CN113839384A - Low-voltage distribution network phase-to-phase relation identification method based on matrix completion - Google Patents

Low-voltage distribution network phase-to-phase relation identification method based on matrix completion Download PDF

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CN113839384A
CN113839384A CN202111122919.4A CN202111122919A CN113839384A CN 113839384 A CN113839384 A CN 113839384A CN 202111122919 A CN202111122919 A CN 202111122919A CN 113839384 A CN113839384 A CN 113839384A
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张勇军
洪文慧
李钦豪
羿应棋
刘斯亮
周来
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South China University of Technology SCUT
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Abstract

The invention discloses a low-voltage distribution network household relation identification method based on matrix completion. The method comprises the following steps: collecting current time sequence data of a low-voltage bus of a target area and a plurality of time periods of all electric meters to form an original current matrix; decomposing, optimizing and combining an original current matrix with data loss by using a singular value threshold decomposition method to form a current completion matrix; calculating the relative error of the completed matrix relative to the original current matrix by a method of solving the ratio of the norm of the difference between the current completed matrix and the original current matrix to the norm of the original current matrix; and identifying the corresponding correlation in the transformer area by adopting a mathematical optimization method. The invention effectively completes the project condition of incomplete data acquired by the low-voltage distribution area by means of a data analysis technology, improves the accuracy of the identification of each household, and is beneficial to intelligent and efficient operation and maintenance of the low-voltage distribution area.

Description

Low-voltage distribution network phase-to-phase relation identification method based on matrix completion
Technical Field
The invention relates to the technical field of electric power low-voltage distribution networks, in particular to a low-voltage distribution network household relation identification method based on matrix completion.
Background
One of the currently applied phase-user identification relationship methods is a phase-user identification algorithm based on the KCL law, which is used for calculating the phase-user relationship of a transformer area by using current data, and therefore, the accuracy of the current data is of great importance. Considering the influence of reasons such as the non-support of the functions of the electric meter, the communication packet loss and the communication failure, the problem that the current data often has data loss is solved urgently, and therefore, how to complement the current matrix with the data loss is generated, so that a more accurate user relationship is generated. Vijay Arya et al, in Phase Identification in Smart Grids (2011IEEE International Conference on Smart Grids communication, Brussels, Belgium: IEEE, 2011: 25-30), perform Phase Identification by a mathematical optimization method, but the method does not consider the possible data missing condition of a current matrix, and has low applicability.
Disclosure of Invention
The invention aims to complement the current matrix with data missing, improve the accuracy of identifying the phase users in the low-voltage transformer area, reduce the error of identifying the phase users in the current data and the low-voltage transformer area, and improve the operation benefit of a power grid enterprise and the satisfaction index of customers.
The object of the invention is achieved by at least one of the following solutions.
The method for identifying the phase-to-household relationship of the low-voltage distribution network based on matrix completion comprises the following steps:
s1, collecting current time sequence data of a target platform area low-voltage bus and all electric meters in multiple time periods to form an original current matrix;
s2, decomposing, optimizing and combining the original current matrix with the data missing condition by using a singular value threshold decomposition method to form a current completion matrix;
s3, calculating the relative error of the completed matrix relative to the original current matrix by a method of solving the ratio of the norm of the difference between the current completed matrix and the original current matrix to the norm of the original current matrix;
and S4, identifying the corresponding correlation in the transformer area by adopting a mathematical optimization method.
Further, in step S1, an original current matrix with the abscissa as the user and the ordinate as the time is formed based on the current data of the actual electric meters in the distribution area according to the current time series data of the target distribution area low voltage bus and the plurality of time intervals of all the electric meters.
