CN112165085A - Time-lag power system efficient characteristic value analysis method and system based on PSOD - Google Patents

Time-lag power system efficient characteristic value analysis method and system based on PSOD Download PDF

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CN112165085A
CN112165085A CN202010759221.2A CN202010759221A CN112165085A CN 112165085 A CN112165085 A CN 112165085A CN 202010759221 A CN202010759221 A CN 202010759221A CN 112165085 A CN112165085 A CN 112165085A
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power system
psod
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CN112165085B (en
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叶华
张思聪
刘玉田
李常刚
王亮
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Shandong University
State Grid Shandong Electric Power Co Ltd
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State Grid Shandong Electric Power Co Ltd
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for ac mains or ac distribution networks
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F7/00—Methods or arrangements for processing data by operating upon the order or content of the data handled
    • G06F7/38—Methods or arrangements for performing computations using exclusively denominational number representation, e.g. using binary, ternary, decimal representation
    • G06F7/48—Methods or arrangements for performing computations using exclusively denominational number representation, e.g. using binary, ternary, decimal representation using non-contact-making devices, e.g. tube, solid state device; using unspecified devices
    • G06F7/544—Methods or arrangements for performing computations using exclusively denominational number representation, e.g. using binary, ternary, decimal representation using non-contact-making devices, e.g. tube, solid state device; using unspecified devices for evaluating functions by calculation
    • G06F7/552—Powers or roots, e.g. Pythagorean sums
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
    • G06Q50/00—Systems or methods specially adapted for specific business sectors, e.g. utilities or tourism
    • G06Q50/06—Electricity, gas or water supply
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for ac mains or ac distribution networks
    • H02J3/24—Arrangements for preventing or reducing oscillations of power in networks
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J2203/00—Indexing scheme relating to details of circuit arrangements for AC mains or AC distribution networks
    • H02J2203/20—Simulating, e g planning, reliability check, modelling or computer assisted design [CAD]
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y04—INFORMATION OR COMMUNICATION TECHNOLOGIES HAVING AN IMPACT ON OTHER TECHNOLOGY AREAS
    • Y04S—SYSTEMS INTEGRATING TECHNOLOGIES RELATED TO POWER NETWORK OPERATION, COMMUNICATION OR INFORMATION TECHNOLOGIES FOR IMPROVING THE ELECTRICAL POWER GENERATION, TRANSMISSION, DISTRIBUTION, MANAGEMENT OR USAGE, i.e. SMART GRIDS
    • Y04S10/00—Systems supporting electrical power generation, transmission or distribution
    • Y04S10/50—Systems or methods supporting the power network operation or management, involving a certain degree of interaction with the load-side end user applications

Abstract

The utility model provides a PSOD-based time-lag power system efficient eigenvalue analysis method and system, which utilizes a PSOD method to obtain a discretization matrix of a partial solver of a time-lag power system according to the acquired running state data of the time-lag power system; calculating and resolving a first eigenvalue of a preset number of the sub-part discretization matrixes by using a subspace method; the first characteristic value is subjected to spectrum mapping, anti-rotation amplification and Newton verification in sequence to obtain a characteristic value of the time-lag power system; in the subspace method, a matrix to be inverted in a discretization matrix of a partial solution operator is expressed in a Schur complement form, an expression of the product of the inverse of the matrix and a vector is obtained by using the Schur complement form, and the product of the inverse of the matrix and the vector is solved by using an inverse power method; the method avoids iterative computation of the matrix inverse-vector product by using an iterative method, can obtain the solution of the matrix inverse-vector product only by using an inverse power method once, and obviously reduces the computation amount of the matrix inverse-vector product.

Description

Time-lag power system efficient characteristic value analysis method and system based on PSOD
Technical Field
The disclosure relates to the technical field of time-lag power systems, in particular to a PSOD-based time-lag power system efficient characteristic value analysis method and system.
Background
The statements in this section merely provide background information related to the present disclosure and may not necessarily constitute prior art.
In order to solve the problem of geographical unbalanced distribution of the energy center and the load center, modern power systems are gradually developed into large-scale interconnected power systems. Although the inter-regional interconnected power grid can better allocate resources and improve the economy and reliability of the operation of the power grid, the low-frequency oscillation generated in the inter-regional interconnected power grid makes the safe and stable operation of the power grid challenging. The conventional Power System Stabilizer (PSS) can only be used to solve the problem of local oscillation of the System, and is difficult to be used to suppress the interval oscillation.
The appearance of Wide-Area Measurement System (WAMS) provides a new solution for suppressing the interval oscillation of large-scale interconnected power systems. The interconnected power grid low-frequency oscillation control based on the wide-area information provided by the WAMS can obtain better damping control performance by introducing the wide-area feedback signal which effectively reflects the inter-area oscillation mode of the interconnected power system, provides a better control means for inhibiting inter-area low-frequency oscillation in the interconnected power grid, and thus improves the power transmission capacity of the system, and has good and wide application prospect.
