Reduced-order method for quickly calculating insulation state of converter transformer

Abstract

A reduced-order method for calculating an insulation state of a converter transformer, includes the steps of: S1, finite element analysis (FEA) of an electric quasi-static field; S2, reduced-order cal

Description

TECHNICAL FIELD The present invention relates to the technical field of the insulation assessment of transformers, and in particular to a reduced-order method for quickly calculating an insulation state of a converter transformer. BACKGROUND The insulation assessment of the converter transformer consumes a large amount of storage resources due to the complexity of the model and the high degree of freedom. To this end, the present invention provides a reduced-order method for quickly calculating an insulation state of a converter transformer, which obtains the time-domain results in a low-cost way. Finally, a converter transformer model is taken as an example, the reduced-order calculation of insulation assessment is performed by proper orthogonal decomposition (POD). SUMMARY In view of the deficiencies of the prior art, the present invention provides a reduced-order method for quickly calculating an insulation state of a converter transformer, which solves the problems mentioned above in the background. In order to achieve the above object, the present invention provides the following technical solutions. A reduced-order method for quickly calculating an insulation state of a converter transformer specifically includes the following steps: SI, finite element analysis (FEA) of an electric quasi-static field: a converter transformer operating at a high temperature and a strong electric field, a valve side of the converter transformer being subjected to a polarity inversion voltage during a polarity inversion test, and a grid side being grounded, belonging to the electric quasi-static field and being represented by the following equation: / d \ V • I oV<p + = 0 where <p is an electric potential; a is a conductivity; and £ is a permittivity; using a Galerkin finite element method after mesh parting is performed on a converter transformer model, to obtain a semi-discrete differential equation: Ks— + Kff(p = F where Ks is a permittivity stiffness matrix; Ka is a conductivity stiffness matrix; and F is a right end term of the second kind of boundary conditions; and obtaining the following semi-discrete scheme under the condition that the Fterm is zero since only the first kind of boundary conditions and axisymmetric boundary conditions exist, ^E~dt = $ using a backward Euler method to discretize a time for the above governing equation, + ^tKa)(pm+1 where m is a time step, namely, at a moment; S2, reduced-order calculation based on POD: summary of singular value decomposition (SVD): using a POD method for reducing an order number of electric potential information, and sampling correct electric potential information along time to obtain an electric potential matrix containing Nt moments, •Pi (ti) <ppwt) V2 (t2) <P2 (h) VN Pi = (ti) ¢2) <P3 (ti) (p^tNt) (tj Vns- (t2) <Pk-l (t3) Vns-1 • ■■ riNP (^Nt) Vns-I (^) L <Pns (t2) <Pns (t3) Vns • <Pns where t, is an / "’ moment, Nt moments are collected by the above sampling matrix, and Ns is the total number of nodes in the model; and cp^ is electric potential information of an ath node at a b& time point; using SVD to extract important features of the matrix, Pi being expressed in the following equation: Pi = USVT where U and V are standard orthogonal bases, and X is a diagonal matrix with singular values, matrix sizes of U, X and V being Ns*Ns, Ns* Nt, and Nt*Nt, respectively; arranging the singular values A, as important information for representing matrix features, in a descending order in S, part of the singular values reflecting all characteristics, and the matrix being approximately described by the first Nd singular values, and defining part of SVD, Pi ~ UXV1 where U and V are taken as standard orthogonal matrices, and sizes of U and PT are changed and reduced from Ns* Ns and Nt^Nt to Ns*Nd and Nd^Nt, I is a diagonal matrix containing Nd singular values A greater than a particular threshold, Nd«Nt, the singular values in the equation being arranged in a descending order as follows: / 1>zv&, and at this time, the matrix U corresponding to Z being a matrix with only Nd column vectors, called an eigenvector of P, pi 1 Z = where U and