Further, step S2 specifically includes the following steps:
s2.1, converting the problem of complementing the original current matrix into an optimization target problem with constraint solving:
Figure BDA0003277659190000021
wherein M is the original current matrix, X is the current complement matrix, | | X | | the luminance*To complement the current with the nuclear norm of the matrix X, min | | | X | | luminance*Means that the nuclear norm of the current completion matrix X is minimized, MijIs an element of the ith row and jth column of the original current matrix, XijComplementing the elements of the ith row and the jth column of the current matrix, wherein W is a known element set after the data missing condition of the original current matrix M occurs, i and j respectively refer to the abscissa and the ordinate of the known element, and s.t.Xij＝Mij(i, j) epsilon W represents a constraint condition when a function of the X nuclear norm of the 0 minimized current completion matrix is solved, wherein the constraint condition is that elements of the original current matrix which are not subjected to the missing condition are kept unchanged;
defining a projection quantity PWComprises the following steps:
Figure BDA0003277659190000022
solving the optimization objective problem with constraints is represented as:
Figure BDA0003277659190000023
s2.2, introducing an intermediate matrix, and performing singular value threshold decomposition (SVT) algorithm on the initially generated intermediate matrix Y00, the current completion matrix X0Starting at 0, τ is a set singular value threshold, τ is selected as a singular value median method, that is, the singular value σ of the original current matrix M isrAnd (3) arranging the values from large to small, wherein r is the number of singular values of the original current matrix M, and then:
τ＝median(σ1，σ2，...，σr)； (3-4)
the median function represents taking the median value in the array;
definition k0Are integers that satisfy the following formula:
Figure BDA0003277659190000031
obtaining an intermediate matrix Y generated by the kth iteration by combining the singular value threshold value decomposition (SVT) principlekThe method comprises the following steps:
Yk＝kδPW(M)k＝1，...，k0； (3-6)
wherein δ is a set iteration step; the resulting k-th iteration generated intermediate matrix YkIs an intermediate matrix Y;
for the intermediate matrix Y with the rank R belongs to Rn1×n2Singular Value Decomposition (SVD) is carried out, the number s of singular values is appointed at first, and then the first s maximum singular values and corresponding singular vectors are calculated; therefore, Y is needed at the k-th iterationk-1Number of singular values of skLet sk＝rk-1+1，rk-1Is Xk-1Number of non-zero singular values in the previous iteration, rk-1＝rank(Xk-1) Then calculate Yk-1S beforekSingular value, skThe predefined integer l needs to be added repeatedly until skAmong the singular values, there are singular values less than τ as follows:
Y＝USV*，S＝diag({σi}l≤i≤r)； (3-7)
wherein σiIs a singular value of the intermediate matrix Y and σiBeing positive, S is the singular value σ of the intermediate matrix YiDiagonal matrix being diagonal, U and V being singular values σ of the intermediate matrix YiCorresponding left and right singular vectors, V＊Is the transpose of V; the transposition of Y and Y are subjected to matrix multiplication to obtain an n multiplied by n square matrix YTY; pair matrix YTAnd performing characteristic decomposition on Y to obtain a characteristic value and a characteristic vector which satisfy the following formula:
(YTY)vg＝λgvg； (3-8)
obtain matrix YTN eigenvalues λ of YgAnd corresponding n feature vectors vgG takes the value of 1-n; will YTAll the eigenvectors of Y are expanded into an n multiplied by n matrix V, and the matrix V is the right singular vector of Y;
the transpose of Y and Y is used for matrix multiplication to obtain a square matrix YY of m x mTOpposite square matrix YYTAnd performing characteristic decomposition to obtain characteristic values and characteristic vectors which satisfy the following formula:
(YYT)ul＝λlul； (3-9)
thereby obtaining a matrix YYTM eigenvalues λ oflAnd corresponding m eigenvectors ulL is 1-m; general matrix YYTAll the eigenvectors are expanded into a matrix U with the size of m multiplied by m, and the matrix U is a left singular vector of the middle matrix Y;
s2.3, introducing a soft threshold operator D for each tau being more than or equal to 0τThe definition is as follows:
Dτ(X)＝UDτ(S)V*，Dτ(S)＝diag({(σi-τ)+})； (3-10)
Dτ(X) according to a set threshold value, taking a value larger than the threshold value for an element in the current completion matrix X, and then forming a matrix; wherein (sigma)i-τ)+Is (sigma)i- τ) is a positive fraction, i.e. (σ)i-τ)+＝max(0，(σi- τ)); the current completion matrix X thus generated for the kth iterationkConvergence to a minimized X Y*And (3) obtaining U, S and V by combining the step S2.2; based on this, a current completion matrix X is obtained.