Wide area signals are transmitted and processed in a WAMS communication network composed of different communication media (such as optical fibers, telephone lines, digital microwaves, satellites, and the like), and communication delay varying from tens of milliseconds to hundreds of milliseconds exists. Communication delay is an important cause of system control law failure, deterioration of operating conditions, and system instability. Therefore, when the closed-loop control of the power system is performed using the wide-area measurement information, the influence of the communication delay must be taken into consideration.
In modern power systems, electromechanical oscillation is a major concern for small disturbance stabilization. Currently, researchers have proposed a variety of electromechanical oscillation mode calculation methods for power systems that take into account time lag. The Chinese invention patent is an electromechanical oscillation mode calculation method of a time-lag power system based on an SOD-PS-R algorithm. Researchers have proposed a time-lag power system characteristic value calculation method based on an SOD-PS-R (Solution Operator discretization-Pseudo Spectrum and Rotation, resolving sub-Pseudo-Spectrum discretization and Rotation) algorithm; researchers also propose a time-lag power system characteristic value calculation method based on an SOD-IRK (SOD Methods with Implicit Runge-Kutta, solver-Runge-Kutta discretization) algorithm. The characteristic value calculation method of the time-lag power system can obtain all characteristic values of which the system damping ratio is smaller than a given value through one-time calculation; some researchers provide a time-lag power system characteristic value calculation method based on a low-order EIGD (Explicit infinite Generator Discretization), wherein partial Discretization of state variables can be applied to a solution subclass Discretization method, so that a time-lag power system characteristic value calculation method based on partial Discretization of a solver, namely a PSOD-IRK and a PSOD-PS-R method, can be obtained, the dimension of a Discretization matrix formed by the SOD-IRK and the SOD-PS-R method when calculating an electromechanical oscillation mode of a time-lag power system can be reduced, and the calculation efficiency of the method is improved.
The inventor of the present disclosure finds that, in the time-lag power system electromechanical oscillation mode calculation method based on solver discretization, an implicit Arnoldi restart method is used for iteratively calculating a characteristic value of a maximum module value of a discretization matrix of a system solver formed by the method, and the method mainly operates on a product of a discretization matrix in a Krylov sub-vector forming process and a Krylov sub-vector, but since a first block row of the solver discretization matrix contains a matrix inverse, an iterative method is usually required to calculate the matrix inverse-vector product, so that the calculation amount is large, and the calculation efficiency is low.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a PSOD-based time-lag power system efficient eigenvalue analysis method and a PSOD-based time-lag power system efficient eigenvalue analysis system, which avoid iterative computation of matrix inverse-vector products by using an iteration method, obtain solutions of the matrix inverse-vector products only by using an inverse power method once, reduce the computation amount of the matrix inverse-vector products, and remarkably improve the computation efficiency of the matrix inverse-vector products in the PSOD-based time-lag power system eigenvalue computation method.
In order to achieve the purpose, the following technical scheme is adopted in the disclosure:
the first aspect of the disclosure provides a time lag power system efficient eigenvalue analysis method based on a PSOD.
A time-lag power system efficient characteristic value analysis method based on PSOD comprises the following steps:
acquiring running state data of a time-lag power system;
obtaining a discretization matrix of a partial solver of the time-lag power system by using a PSOD method;
calculating and resolving a first eigenvalue of a preset number of the sub-part discretization matrixes by using a subspace method;
the first characteristic value is subjected to spectrum mapping, anti-rotation amplification and Newton verification in sequence to obtain a characteristic value of the time-lag power system;
in the subspace method, a matrix to be inverted in a discretization matrix of a partial solution operator is expressed in a Schur complement form, an expression of a product of the matrix inversion and a vector is obtained by using the Schur complement form, and the product of the matrix inversion and the vector is solved by using an inverse power method.
A second aspect of the present disclosure provides a PSOD-based time lag power system efficient eigenvalue analysis system.
A time-lapse power system efficient eigenvalue analysis system based on PSOD comprises:
a data acquisition module configured to: acquiring running state data of a time-lag power system;
a pre-processing module configured to: obtaining a discretization matrix of a partial solver of the time-lag power system by using a PSOD method;
a processing module configured to: calculating and resolving a first eigenvalue of a preset number of the sub-part discretization matrixes by using a subspace method; in the subspace method, a matrix to be inverted in a discretization matrix of a partial solution operator is represented in a Schur complement form, an expression of a product of the matrix inversion and a vector is obtained by using the Schur complement form, and the product of the matrix inversion and the vector is solved by using an inverse power method;
a eigenvalue analysis module configured to: and sequentially carrying out spectrum mapping, derotation amplification and Newton verification on the first characteristic value to obtain a characteristic value of the time-lag power system.