V are also eigenvectors of PTP and PPT, and a2 is an eigenvalue of PTP and PPT, physically, the eigenvalue A2 representing a generalized energy of a physical field captured by a corresponding eigenvector, and also representing a degree of the amount of information of a physical field represented in sample data contained in the eigenvector; S3, analysis of transient insulation characteristics based on POD performing SVD on the collected electric potential sample matrix Pi to obtain a standard orthogonal basis U containing Nd column vectors, at this time, an electric potential vector at a certain moment being represented by U and a reduced-order electric potential vector, that is, POD being used to reconstruct a node electric potential to obtain the following equation: = Ua Pd K-i L Vns J where, u = Un U21 U31 U12 U22 U32 "■ u1Nd ■" U1Nd ■" U1Nd , a = u(Ns-l}l u(Ns-i)2 ■ U^Ns_1^)Nd - uNsl UNs2 UNsNd - L Nd J where (p is an electric potential column vector at a tt moment and is composed of electric potential values of various nodes in the model, (p being represented as the multiplication of two matrices, U being an eigenvector obtained in a previous section, and a being a vector reduced by the electric potential vector; and S4, insulation assessment reduced-order based on POD: an online stage being generally a reduced-order calculation, that is, solving according to the reduced-order governing equation, during which eigenvectors and interpolation matrices provided by a sample space are used; and an off-line stage being generally a finite element calculation of a full-order model, providing selectable samples for online calculation through the results of the offline calculation, and being limited by a calculation speed of the full-order model, in which a model in the off-line stage generally only calculates part of the time step and provides usable parameters for online calculation through the calculated results. Advantageous Effects The present invention provides a reduced-order method for quickly calculating an insulation state of a converter transformer. Compared with the prior art, the present invention has the following advantageous effects. The reduced-order method for quickly calculating an insulation state of a converter transformer can calculate time domain results quickly. Finally, a converter transformer model is taken as an example, the reduced-order calculation of insulation assessment is performed by POD, avoiding the high degree of freedom in the insulation assessment of the converter transformer due to the complexity of the model at present, thus reducing the consumption of storage resources. BRIEF DESCRIPTION OF THE DRAWINGS FIG. lisa calculation flow of the present invention; FIG. 2 is a schematic diagram of POD of the present invention; and FIG. 3 is a schematic diagram of a computational example of the present invention. DETAILED DESCRIPTION Technical solutions in examples of the present invention are described clearly and completely in the following with reference to the attached drawings in the examples of the present invention. Obviously, all the described examples are only some, rather than all examples of the present invention. Based on the examples in the present invention, all other examples obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present invention. Referring to FIGS. 1-3, the present invention provides a technical solution: a reduced-order method for quickly calculating an insulation state of a converter transformer, specifically including the following steps. In SI, FEA of an electric quasi-static field: a valve side of a converter transformer is subjected to a polarity inversion voltage during a polarity inversion test, and a grid side is grounded, belonging to the electric quasi-static field and being represented by the following equation: V ■ + = 0 (1.1) where (p is an electric potential; a is a conductivity; and £ is a permittivity, and is less affected by a field strength and a temperature, so the coupling relationship is not considered; a Galerkin finite element method is used after mesh parting is performed on a converter transformer model, to obtain a semi-discrete differential equation: Ke^ + Kff(p = F (1.2) where Ks is a permittivity stiffness matrix; Ka is a conductivity stiffness matrix; and F is a right end term of the second kind of boundary conditions; and the following semi-discrete scheme is obtained under the condition