Further, in step S3, the relative error of the current completion matrix with respect to the original current matrix is calculated by a method of calculating the ratio of the norm of the difference between the current completion matrix and the original current matrix to the norm of the original current matrix, which is specifically as follows:
Figure BDA0003277659190000041
wherein M is an original current matrix, X represents a current completion matrix, | | | | FWatch (A)The F-norm of the matrix is shown, and r represents the relative error of the current completion matrix with respect to the original current matrix.
Further, in step S4, the basic principle of the mathematical optimization method is the current conservation law (KCL) in the circuit, which is: at any instant, the sum of the currents flowing to a node is constantly equal to the sum of the currents flowing from the node;
the mathematical optimization method introduces a binary variable of 0-1 to represent the attribution relationship of the electric meter and the phase line, and the objective function is that the sum of the total current of each phase and the electric meter power of the phase obtained according to the binary variable is basically equal at each moment:
Figure BDA0003277659190000051
wherein the content of the first and second substances,
Figure BDA0003277659190000052
show the homeLow-voltage side phase sequence of distribution transformer
Figure BDA0003277659190000053
The current error at the time t above,
Figure BDA0003277659190000054
distributing phase sequence to low-voltage side for attribution
Figure BDA0003277659190000055
Collecting all the electric meters;
Figure BDA0003277659190000056
distributing phase sequence to low-voltage side for attribution
Figure BDA0003277659190000057
The current value of the mth meter at the time t.
Further, in step S4, after introducing the binary variable of 0-1, if the meter g belongs to the second meter
Figure BDA0003277659190000058
Phase, then
Figure BDA0003277659190000059
Otherwise
Figure BDA00032776591900000510
Formula (5-1) to:
Figure BDA00032776591900000511
wherein the content of the first and second substances,
Figure BDA00032776591900000512
is the current value of the g-th ammeter at the moment t,
Figure BDA00032776591900000513
is to indicate the g-th electric meter and the
Figure BDA00032776591900000514
The variable 0-1, G ═ 1, 2, of the phase attribution, G, represents the total number of meters.
Further, in step S4, let
Figure BDA00032776591900000515
Figure BDA00032776591900000516
Wherein, T is 1, 2.. times.t; t represents the number of time instants;
Figure BDA00032776591900000517
refer to the phase sequence
Figure BDA00032776591900000518
The affiliation of the G-th meter in (c),
Figure BDA00032776591900000519
is shown as
Figure BDA00032776591900000520
The current error of the phase at the time T,
Figure BDA00032776591900000521
phase sequence at time T
Figure BDA00032776591900000522
A terminal current; the phase sequence definition ammeter current matrix Q, the ammeter phase sequence matrix Z, the current error matrix xi and the phase sequence current I are as follows:
Figure BDA00032776591900000523
wherein, A, C is phase sequence
Figure BDA00032776591900000524
And A is less than C;
the 'phase-user' relation recognition problem is converted into a 0-1 variable solving problem about a current completion matrix X, and the concrete model for solving the current completion matrix X is the minimum value of two-norm square of a matrix obtained by multiplying a solving objective function by a terminal current matrix and a current matrix of the ammeter at each moment and a phase sequence attribution relation matrix point; the method comprises the following specific steps:
Figure BDA0003277659190000061
wherein Z represents the affiliation of the electricity meter and the phase sequence.
The invention has the beneficial effects that:
(1) the missing current data is supplemented by a matrix supplementing method of singular value threshold decomposition (SVT), so that the accuracy of the current data is improved, and the intelligent identification accuracy of the low-voltage distribution area topology is improved in an assisting manner.
