted and the reduced-order model.

[0202] In one embodiment, the state space model includes a system matrix, an input matrix and an output matrix. The Laguerre coefficient acquisition module 204 of the present application includes:

[0203] The first coefficient calculation unit is used to calculate each of the first Laguerre coefficients based on the following expression:

[0204]

[0205] In the formula, is the coefficient of the first Laguerre expansion corresponding to the state impulse response of the state-space model; is the coefficient of the i+1th Laguerre expansion corresponding to the state impulse response of the state-space model; for dimensional identity matrix, is the number of differential variables in the linear differential algebraic equation of the power system, and the state space model is an equivalent model of the linear differential algebraic equation; is the system matrix, is the input matrix, is the predetermined number of Laguerre expansion terms;

[0206] The second coefficient calculation unit is used to calculate each of the second Laguerre coefficients based on the following expression:

[0207]

[0208] In the formula, is the coefficient of the first Laguerre expansion corresponding to the state impulse response of the dual model; is the coefficient of the i+1th Laguerre expansion corresponding to the state impulse response of the dual model; for The transposed matrix of for The transposed matrix of is the output matrix.

[0209] In one embodiment, the low-rank decomposition module 206 of the present application includes:

[0210] An integral matrix construction unit, used to calculate the integral result of the product of the Laguerre polynomial within a preset time interval, and to construct an integral matrix according to the integral result;

[0211] A Cholesky decomposition unit, used for performing Cholesky decomposition on the integral matrix to obtain a first lower triangular matrix;

[0212] A first low-rank decomposition unit, configured to obtain the first low-rank decomposition factor matrix according to the first lower triangular matrix and each of the first Laguerre coefficients;

[0213] The second low-rank decomposition unit is used to obtain the second low-rank decomposition factor matrix according to the first lower triangular matrix and each of the second Laguerre coefficients.

[0214] In one embodiment, the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix are obtained based on the following expressions:

[0215]

[0216] In the formula, is the first low-rank decomposition factor matrix; is a coefficient matrix composed of the first Laguerre coefficients; is the first lower triangular matrix; for dimensional identity matrix, is the number of inputs of the power system; is the second low-rank decomposition factor matrix; is a coefficient matrix composed of each of the second Laguerre coefficients; for dimensional identity matrix, is the number of outputs of the power system; is the Kronecker product operator.

[0217] In one embodiment, the order reduction module 208 of the present application includes:

[0218] SVD decomposition unit is used to The matrix is ​​decomposed by SVD, and the left truncation matrix is ​​obtained according to the SVD decomposition result and the preset reduction order. , diagonal truncation matrix and the right intercept matrix ;in, is the first low-rank decomposition factor matrix, for The transposed matrix of is the second low-rank decomposition factor matrix;

[0219] A projection matrix calculation unit is used to calculate the left interception matrix according to the left interception matrix. , the diagonal intercept matrix and the right intercept matrix , calculate the left projection matrix and the right projection matrix respectively;

[0220] The order reduction unit is used to obtain the reduced-order model according to the left projection matrix and the right projection matrix.

[0221] In one embodiment, the projection matrix calculation unit of the present application is used to calculate the left projection matrix and the right projection matrix respectively based on the following expressions:

[0222]

[0223] In the formula, is the left projection matrix; is the diagonal intercept matrix; for The transposed matrix of is the left intercept matrix; for The transposed matrix of is the second low-rank decomposition factor matrix; is the right projection matrix; is the first low-rank decomposition factor matrix; is the right truncation matrix.

[0224] In one embodiment, the state-space model includes the system matrix , input matrix , output matrix and the constant matrix The reduced-order units of this application include:

[0225] The reduced-order matrix acquisition unit is used to obtain the reduced-order system matrix based on the following expressions , the reduced-order input matrix and the reduced-order output matrix :

[0226]

[0227] In the formula, is the left projection matrix, is the right projection matrix;

[0228] A reduced-order model construction unit is used to construct the reduced-order model according to the reduced-order system matrix, the reduced-order input matrix, the reduced-order output matrix and the constant matrix.

[0229] In one embodiment, the present application also provides a storage medium storing computer-readable instructions. When the computer-readable instructions are executed by one or more processors, the one or more processors execute the steps of the power system time domain model reduction method based on Laguerre polynomials and approximate Gram matrices in any embodiment.

[0230] In one embodiment, the present application also provides a computer device having computer-readable instructions stored therein. When the computer-readable instructions are executed by one or more processors, the one or more processors execute the steps of the power system time domain model reduction method based on Laguerre polynomials and approximate Gram matrices as in any embodiment.