A third aspect of the present disclosure provides a medium having stored thereon a program that, when executed by a processor, implements the steps in the PSOD-based time lapse power system efficient eigenvalue analysis method according to the first aspect of the present disclosure.
A fourth aspect of the present disclosure provides an electronic device comprising a memory, a processor and a program stored on the memory and executable on the processor, the processor implementing the steps of the PSOD-based skew power system efficient eigenvalue analysis method according to the first aspect of the present disclosure when executing the program.
Compared with the prior art, the beneficial effect of this disclosure is:
1. according to the method, the system, the medium and the electronic equipment, the matrix to be inverted in the time-lag power system analysis method based on the discretization of the solver is expressed in a Schur complement form by utilizing the system state matrix, the augmented state matrix expression of the time-lag state matrix and three properties of Kronecker product, then a solution of matrix inverse-vector product can be calculated only by one-time inverse power method, and the efficiency of matrix inverse-vector product operation in the time-lag power system characteristic value calculation method based on PSOD is remarkably improved.
2. The method, the system, the medium and the electronic equipment avoid the phenomenon that the existing time-lag power system analysis method based on the discretization of the solver calculates the matrix inverse-vector product by using an iteration method, and obviously improve the calculation efficiency of the method; the method has good applicability and can be applied to time-lag power system characteristic value calculation methods based on partial solver discretization, including but not limited to PSOD-IRK and PSOD-PS-R methods.
Drawings
Fig. 1 is a schematic flow chart of a PSOD-based time lag power system efficient eigenvalue analysis method provided in embodiment 1 of the present disclosure.
Detailed Description
It should be noted that the following detailed description is exemplary and is intended to provide further explanation of the disclosure. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this disclosure belongs.
It is noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of example embodiments according to the present disclosure. As used herein, the singular forms "a", "an" and "the" are intended to include the plural forms as well, and it should be understood that when the terms "comprises" and/or "comprising" are used in this specification, they specify the presence of stated features, steps, operations, devices, components, and/or combinations thereof, unless the context clearly indicates otherwise.
The embodiments and features of the embodiments in the present application may be combined with each other without conflict.
Example 1:
as shown in fig. 1, an embodiment 1 of the present disclosure provides a PSOD-based time lag power system efficient eigenvalue analysis method, including the following steps:
step (1): obtaining a discretization matrix of a partial solver of the time-lag power system by using a PSOD method;
step (2): expanding a system state matrix and a time-lag state matrix in a matrix to be inverted in the discretization matrix of the partial solver by using an augmented state matrix expression of the system state matrix and the time-lag state matrix;
and (3): further representing the matrix to be inverted into a Schur complement form by utilizing the distribution rate, the mixed product and the inverse property of the Kronecker product;
and (4): calculating and resolving set characteristic values with the largest sub-part discretization matrix modulus value by using a subspace method, wherein the product of the inverse of a matrix to be inverted and a vector is expressed as an augmentation form, and the calculation is performed by using an inverse power method;
and (5): and converting the eigenvalue of the solved sub-part discretization matrix into the eigenvalue of the time-lag power system through the mapping relation between the eigenvalue of the operator and the eigenvalue of the time-lag power system, and obtaining the accurate eigenvalue of the time-lag power system through inverse rotation amplification and Newton verification.
In the step (1), the solver discretization matrix of the time-lag power system obtained by the PSOD method includes, but is not limited to, a partial solver discretization matrix formed by applying the PSOD-IRK method and the PSOD-PS-R method.
In the step (1), a time-lag power system stability judging method based on PSOD-IRK is adopted to generate a discretization matrix
Figure BDA0002612586990000061
Can be expressed as:
Figure BDA0002612586990000062
in the formula (1), P' is P/alpha, and P is a time lag interval [ -tau ]max,0]Number of divided sub-intervals, τmaxAlpha is a magnification factor parameter in the rotation-magnification preprocessing and is usually 2 or 3; s is the stage number of the IRK method; n is2The number of time-lag related state variables in the discretization method of the partial solution operator is n1And the number n of the system state variables satisfies n ═ n1+n2(ii) a In the second row of blocks, the first block,
Figure BDA0002612586990000063
is (P' -2) sn2A dimension unit matrix;
Figure BDA0002612586990000064
is (P' -2) sn2X sn-dimensional zero matrix; in the first block row
Figure BDA0002612586990000065
And second block row
Figure BDA0002612586990000066
Can be expressed in the following form:
Figure BDA0002612586990000067
Figure BDA0002612586990000071
Figure BDA0002612586990000072
Figure BDA0002612586990000073
Figure BDA0002612586990000074
in the formula (I), the compound is shown in the specification,
Figure BDA0002612586990000075
representing a Kronecker product operation;
Figure BDA0002612586990000076
coefficient matrix of Butcher table for s-level IRK method.