(2) The invention does not need to add an acquisition terminal in the low-voltage distribution network and has the characteristics of low cost and small engineering quantity.
(3) The invention designs a low-voltage distribution network household relation identification method based on matrix completion by comprehensively utilizing measurement data of a distribution transformer low-voltage side metering table and a user electric meter. The current data completion processing is beneficial to improving the accuracy of identifying the low-voltage distribution area topology by a data analysis method and reducing errors in the identification process. Therefore, the invention is beneficial to reducing the labor cost of the power company and improving the efficiency.
Drawings
Fig. 1 is a flowchart of a low-voltage distribution network household relationship identification method based on matrix completion in the embodiment of the present invention.
FIG. 2 is a graph of the matrix average relative recovery error at different dropout rates in an embodiment of the present invention.
Fig. 3 is a graph of the average pre-and post-missing user identification accuracy rate for different missing rates in the embodiment of the present invention.
Fig. 4 is a graph of average subscriber identification accuracy before and after completion of different sets of matrices at different missing time periods in the embodiment of the present invention.
FIG. 5 is a diagram of relative recovery errors of different matrix completion methods for different matrix dropout rates according to an embodiment of the present invention.
FIG. 6 is a diagram of relative recovery errors of different matrix completion methods during different matrix missing periods according to an embodiment of the present invention.
Detailed Description
The following description of the embodiments of the present invention is provided in connection with the accompanying drawings and examples.
Example (b):
the method for identifying the phase-to-household relationship of the low-voltage distribution network based on matrix completion, as shown in fig. 1, comprises the following steps:
s1, collecting current time sequence data of a target platform area low-voltage bus and all electric meters in multiple time periods to form an original current matrix;
and forming an original current matrix with the abscissa as the user and the ordinate as the time on the basis of the current data of the electric meters in the actual distribution area according to the current time sequence data of the low-voltage bus of the target distribution area and a plurality of time periods of all the electric meters.
S2, decomposing, optimizing and combining the original current matrix with data missing by using a singular value threshold decomposition method to form a current completion matrix, which specifically comprises the following steps:
s2.1, converting the problem of complementing the original current matrix into an optimization target problem with constraint solving:
Figure BDA0003277659190000071
wherein M is the original current matrix, X is the current complement matrix, | | X | | the luminance*To complement the current with the nuclear norm of the matrix X, min | | | X | | luminance*Means that the nuclear norm of the current completion matrix X is minimized, MijIs an element of the ith row and jth column of the original current matrix, XijComplementing the elements of the ith row and the jth column of the current matrix, wherein W is a known element set after the data missing condition of the original current matrix M occurs, i and j respectively refer to the abscissa and the ordinate of the known element, and s.t.Xij＝Mij(i, j) e W represents the constraint on solving the function of the 0-minimum current completion matrix X kernel norm that the original current matrix does not haveThe elements where the deletion occurs remain unchanged;
defining a projection quantity PWComprises the following steps:
Figure BDA0003277659190000072
solving the optimization objective problem with constraints is represented as:
Figure BDA0003277659190000081
s2.2, introducing an intermediate matrix, and performing singular value threshold decomposition (SVT) algorithm on the initially generated intermediate matrix Y00, the current completion matrix X0Starting at 0, τ is a set singular value threshold, τ is selected as a singular value median method, that is, the singular value σ of the original current matrix M isrAnd (3) arranging the values from large to small, wherein r is the number of singular values of the original current matrix M, and then:
τ＝median(σ1，σ2，...，σr)； (3-4)
the median function represents taking the median value in the array;
definition k0Are integers that satisfy the following formula:
Figure BDA0003277659190000082