that the F term is zero since only the first kind of boundary conditions and axisymmetric boundary conditions exist, Ks^ + Ka<p = 0 (1.3) a backward Euler method is used to discretize a time for the above governing equation, (^s + MKa)(pm+] = Ks(pm (1.4) where m is a time step, namely, at a moment. In S2, reduced-order calculation based on POD: summary of SVD: a POD method is used for reducing an order number of electric potential information, POD being a numerical method for reducing computational complexity, and correct electric potential information is sampled along time to obtain an electric potential matrix containing Nt moments, Pi = *P2 dd <p3 (t2) ^1 P2) <P2 (t2) <?3 (tl) dd dd ,ndm-d - d N t—d AN (p^Nt) (1.5) where is an zth mo (tl) (^2) (£3) Nt— 1) Nt) Vns-1 Vns-I Vns-1 - Vns-I Vns-1 (tl) (^2) (tg) Nt— 1) (t Nt) <PNs - ^Ns ^Ns ment, Nt moments are collected by the above sai npling matrix, and Ns is the total number of nodes in the model; and <Pa is electric potential information of an node at a 6th time point; SVD is used to extract important features of the matrix, Pi being represented by the following equation: Pi = USVT (1.6) where U and V are standard orthogonal bases, and Z is a diagonal matrix with singular values, matrix sizes of U, S and V being Ns^Ns, Ns* Nt, and Nt*Nt, respectively; Obviously, this decomposition does not reduce an order number of the decomposed matrix. A schematic diagram is shown in FIG. 2. However, the singular values 2, as important information for representing matrix features, are arranged in a descending order in X, part of the singular values reflecting all characteristics. For example, in the case where the singular value reduction is extremely fast, the first 10% or even 1% of the singular value accounts for more than 99% of the total singular value information. That is, the first Nd singular values can be used to approximately describe the matrix, and part of SVD is defined herein: Pi * USVT (1.7) where U and V are taken as standard orthogonal matrices, and sizes of U and VT are changed and reduced from Ns* Ns and Nt^Nt to Ns*Nd and Nd*Nt\ I is a diagonal matrix containing Nd singular values 2 greater than a particular threshold, Nd«Nt. As shown in Equation (1.7), the singular values in the equation are arranged in a descending order: Ar>A2>Nt>...>ANd. At this time, the matrix U corresponding to I is a matrix with only Nd column vectors, called an eigenvector of P. where U and V are also eigenvectors of PTP and PPT, and A2 is an eigenvalue of PTP and PPT, physically, the eigenvalue A2 representing a generalized energy of a physical field captured by a corresponding eigenvector, and also representing a degree of the amount of information of a physical field represented in sample data contained in the eigenvector. Therefore, in the standard orthogonal matrix U, a generalized energy captured by the POD orthogonal basis composed of the Nd eigenvectors is: Q(Nd) = ZidA2(1.9). The whole sample data actually contains Ns pieces of feature information, and the whole sample system contains all the generalized energies: 0( Ns) = A2(l .10). Therefore, the error of the system can be determined by the generalized energy captured by the eigenvector and the generalized energy existing in the whole system. The defined truncation error is as follows: _ 0(jvs)-0(wd) _ X &(Ns) Obviously, when POD orthonormal bases are selected, the more orthonormal bases selected, the more information contained and the less truncation error obtained. However, when the singular values are arranged in a descending order, a small part of eigenvectors can contain most information, so when the number of eigenvectors is selected, it is necessary to select an appropriate number with reference to the rate of singular value reduction. The calculation accuracy is ensured without increasing the calculation burden. In S3, analysis of transient insulation characteristics based on POD A transient insulation characteristic analysis equation is obtained, namely, Equation (1.4). In a previous section, the electric potential sample matrix Pi is collected and subjected to SVD to obtain the standard orthonormal basis U containing Nd column vectors. At this time, an electric potential vector at a certain moment is represented by U and a reduced-order electric potential vector, that is, POD is used to reconstruct a node electric potential to obtain the following equation: <p = r di) i dd dd ^3 = Ua (1.12) do ns-i ^dt) L Vns J where, U = Un U21 U31 U12 U22 ^32 ■" ulNd ■" ulNd ■" ulNd , a = (1.13) U(Ns-l)2 ” U(Ns-l)Nd ndd - uNsl UNs2 uNsNd - ^Nd 1 where (p is an electric potential column vector at a moment and is composed of electric potential values of various nodes in the model, (p is represented as the multiplication of two matrices, U is an eigenvector obtained in a previous section, and a is a vector reduced by the electric potential vector, having Nd elements. Obviously, a can represent electric potential information at this moment herein, so that a vector recording the electric potential information has been reduced from a Ns line to a Nd line. Equation (1.12) is substituted into Equation (1.4), and (p in the equation is replaced by a product of U and a, that is, a reduced-order vector of the electric potential vector is substituted to obtain the following equation: ^tKaUocm+i KEUam(l. 14) Obviously, in the above equation, although the electric potential vector is reduced from Ns to Nd, a permittivity stiffness matrix and a conductivity stiffness matrix are still Ns* Ns, and the involved calculation is still the solution between Ns rows, which has little contribution to the calculation speed. Therefore, for Equation (1.14), a left side of each term in the equation is multiplied by the matrix U\ and an order number of the governing equation is reduced from Ns to Nd, which is expressed as follows: UTKsUam+l + &tUTKaUam+1 = UTKsUam(l. 15) It is obvious that in the equation, the permittivity stiffness matrix KE and the conductivity stiffness matrix Kff have been transformed from a square matrix of Ns^Ns to a reduced square matrix UTKEU and iNK^U, a size of which is Nd^Nd. At the same time, as an electric potential vector to be solved a, it is also reduced to a Nd term. In this calculation, compared with Equation (1.4), an overall order number has been reduced from Ns to Nd, and it is obvious that the calculation difficulty is also significantly reduced. In S4, insulation assessment reduced-order based on POD: most reduced-order models (ROMs) generally involve two stages: online stage and off-line stage. The online stage is generally a reduced-order calculation, that is, solving according to the reduced-order governing equation, during which eigenvectors and interpolation matrices provided by a sample space are used; and the off-line stage is generally a finite element calculation of a fullorder model, provides selectable samples for online calculation through the results of the off-line calculation, and is limited by a calculation speed of the full-order model, in which a model in the off-line stage generally only calculates part of the time step and provides usable parameters for online calculation through the calculated results. The calculation flow of a ROM based on POD is shown in FIG. 1, in which "Start 1" is an off-line processing part. In the off-line part, it is necessary to calculate the first Nt time steps of a full-order finite element model, obtain Nt electric potential information samples, construct a matrix Pi, and obtain an eigenvector U through SVD. This part is used for order reduction of an electric potential vector. “Start 2” is an online processing part that selects an interpolation matrix with the same behavior to interpolate and approximate a stiffness matrix according to the condition of excitation change for solving. At the same time, what is not described in detail in this specification belongs to the prior art known to those skilled in the art. It is to be noted that in this specification, relational terms such as "first" and "second" 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 an actual relationship or order between these entities or operations. Moreover, the terms "including", "containing" or any other variations are intended to cover non-exclusive inclusion, so that a process, method, article or device including a series of elements includes not only those elements, but also other elements not explicitly listed or elements inherent to such a process, method, article or device. While examples of the present invention have been shown and described, it is understood by those skilled in the art that various changes, modifications, substitutions and alterations may be 9 made herein without departing from the principles and spirit of the present invention, the scope of which is defined by the appended claims and equivalents thereof.