[0231] Indicatively, Fig. 9 The internal structure diagram of a computer device provided in an embodiment of the present application is shown in FIG. 1 . In one example, the computer device may be a server. Fig. 9The computer device 900 includes a processing component 902, which further includes one or more processors, and a memory resource represented by a memory 901, for storing instructions executable by the processing component 902, such as an application. The application stored in the memory 901 may include one or more modules, each corresponding to a set of instructions. In addition, the processing component 902 is configured to execute instructions to perform the steps of the power system time domain model reduction method based on Laguerre polynomials and approximate Gram matrices described in any of the above embodiments.

[0232] The computer device 900 may further include a power supply component 903 configured to perform power management of the computer device 900, a wired or wireless network interface 904 configured to connect the computer device 900 to a network, and an input / output (I / O) interface 905. The computer device 900 may operate based on an operating system stored in the memory 901, such as Windows Server TM, Mac OS X TM, Unix TM, Linux TM, Free BSD TM, or the like.

[0233] Those skilled in the art will understand that the internal structure of the computer device shown in the present application is merely a block diagram of a partial structure related to the scheme of the present application, and does not constitute a limitation on the computer device to which the scheme of the present application is applied. The specific computer device may include more or fewer components than those shown in the figure, or combine certain components, or have a different arrangement of components.

[0234] Finally, it should be noted that, in this article, relational terms such as first and second, etc. are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply that there is any such actual relationship or order between these entities or operations. Moreover, the term "include", "comprise" or any other variant thereof is intended to cover non-exclusive inclusion, so that the process, method, article or equipment including a series of elements not only includes those elements, but also includes other elements not clearly listed, or also includes elements inherent to such process, method, article or equipment. In the absence of more restrictions, the elements defined by the sentence "including one..." do not exclude the existence of other identical elements in the process, method, article or equipment including the elements. Herein, "one", "one", "said", "the" and "it" may also include plural forms, unless the context clearly indicates another way. A plurality refers to at least two cases, such as 2, 3, 5 or 8, etc. "And / or" includes any and all combinations of the relevant listed items.

[0235] The various embodiments in this specification are described in a progressive manner, and each embodiment focuses on the differences from other embodiments. The various embodiments can be combined as needed, and the same or similar parts can refer to each other.

[0236] The above description of the disclosed embodiments enables those skilled in the art to implement or use the present application. Various modifications to these embodiments will be apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the present application. Therefore, the present application will not be limited to the embodiments shown herein, but will conform to the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for reducing the order of a power system time domain model based on Laguerre polynomials and approximate Gram matrices, characterized in that: include: Obtaining a state space model of the power system; Using a Laguerre polynomial approximation algorithm, respectively calculating each first Laguerre coefficient corresponding to the state impulse response of the state-space model and each second Laguerre coefficient corresponding to the state impulse response of the dual model; wherein the dual model is dual to the state-space model; Calculate a first low-rank decomposition factor matrix corresponding to a controllable Gram matrix according to each of the first Laguerre coefficients, and calculate a second low-rank decomposition factor matrix corresponding to an observable Gram matrix according to each of the second Laguerre coefficients; Based on the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix, obtaining a reduced-order model corresponding to the state-space model; In response to a system response prediction request, an input vector to be predicted of the power system is received, and a time domain response vector of the power system is predicted according to the input vector to be predicted and the reduced-order model.

2. The method according to claim 1, characterized in that The state space model includes a system matrix, an input matrix and an output matrix; The Laguerre polynomial approximation algorithm is used to calculate the first Laguerre coefficients corresponding to the state impulse response of the state space model and the second Laguerre coefficients corresponding to the state impulse response of the dual model, including: The first Laguerre coefficients are calculated based on the following expressions: In the formula, is the coefficient of the first Laguerre expansion corresponding to the state impulse response of the state-space model; is the coefficient of the i+1th Laguerre expansion corresponding to the state impulse response of the state-space model; for dimensional identity matrix, is the number of differential variables in the linear differential algebraic equation of the power system, and the state space model is an equivalent model of the linear differential algebraic equation; is the system matrix, is the input matrix, is the predetermined number of Laguerre expansion terms; The second Laguerre coefficients are calculated based on the following expressions: In the formula, is the coefficient of the first Laguerre expansion corresponding to the state impulse response of the dual model; is the coefficient of the i+1th Laguerre expansion corresponding to the state impulse response of the dual model; for The transposed matrix of for The transposed matrix of is the output matrix.