obtaining an intermediate matrix Y generated by the kth iteration by combining the singular value threshold value decomposition (SVT) principlekThe method comprises the following steps:
Yk＝kδPW(M)k＝1，...，k0； (3-6)
wherein δ is a set iteration step; the resulting k-th iteration generated intermediate matrix YkIs an intermediate matrix Y;
for the intermediate matrix Y with the rank R belongs to Rn1×n2Singular Value Decomposition (SVD) is carried out, the number s of singular values is firstly specified, and then the first s maximum singular values are calculatedAnd corresponding singular vectors; therefore, Y is needed at the k-th iterationk-1Number of singular values of skLet sk＝rk-1+1，rk-1Is Xk-1Number of non-zero singular values in the previous iteration, rk-1＝rank(Xk-1) Then calculate Yk-1S beforekSingular value, skThe predefined integer l needs to be added repeatedly until skAmong the singular values, there are singular values less than τ as follows:
Y＝UST*，S＝diag({σi}l≤i≤r)； (3-7)
wherein σiIs a singular value of the intermediate matrix Y and σiBeing positive, S is the singular value σ of the intermediate matrix YiDiagonal matrix being diagonal, U and V being singular values σ of the intermediate matrix YiCorresponding left and right singular vectors, V＊Is the transpose of V; the transposition of Y and Y are subjected to matrix multiplication to obtain an n multiplied by n square matrix YTY; pair matrix YTAnd performing characteristic decomposition on Y to obtain a characteristic value and a characteristic vector which satisfy the following formula:
(YTY)vg＝λgvg； (3-8)
obtain matrix YTN eigenvalues λ of YgAnd corresponding n feature vectors vgG takes the value of 1-n; will YTAll the eigenvectors of Y are expanded into an n multiplied by n matrix V, and the matrix V is the right singular vector of Y;
the transpose of Y and Y is used for matrix multiplication to obtain a square matrix YY of m x mTOpposite square matrix YYTAnd performing characteristic decomposition to obtain characteristic values and characteristic vectors which satisfy the following formula:
(YYT)ul＝λlul； (3-9)
thereby obtaining a matrix YYTM eigenvalues λ oflAnd corresponding m eigenvectors ulL is 1-m; general matrix YYTAll the eigenvectors are expanded into a matrix U with the size of m multiplied by m, and the matrix U is a left singular vector of the middle matrix Y;
s2.3, introducing a soft threshold operator D for each tau being more than or equal to 0τThe definition is as follows:
Dτ(X)＝UDτ(S)V*，Dτ(S)＝diag({(σi-τ)+})； (3-10)
Dτ(X) according to a set threshold value, taking a value larger than the threshold value for an element in the current completion matrix X, and then forming a matrix; wherein (sigma)i-τ)+Is (sigma)i- τ) is a positive fraction, i.e. (σ)i-τ)+＝max(0，(σi- τ)); the current completion matrix X thus generated for the kth iterationkConvergence to a minimized X Y*And (3) obtaining U, S and V by combining the step S2.2; based on this, a current completion matrix X is obtained.
S3, calculating a relative error of the current completion matrix with respect to the original current matrix by a method of calculating a ratio of a norm of a difference between the current completion matrix and the original current matrix to a norm of the original current matrix, which is specifically as follows:
Figure BDA0003277659190000091
wherein M is an original current matrix, X represents a current complement matrix, | | | | | YFThe F-norm of the matrix is represented and r represents the relative error of the current completion matrix with respect to the original current matrix.
S4, identifying the corresponding correlation in the transformer area by adopting a mathematical optimization method;
the basic principle of the mathematical optimization method is the current conservation law (KCL) in the circuit, which is as follows: at any instant, the sum of the currents flowing to a node is constantly equal to the sum of the currents flowing from the node;
the mathematical optimization method introduces a binary variable of 0-1 to represent the attribution relationship of the electric meter and the phase line, and the objective function is that the sum of the total current of each phase and the electric meter power of the phase obtained according to the binary variable is basically equal at each moment:
Figure BDA0003277659190000101
wherein the content of the first and second substances,
Figure BDA0003277659190000102
indicating the phase sequence of the low-voltage side of the home distribution transformer
Figure BDA0003277659190000103
The current error at the time t above,
Figure BDA0003277659190000104
distributing phase sequence to low-voltage side for attribution
Figure BDA0003277659190000105
Collecting all the electric meters;
Figure BDA0003277659190000106
distributing phase sequence to low-voltage side for attribution
Figure BDA0003277659190000107
The current value of the mth meter at the time t.