Claims

1. A reduced-order method for quickly calculating an insulation state of a converter transformer, specifically comprising the following steps:SI, finite element analysis (FEA) of an electric quasi-static field:a converter transformer operating at a high temperature and a strong electric field, a valve side of the converter transformer being subjected to a polarity inversion voltage during a polarity inversion test, and a grid side being grounded, belonging to the electric quasi-static field and being represented by the following equation: / d \V ■ + — = 0where (p is an electric potential; o is a conductivity; and £ is a permittivity;using a Galerkin finite element method after mesh parting is performed on a converter transformer model, to obtain a semi-discrete differential equation:— + Ko(p = Fwhere KE is a permittivity stiffness matrix; Ka is a conductivity stiffness matrix; and F is a right end term of the second kind of boundary conditions; andobtaining the following semi-discrete scheme under the condition that the Fterm is zero since only the first kind of boundary conditions and axisymmetric boundary conditions exist,d(pKs + Ka(p = 0using a backward Euler method to discretize a time for the above governing equation, ( / f£ + At K(p KE(pmwhere m is a time step, namely, at a AtL moment;S2, reduced-order calculation based on proper orthogonal decomposition (POD):summary of singular value decomposition (SVD):using a POD method for reducing an order number of electric potential information, and sampling correct electric potential information along time to obtain an electric potential matrix containing Nt moments,r PP <Pi (t2) dP <Pi •• cp^Nt)' ^2 (t2) <P2 dP n •• cp^-^ (p^ Pi = dP n (t2) PP n ■■ (p^m-} <p^ mdP VNs-l (t2) <PNs-l (tp ns-i • ■■ ppp ppp .d^P L VNs (t2) ns ap ns • ■■ PPN pppwhere p is an fl1 moment, Nt moments are collected by the above sampling matrix, and Ns is the total number of nodes in the model; and (p^ is electric potential information of an node at a bth time point;using S VD to extract important features of the matrix, Pi being represented by the following equation:Pi = USV7where U and V are standard orthogonal bases, and Z is a diagonal matrix with singular values, matrix sizes of U, S and V being Ns*Ns, Ns^Ni, and Nt*Nt, respectively; andarranging the singular values A, as important information for representing matrix features, in a descending order in N part of the singular values reflecting all characteristics, and the matrix being approximately described by the first Nd singular values, and defining part of SVD,Pi ~ [72Vtwhere U and V are taken as standard orthogonal matrices, and sizes of U and kT are changed and reduced from Ns^Ns and Nt^Nt to Ns^Nd and Nd^Nt, S is a diagonal matrix containing Nd singular values A greater than a particular threshold, Nd«Nt, the singular values in the equation being arranged in a descending order as follows: A / >A2>A.!>...>Am / , and at this time, the matrix U corresponding to Z being a matrix with only Nd column vectors, called an eigenvector of P,P1 1y. A 2^Nd-where U and V are also eigenvectors of PTP and PP\ and A2 is an eigenvalue of PTP and PPT, physically, the eigenvalue A2 representing a generalized energy of a physical field captured by a corresponding eigenvector, and also representing a degree of the amount of information of a physical field represented in sample data contained in the eigenvector;S3, analysis of transient insulation characteristics based on PODperforming SVD on the collected electric potential sample matrix Pi to obtain a standard orthogonal basis U containing Nd column vectors, at this time, an electric potential vector at a certain moment being represented by U and a reduced-order electric potential vector, that is, POD being used to reconstruct a node electric potential to obtain the following equation:<P = r di) i di) V2 dD = Ua wherein, U = M N M h ro on 3 3 3 t-h oq on 3 3 3 di) Vns-1 dd L <p(Nls} J ” ulNd ai U1N _ adi) * , (X — u(Ns-l - uNsl uNs2 )2 u(Ns-l)Nd Nfi) UNsNd Jwhere <p is an electric potential column vector at a moment and is composed of electric potential values of various nodes in the model, (p being represented as the multiplication of two matrices, U being an eigenvector obtained in a previous section, and a being a vector reduced by the electric potential vector; andS4, insulation assessment reduced-order based on POD:an online stage being generally a reduced-order calculation, that is, solving according to the reduced-order governing equation, during which eigenvectors and interpolation matrices provided by a sample space are used; and an off-line stage being generally a finite element calculation of a full-order model, providing selectable samples for online calculation through the results of the offline calculation, and being limited by a calculation speed of the full-order model, wherein a model in the off-line stage generally only calculates part of the time step and provides usable parameters for online calculation through the calculated results.14

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