In the formula (2), IsnIs sn-dimensional unit matrix;
Figure BDA0002612586990000077
theta is the rotation of the coordinate axis in the rotation-magnification preprocessingAn angle, determined by a given damping ratio bound ζ, and satisfying θ ═ arcsin ζ;
Figure BDA0002612586990000078
is a state matrix of the system and is an n-dimensional dense matrix;
Figure BDA0002612586990000079
is a coefficient matrix A and a matrix of a Butcher table
Figure BDA00026125869900000710
A correlated constant coefficient matrix.
In the formula (3), 0sn×(s-1)nIs sn x (s-1) n-dimensional zero matrix; 1sIs an s-dimensional column vector with elements all being 1;
Figure BDA00026125869900000711
Figure BDA00026125869900000712
is a time lag constant taui(i is 1,2, …, m) related Lagrange interpolation polynomial coefficient matrixes, wherein m is the number of time-lag constants;
Figure BDA00026125869900000713
Isis an s-dimensional identity matrix;
Figure BDA00026125869900000714
Figure BDA00026125869900000715
time lag state matrix for a system
Figure BDA00026125869900000716
A matrix formed of columns corresponding to the time-lag related state variables;
Figure BDA00026125869900000717
is a coefficient matrix of a Butcher table
Figure BDA00026125869900000718
Figure BDA00026125869900000719
Matrix array
Figure BDA00026125869900000720
A correlated constant coefficient matrix.
In the formula (4), the reaction mixture is,
Figure BDA00026125869900000721
Figure BDA00026125869900000722
is n1A dimension unit matrix;
Figure BDA00026125869900000723
is n2×n1A zero-dimensional matrix.
In the formula (5), the reaction mixture is,
Figure BDA00026125869900000724
Figure BDA00026125869900000725
is n1×n2A zero-dimensional matrix;
Figure BDA00026125869900000726
is n2A dimension unit matrix.
In the step (1), a time-lag power system electromechanical oscillation mode calculation method based on a PSOD-PS-R algorithm is adopted to generate a discretization matrix
Figure BDA00026125869900000727
Can be expressed as:
T′M,N＝Π′M+Π′M,N(INn-Σ′N)-1Σ′M,N (7)
in formula (7):
Figure BDA0002612586990000081
Figure BDA0002612586990000082
Figure BDA0002612586990000083
Figure BDA0002612586990000084
in the formula, M and N are given discretization parameters and are positive integers;
Figure BDA0002612586990000085
τmaxis the maximum time lag constant, alpha is the magnification factor parameter in the rotation-magnification pretreatment, and h is the given transfer step length;
Figure BDA0002612586990000086
and theta is the rotation angle of the coordinate axis in the rotation-magnification preprocessing.
In the formula (8), the reaction mixture is,
Figure BDA0002612586990000087
is n1A dimension unit matrix;
Figure BDA0002612586990000088
is a constant coefficient matrix related to the discretization parameter M, N and the transfer step h;
Figure BDA0002612586990000089
is n2A dimension unit matrix;
Figure BDA00026125869900000810
is (n + Q' Mn) related to discretization parameter M, N and transfer step h2) A sparse matrix of dimensional coefficients.
In the formula (9), the reaction mixture is,
Figure BDA00026125869900000811
represents matrix U'M,NThe first row of (a) is an N-dimensional row vector;
Figure BDA00026125869900000812
the M multiplied by N Lagrange interpolation polynomial coefficient integral matrix related to the discretization parameter N and the transfer step h;
Figure BDA00026125869900000813
is ((Q' -1) M +1) n2A zero matrix of x Nn dimensions;
Figure BDA00026125869900000814
the dimension determined by the discretization parameter N and the transfer step h is (N + Q' Mn)2) A constant coefficient matrix of xn.
In the formula (10), the compound represented by the formula (10),
Figure BDA00026125869900000815
is compared with a discretization parameter N, a transfer step h and a time lag constant taui(i ═ 1,2, …, m) of the associated N-dimensional lagrange interpolation polynomial coefficient integral matrix;
Figure BDA00026125869900000816
is a matrix of dimension Nn, the elements of which are
Figure BDA00026125869900000817
And system skew state matrix
Figure BDA0002612586990000091
And (6) determining.
In the formula (11), the reaction mixture is,
Figure BDA0002612586990000092
is an N-dimensional row vector with elements all being 1;
Figure BDA0002612586990000093
is a system state matrix
Figure BDA0002612586990000094
Wherein the time-lag independent state variables correspond to a matrix of columns,
Figure BDA0002612586990000095
is a system state matrix
Figure BDA0002612586990000096
A matrix formed by corresponding columns of the middle and time-lag state variables;
Figure BDA0002612586990000097
an N × (Q' M +1) -dimensional matrix with the first column all being 1 and the other elements all being 0;
Figure BDA0002612586990000098
is compared with a discretization parameter N, a transfer step h and a time lag constant taui(i 1,2, …, M) a matrix of related nx (Q' M +1) dimensional lagrange polynomial coefficients;
Figure BDA0002612586990000099
is Nnx (n + Q' Mn)2) Dimension matrix of elements of
Figure BDA00026125869900000910
And system skew state matrix
Figure BDA00026125869900000911
And (6) determining.