After the binary variable of 0-1 is introduced, if the electric meter g belongs to the second
Figure BDA0003277659190000108
Phase, then
Figure BDA0003277659190000109
Otherwise
Figure BDA00032776591900001010
Formula (5-1) to:
Figure BDA00032776591900001011
wherein the content of the first and second substances,
Figure BDA00032776591900001012
for the g meter atthe current value at the time t is,
Figure BDA00032776591900001013
is to indicate the g-th electric meter and the
Figure BDA00032776591900001014
The variable 0-1, G ═ 1, 2, of the phase attribution, G, represents the total number of meters.
Order to
Figure BDA00032776591900001015
Wherein, T is 1, 2.. times.t; t represents the number of time instants;
Figure BDA00032776591900001016
refer to the phase sequence
Figure BDA00032776591900001017
The affiliation of the G-th meter in (c),
Figure BDA00032776591900001018
is shown as
Figure BDA00032776591900001019
The current error of the phase at the time T,
Figure BDA00032776591900001020
phase sequence at time T
Figure BDA00032776591900001021
A terminal current; the phase sequence definition ammeter current matrix Q, the ammeter phase sequence matrix Z, the current error matrix xi and the phase sequence current I are as follows:
Figure BDA0003277659190000111
wherein, A, C is phase sequence
Figure BDA0003277659190000112
And A is less than C;
the 'phase-user' relation recognition problem is converted into a 0-1 variable solving problem about a current completion matrix X, and the concrete model for solving the current completion matrix X is the minimum value of two-norm square of a matrix obtained by multiplying a solving objective function by a terminal current matrix and a current matrix of the ammeter at each moment and a phase sequence attribution relation matrix point; the method comprises the following specific steps:
Figure BDA0003277659190000113
wherein Z represents the affiliation of the electricity meter and the phase sequence.
In order to verify the effectiveness of the completion algorithm, in this embodiment, the accuracy obtained by comparing the house-to-house relationship obtained by the current matrix in which the data is missing and the processed current completion matrix data with the real house-to-house relationship is used as an evaluation index of the recognition effect of the matrix completion on the house-to-house relationship, that is, the accuracy obtained by comparing the house-to-house relationship generated by the missing data with the house-to-house relationship generated by the completed data is compared, so that the parameter can be used to evaluate the recognition effect of the completion algorithm on the house-to-house relationship:
Figure BDA0003277659190000114
wherein, λ refers to the accuracy rate obtained by comparing the correlation generated by the processed data with the real correlation, nSame elementThe number n of correctly identified electric meters is obtained by comparing the correlation generated by the processed data with the real correlationTotal number ofRefers to the total number of meters in the household relationship.
In this embodiment, when the low-voltage station area is randomly deleted, three groups of current matrices are selected, where there are two matrices in each group, that is, 6 matrices in total are respectively subjected to simulation under different deletion rates, and the deletion rate is 1% to 9%, and when the time interval deletion is performed, the deletion rate is 50 time intervals, 100 time intervals, and 150 time intervals. R in the table refers to the relative recovery error. As can be seen from FIG. 2, when the matrix defect rate is within 9%, the average relative recovery error is not more than 0.25, which indicates that the completion accuracy is high and stable; as can be seen from fig. 3, the phase sequence identification accuracy obtained according to the current completion matrix is improved by about 15% at most compared with the current matrix in the case of data loss, and the lowest phase sequence identification accuracy of the current completion matrix is also about 80%, which indicates that the effect of the matrix completion method applied to the phase sequence identification method is better; fig. 4 shows that, when a matrix is missing in a time period, the phase sequence identification accuracy obtained according to the current completion matrix is improved by about 20% at most compared with that of the current matrix in the case of data missing, and the phase sequence identification accuracy obtained according to the current completion matrix is about 90% at least, which indicates that the current completion matrix has a good completion effect in the case of time period missing; FIG. 5 shows that when the loss rate is 1% -9%, the singular value threshold decomposition method, the robust matrix completion method with abnormal values and sparse noise, and the basic sub-gradient descent method are compared, and the singular value threshold decomposition method has the lowest relative error and is most stable; fig. 6 shows that when the missing time period is 50-150, three data completion algorithms of a singular value threshold decomposition method, a robust matrix completion method with abnormal values and sparse noise and a basic sub-gradient descent method are compared, the relative error of the singular value threshold decomposition method is far lower than that of the other two methods, and the effect is very stable.