3. The method according to claim 1, characterized in that The step of calculating a first low-rank decomposition factor matrix corresponding to a controllable Gram matrix according to each of the first Laguerre coefficients, and calculating a second low-rank decomposition factor matrix corresponding to an observable Gram matrix according to each of the second Laguerre coefficients, comprises: Calculating the integral result of the product of the Laguerre polynomial within a preset time interval, and constructing an integral matrix according to the integral result; Performing Cholesky decomposition on the integral matrix to obtain a first lower triangular matrix; Obtaining the first low-rank decomposition factor matrix according to the first lower triangular matrix and each of the first Laguerre coefficients; The second low-rank decomposition factor matrix is ​​obtained according to the first lower triangular matrix and each of the second Laguerre coefficients.

4. The method according to claim 3, characterized in that The first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix are obtained based on the following expressions: In the formula, is the first low-rank decomposition factor matrix; is a coefficient matrix composed of the first Laguerre coefficients; is the first lower triangular matrix; for dimensional identity matrix, is the number of inputs of the power system; is the second low-rank decomposition factor matrix; is a coefficient matrix composed of each of the second Laguerre coefficients; for dimensional identity matrix, is the number of outputs of the power system; is the Kronecker product operator.

5. The method according to any one of claims 1 to 4, characterized in that: The step of obtaining a reduced-order model corresponding to the state-space model based on the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix includes: right The matrix is ​​decomposed by SVD, and the left truncation matrix is ​​obtained according to the SVD decomposition result and the preset reduction order. , diagonal truncation matrix and the right intercept matrix ;in, is the first low-rank decomposition factor matrix, for The transposed matrix of is the second low-rank decomposition factor matrix; According to the left interception matrix , the diagonal intercept matrix and the right intercept matrix , calculate the left projection matrix and the right projection matrix respectively; The reduced-order model is obtained according to the left projection matrix and the right projection matrix.

6. The method according to claim 5, characterized in that The left projection matrix and the right projection matrix are obtained based on the following expressions: In the formula, is the left projection matrix; is the diagonal intercept matrix; for The transposed matrix of is the left intercept matrix; for The transposed matrix of is the second low-rank decomposition factor matrix; is the right projection matrix; is the first low-rank decomposition factor matrix; is the right truncation matrix.

7. The method according to claim 5, characterized in that The state-space model includes the system matrix , input matrix , output matrix and the constant matrix ; The step of obtaining the reduced-order model according to the left projection matrix and the right projection matrix comprises: The reduced-order system matrices are obtained based on the following expressions: , the reduced-order input matrix and the reduced-order output matrix : In the formula, is the left projection matrix, is the right projection matrix; The reduced-order model is constructed according to the reduced-order system matrix, the reduced-order input matrix, the reduced-order output matrix and the constant matrix.

8. A device for reducing the order of a power system time domain model based on Laguerre polynomials and approximate Gram matrices, characterized in that: include: A state space model acquisition module, used for acquiring a state space model of a power system; A Laguerre coefficient acquisition module, used to calculate each first Laguerre coefficient corresponding to the state impulse response of the state space model and each second Laguerre coefficient corresponding to the state impulse response of the dual model by using a Laguerre polynomial approximation algorithm; wherein the dual model is dual to the state space model; A low-rank decomposition module, used to calculate a first low-rank decomposition factor matrix corresponding to a controllable Gram matrix according to each of the first Laguerre coefficients, and to calculate a second low-rank decomposition factor matrix corresponding to an observable Gram matrix according to each of the second Laguerre coefficients; A reduction module, used for obtaining a reduced-order model corresponding to the state-space model based on the first low-rank decomposition factor matrix and the second low-rank decomposition factor matrix; The response prediction module is used to receive the input vector to be predicted of the power system in response to the system response prediction request, and predict the time domain response vector of the power system according to the input vector to be predicted and the reduced-order model.

9. A storage medium, characterized in that: The storage medium stores computer-readable instructions, and when the computer-readable instructions are executed by one or more processors, the one or more processors execute the steps of the power system time domain model reduction method based on Laguerre polynomials and approximate Gram matrices as described in any one of claims 1 to 7.

10. A computer device, characterized in that: include: one or more processors, and memory; The memory stores computer-readable instructions, and when the computer-readable instructions are executed by the one or more processors, the steps of the power system time domain model reduction method based on Laguerre polynomials and approximate Gram matrices as described in any one of claims 1 to 7 are performed.

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