In the step (2), the system state matrix and the augmented state matrix expression of the time lag state matrix are as follows:
Figure BDA00026125869900000912
in the formula:
Figure BDA00026125869900000913
and
Figure BDA00026125869900000914
being highly sparse systemsA system augmentation state matrix, wherein l is the number of system algebraic variables; a. thei(i-0, 1, …, m) is an n-dimensional matrix, Bi(i-0, 1, …, m) is a matrix of dimension n × l, C0Is a l × n dimensional matrix, D0Is a matrix of dimension l.
In the step (2), the matrix to be inverted in the partial solver discretization matrix includes, but is not limited to, the partial solver discretization matrix formed by the PSOD-IRK method and the PSOD-PS-R method
Figure BDA00026125869900000915
And
Figure BDA00026125869900000916
in the step (2), an augmented state matrix expression of the system state matrix and the time-lag state matrix is utilized, namely, the formula (12) is substituted into the matrix RP′sAnd (I)Nn-Σ′N) The expansion can be obtained as follows:
Figure BDA00026125869900000917
Figure BDA0002612586990000101
of formula (II) to'0＝αA0e-jθ(i ═ 0,1, …, m) is an n-dimensional matrix; b'0＝αB0e-jθ(i ═ 0,1, …, m) is a matrix of dimension n × l.
In the step (3), the distribution rate, the mixing product and the inverse property of the Kronecker product are as follows:
Figure BDA0002612586990000102
Figure BDA0002612586990000103
Figure BDA0002612586990000104
in the formula, X, Y, Z, U, V and W are arbitrary matrices that can perform corresponding matrix operations.
In the step (3), three properties of the Kronecker product represented by the formulas (15) to (17) are substituted into the matrix to be inverted R in the discretization matrix formed by the PSOD-IRK method and the PSOD-PS-R method in the step (1)P′sAnd (I)Nn-Σ′N) They can be expressed as:
Figure BDA0002612586990000105
Figure BDA0002612586990000106
in the formula (18), the reaction mixture,
Figure BDA0002612586990000107
is the inverse of the s-dimensional unit matrix; i issIs an s-dimensional identity matrix;
Figure BDA0002612586990000108
is a matrix of the sn dimension,
Figure BDA0002612586990000109
is a sn x sl dimensional matrix,
Figure BDA0002612586990000111
is a matrix of sl x sn dimensions,
Figure BDA0002612586990000112
the four matrices are highly sparse matrices, namely sl dimensional matrices.
In the formula (19), the compound represented by the formula (I),
Figure BDA0002612586990000113
is the inverse moment of an N-dimensional unit matrixArraying; i isNIs an N-dimensional unit matrix;
Figure BDA0002612586990000114
is a matrix of the dimension of Nn,
Figure BDA0002612586990000115
is a matrix with dimensions of Nn multiplied by Nl,
Figure BDA0002612586990000116
is a matrix of dimension Nl multiplied by Nn,
Figure BDA0002612586990000117
the four matrices are highly sparse matrices, being Nl dimensional matrices.
The formula (18) is a matrix R to be inverted in the discretization matrix formed by the PSOD-IRK methodP′sThe Schur complement of (1); the formula (19) is a matrix to be inverted (I) in the discretization matrix formed by the PSOD-PS-R methodNn-Σ′N) Schur complement form (n).
In the step (4), for a large-scale time-lag power system, the dimensionality of the discretization matrix of the partial solver is huge. Therefore, it is necessary to calculate and solve the set number of eigenvalues μ ″ with the largest modulus of the sub-part discretization matrix using a subspace method, wherein the main operation is to iterate the matrix-vector product in the formation of Krylov subvectors.
Suppose that at the kth iteration, a partial solver discretization matrix T needs to be computedP′sAnd T'M,NRespectively with vector qk-1And pk-1Product of (i), i.e. qk＝TP′s·qk-1And pk＝T′M,N·pk-1The method comprises the following steps:
step (4-1): obtaining an expression of the product of the inverse of the matrix to be inverted and the vector;
step (4-2): expressing the matrix inverse-vector product in the step (4-1) as a form of an augmentation matrix;
step (4-3): the inverse-vector product of the matrix is calculated using the inverse power method.
Step (4-4): to obtain qkAnd pkThe solution of (1).
In the step (4-1), the matrix to be inverted R in the discretization matrix formed by the PSOD-IRK and PSOD-PS-R methods in the step (2) is determined by using the formula (18)P′sAnd (I)Nn-Σ′N) The product operations with the vectors are respectively expressed as
Figure BDA0002612586990000121
In the formula (20), the reaction mixture is,
Figure BDA0002612586990000122
are sn-dimensional column vectors;
Figure BDA0002612586990000123
Figure BDA0002612586990000124
are all Nn-dimensional column vectors.
In the step (4-2), the formula (20) may be expressed as an amplification matrix in the form of:
Figure BDA0002612586990000125
in the formula (21), the compound represented by the formula,
Figure BDA0002612586990000126
is an sl-dimensional column vector;
Figure BDA0002612586990000127
is an Nl dimension column vector; 0sl×1Is sl-dimensional zero vector; 0Nl×1Is an Nl-dimensional zero vector.
In the step (4-3), the following equation (21) is calculated by the inverse power method:
Figure BDA0002612586990000128
in the formula (22), lu is sparse trigonometric decomposition operation; is prepared byA left divide operation, where a left divide of one matrix by another is equivalent to the inverse of the matrix multiplied by another matrix;
Figure BDA0002612586990000129
is a pair AsPerforming sparse trigonometric decomposition to obtain a lower triangular matrix and an upper triangular matrix of which the dimensionalities are sn;
Figure BDA00026125869900001210
is a pair AsWhen sparse triangular decomposition is carried out, the dimensionalities are left-multiplication and right-multiplication permutation matrixes of sn;
Figure BDA00026125869900001211
is a sn × sl dimensional matrix;
Figure BDA00026125869900001212
is a pair of
Figure BDA00026125869900001213
Performing sparse trigonometric decomposition to obtain a lower triangular matrix and an upper triangular matrix with the dimensionalities of sl;
Figure BDA00026125869900001214
is a pair of
Figure BDA00026125869900001215
When sparse triangular decomposition is carried out, left-multiplication and right-multiplication permutation matrixes with the dimensionality being sl are used;
Figure BDA00026125869900001216
is a sn-dimensional column vector;
Figure BDA00026125869900001217
is an sl-dimensional column vector;
Figure BDA00026125869900001218
is a pair ANPerforming sparse trigonometric decomposition to obtain a lower triangular matrix and an upper triangular matrix, wherein the dimensionalities of the lower triangular matrix and the dimensionalities of the upper triangular matrix are Nn;
Figure BDA00026125869900001219
is a pair ANWhen sparse triangular decomposition is carried out, the dimensionalities are left-multiplication and right-multiplication permutation matrixes of sn;
Figure BDA00026125869900001220
is a matrix with dimensions of NnxNl;
Figure BDA0002612586990000131
is a pair of
Figure BDA0002612586990000132
Performing sparse trigonometric decomposition to obtain a lower triangular matrix and an upper triangular matrix, wherein the dimensionalities of the lower triangular matrix and the dimensionalities of the upper triangular matrix are Nl;
Figure BDA0002612586990000133
is a pair of
Figure BDA0002612586990000134
When sparse triangular decomposition is carried out, the dimensionalities are left-multiplication and right-multiplication permutation matrixes of Nl;
Figure BDA0002612586990000135
is an Nn-dimensional column vector;
Figure BDA0002612586990000136
is an Nl dimensional column vector.
The solution q of matrix inverse-vector product can be efficiently calculated by the stepsk(1: sn,1) and r, so that q can be obtainedkAnd pkIn the form:
Figure BDA0002612586990000137
in the step (5), after the mu' is obtained through calculation, the characteristic value lambda of the time-lag power system is obtained after spectrum mapping, reverse rotation amplification and Newton verification are sequentially carried out, and before the Newton verification, the approximate value of the lambda is obtained
Figure BDA0002612586990000138
Is calculated byThe formula is as follows:
Figure BDA0002612586990000139
can be checked by Newton's test
Figure BDA00026125869900001310
And obtaining an accurate characteristic value lambda of the time-lag power system.
In order to verify the effect of the method for calculating the high-efficiency characteristic value of the large-scale time-lag power system based on the PSOD, all analyses are performed in Matlab and on an Intel 2.8GHz 8GB RAM desktop computer on the basis of the Shandong power grid system.
The parameters of the Shandong power grid in a certain horizontal year are as follows: 114 synchronous generators, 516 buses, 936 transformer branches and transmission lines and 299 loads. The wide-area LQR is arranged on two units of the chat factory, the wide-area feedback signals are respectively the rotating speed difference and the power angle difference of one unit of the Weihai factory to the two units of the chat factory, and the control gains are both 40 and 0.125. The dimension of the system state variable is 1128, and the dimension of the system algebraic variable is 5765.
To verify the effect of the method proposed in this embodiment, first, the calculation efficiency of the PSOD-IRK method to which the method proposed in this embodiment is applied is compared with that of the original PSOD-IRK method. Table 1 shows the time for solving the matrix inverse-vector product and the time for calculating the eigenvalues of the time-lag system when the parameters are identical for both methods. Then, the calculation efficiency of the PSOD-PS-R method using the method proposed in this example was compared with that of the original PSOD-PS-R method. Table 2 shows the time for solving the matrix inverse-vector product and the time for calculating the eigenvalues of the time-lag system when the parameters are identical for both methods. The number of characteristic values calculated by the two methods is r-10, the rotation angle is theta-2.87 degrees, and the magnification is alpha-2. As can be easily found from tables 1 and 2, the calculation efficiency of the matrix inverse-vector product in the PSOD-IRK and PSOD-PS-R methods can be significantly improved by the method provided by this embodiment, and the time required for calculating the time lag system characteristic value is greatly reduced by the PSOD-IRK and PSOD-PS-R methods after the method provided by this embodiment is applied to efficiently calculate the matrix inverse-vector product.
Table 1:
Figure BDA0002612586990000141
table 2:
Figure BDA0002612586990000142
Figure BDA0002612586990000151
example 2:
the embodiment 2 of the present disclosure provides a time lag electric power system efficient eigenvalue analysis system based on a PSOD, including:
a data acquisition module configured to: acquiring running state data of a time-lag power system;
a pre-processing module configured to: obtaining a discretization matrix of a partial solver of the time-lag power system by using a PSOD method;
a processing module configured to: calculating and resolving a first eigenvalue of a preset number of the sub-part discretization matrixes by using a subspace method; in the subspace method, a matrix to be inverted in a discretization matrix of a partial solution operator is represented in a Schur complement form, an expression of a product of the matrix inversion and a vector is obtained by using the Schur complement form, and the product of the matrix inversion and the vector is solved by using an inverse power method;
a eigenvalue analysis module configured to: and sequentially carrying out spectrum mapping, derotation amplification and Newton verification on the first characteristic value to obtain a characteristic value of the time-lag power system.
The working method of the system is the same as the PSOD-based time lag power system efficient characteristic value analysis method provided in embodiment 1, and details are not repeated here.
Example 3:
the embodiment 3 of the present disclosure provides a medium, on which a program is stored, which when executed by a processor, implements the steps in the PSOD-based time lag power system efficient eigenvalue analysis method according to the embodiment 1 of the present disclosure, where the steps are:
step (1): obtaining a discretization matrix of a partial solver of the time-lag power system by using a PSOD method;
step (2): expanding a system state matrix and a time-lag state matrix in a matrix to be inverted in the discretization matrix of the partial solver by using an augmented state matrix expression of the system state matrix and the time-lag state matrix;
and (3): further representing the matrix to be inverted into a Schur complement form by utilizing the distribution rate, the mixed product and the inverse property of the Kronecker product;
and (4): calculating and resolving a characteristic value of a preset number with the largest sub-part discretization matrix modulus by using a subspace method, wherein the product of the inverse of a matrix to be inverted and a vector is expressed as an augmentation form, and the calculation is performed by using an inverse power method;
and (5): and converting the eigenvalue of the solved sub-part discretization matrix into the eigenvalue of the time-lag power system through the mapping relation between the eigenvalue of the operator and the eigenvalue of the time-lag power system, and obtaining the accurate eigenvalue of the time-lag power system through inverse rotation amplification and Newton verification.
The detailed steps are the same as those in the PSOD-based time lag power system efficient eigenvalue analysis method provided in embodiment 1, and are not described herein again.
Example 4:
the embodiment 4 of the present disclosure provides an electronic device, which includes a memory, a processor, and a program stored in the memory and executable on the processor, where the processor executes the program to implement the steps in the PSOD-based skew power system efficient eigenvalue analysis method according to embodiment 1 of the present disclosure, where the steps are:
step (1): obtaining a discretization matrix of a partial solver of the time-lag power system by using a PSOD method;
step (2): expanding a system state matrix and a time-lag state matrix in a matrix to be inverted in the discretization matrix of the partial solver by using an augmented state matrix expression of the system state matrix and the time-lag state matrix;
and (3): further representing the matrix to be inverted into a Schur complement form by utilizing the distribution rate, the mixed product and the inverse property of the Kronecker product;
and (4): calculating and resolving a characteristic value of a preset number with the largest sub-part discretization matrix modulus by using a subspace method, wherein the product of the inverse of a matrix to be inverted and a vector is expressed as an augmentation form, and the calculation is performed by using an inverse power method;
and (5): and converting the eigenvalue of the solved sub-part discretization matrix into the eigenvalue of the time-lag power system through the mapping relation between the eigenvalue of the operator and the eigenvalue of the time-lag power system, and obtaining the accurate eigenvalue of the time-lag power system through inverse rotation amplification and Newton verification.
As will be appreciated by one skilled in the art, embodiments of the present disclosure may be provided as a method, system, or computer program product. Accordingly, the present disclosure may take the form of a hardware embodiment, a software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present disclosure may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, optical storage, and the like) having computer-usable program code embodied therein.
The present disclosure is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the disclosure. It will be understood that each flow and/or block of the flow diagrams and/or block diagrams, and combinations of flows and/or blocks in the flow diagrams and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
It will be understood by those skilled in the art that all or part of the processes of the methods of the embodiments described above can be implemented by a computer program, which can be stored in a computer-readable storage medium, and when executed, can include the processes of the embodiments of the methods described above. The storage medium may be a magnetic disk, an optical disk, a Read-Only Memory (ROM), a Random Access Memory (RAM), or the like.
The above description is only a preferred embodiment of the present disclosure and is not intended to limit the present disclosure, and various modifications and changes may be made to the present disclosure by those skilled in the art. Any modification, equivalent replacement, improvement and the like made within the spirit and principle of the present disclosure should be included in the protection scope of the present disclosure.

Claims (10)

1. A time lag power system efficient characteristic value analysis method based on PSOD is characterized by comprising the following steps:
acquiring running state data of a time-lag power system;
obtaining a discretization matrix of a partial solver of the time-lag power system by using a PSOD method;
calculating and resolving a first eigenvalue of a preset number of the sub-part discretization matrixes by using a subspace method;
the first characteristic value is subjected to spectrum mapping, anti-rotation amplification and Newton verification in sequence to obtain a characteristic value of the time-lag power system;
in the subspace method, a matrix to be inverted in a discretization matrix of a partial solution operator is expressed in a Schur complement form, an expression of a product of the matrix inversion and a vector is obtained by using the Schur complement form, and the product of the matrix inversion and the vector is solved by using an inverse power method.
2. The PSOD-based time-lapse electric power system efficient eigenvalue analysis method of claim 1, wherein an augmented state matrix expression of a system state matrix and a time-lapse state matrix is utilized, and substituted into a matrix to be inverted in a partial solver discretization matrix to obtain an expansion.
3. The PSOD-based time-lapse power system efficient eigenvalue analysis method of claim 2 wherein the to-be-inverted matrix is represented in the form of Schur's complement using the distribution ratio, the mixed product and the inversion property of the Kronecker product.
4. The PSOD-based time-lapse power system efficient eigenvalue analysis method of claim 3, wherein the distribution ratio of Kronecker product, the mixture product and the inverse property formula are substituted into the expansion of the matrix to be inverted to obtain the Schur complement form of the matrix to be inverted.
5. The PSOD-based time-lag power system efficient eigenvalue analysis method of claim 1, wherein an inverse power method is used to solve a matrix inverse-vector product, specifically, an expression for obtaining the inverse-vector product of a matrix to be inverted is expressed in the form of an augmented matrix, and the inverse power method is used to calculate the product of the matrix and the vector.
6. The PSOD-based time-lag power system efficient eigenvalue analysis method as claimed in claim 1, wherein the first eigenvalue of the solved sub-part discretization matrix is converted into the eigenvalue of the time-lag power system through the mapping relation between the eigenvalue of the solver and the eigenvalue of the time-lag power system, and the final eigenvalue of the time-lag power system is obtained through anti-rotation amplification and Newton verification.
7. The PSOD-based time-lag power system efficient eigenvalue analysis method of claim 1, wherein before Newton's verification, the approximate value of the time-lag power system is calculated in the following way:
Figure FDA0002612586980000021
wherein alpha is a magnification factor parameter in the rotation-amplification pretreatment, h is a given transfer step length, theta is a rotation angle of a coordinate axis in the rotation-amplification pretreatment, and mu' is a first characteristic value.
8. A time-lapse power system efficient eigenvalue analysis system based on PSOD, comprising:
a data acquisition module configured to: acquiring running state data of a time-lag power system;
a pre-processing module configured to: obtaining a discretization matrix of a partial solver of the time-lag power system by using a PSOD method;
a processing module configured to: calculating and resolving a first eigenvalue of a preset number of the sub-part discretization matrixes by using a subspace method; in the subspace method, expressing a matrix to be inverted in a discretization matrix of a partial solver as a Schur complement form, obtaining an expression of the product of the inverse of the matrix and a vector by using the Schur complement form, and solving the product of the inverse of the matrix and the vector by using an inverse power method;
a eigenvalue analysis module configured to: and sequentially carrying out spectrum mapping, derotation amplification and Newton verification on the first characteristic value to obtain a characteristic value of the time-lag power system.
9. A medium having a program stored thereon, wherein the program, when executed by a processor, implements the steps in the PSOD-based time lapse power system efficient eigenvalue analysis method of any of claims 1-7.
10. An electronic device comprising a memory, a processor, and a program stored on the memory and executable on the processor, wherein the processor when executing the program implements the steps in the PSOD-based skew power system high efficiency eigenvalue analysis method of any of claims 1-7.
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