The MATALB program is used to calculate the relative error r of the current completion matrix with respect to the original current matrix in step S3, and the results of the relative recovery errors of the current completion matrix with respect to the original current matrix at different miss rates are shown in table 1.
TABLE 1 relative recovery error for each matrix at different miss rates
Rate of absence 9％ 8％ 7％ 6％ 5％ 4％ 3％ 2％ 1％
r (matrix one) 0.25 0.24 0.22 0.20 0.19 0.16 0.15 0.12 0.09
r (matrix two) 0.25 0.24 0.22 0.21 0.19 0.16 0.15 0.12 0.09
r (three matrix) 0.25 0.24 0.22 0.21 0.19 0.17 0.15 0.12 0.09
r (matrix one) 0.18 0.16 0.15 0.13 0.12 0.11 0.09 0.07 0.04
r (matrix five) 0.18 0.17 0.15 0.14 0.12 0.11 0.09 0.07 0.04
r (six matrix) 0.18 0.17 0.15 0.14 0.12 0.11 0.09 0.07 0.04
The relative recovery error results of the current completion matrix relative to the original current matrix in the absence of different periods are shown in table 2.
TABLE 2 relative recovery error for matrix completion at different periods of absence
Absence period 50 100 150
r (matrix one) 0.0501 0.0457 0.0502
r (matrix two) 0.0047 0.0046 0.0032
r (three matrix) 0.0015 0.0023 0.0021
r (matrix four) 0.0425 0.0563 0.0534
r (matrix five) 0.0509 0.0528 0.0448
r (six matrix) 0.0501 0.0457 0.0502
The MATALB program is used for solving the current matrix under the condition of data loss and the corresponding phase identification accuracy of the current completion matrix, and the results of the phase identification accuracy before and after matrix completion under different loss rates are shown in Table 3.
TABLE 3 accuracy of phase identification before and after matrix completion under different deficiency rates
Rate of absence 9％ 8％ 7％ 6％ 5％ 4％ 3％ 2％ 1％
Acc misses of the first group 54％ 57％ 71％ 69％ 70％ 77％ 83％ 84％ 97％
Acc completion of the first group 73％ 75％ 76％ 84％ 86％ 90％ 93％ 96％ 99％
Second set of Acc misses 61％ 54％ 63％ 72％ 79％ 83％ 86％ 92％ 95％
Acc completion of the second group 77％ 78％ 83％ 85％ 89％ 93％ 95％ 97％ 98％
Third Acc deletion 71％ 76％ 72％ 79％ 83％ 82％ 83％ 88％ 92％
Acc completion of the third group 82％ 83％ 87％ 88％ 89％ 93％ 97％ 97％ 99％
The results of the phase-to-phase identification accuracy of the current matrix and the current completion matrix in which data loss occurs at different loss periods are shown in table 4.
TABLE 4 accuracy of identification of the front and back facies users by matrix completion under different missing periods
Number of missing sessions 50 100 150
Acc misses of the first group 95％ 84％ 72％
Acc completion of the first group 98％ 93％ 88％
Second set of Acc misses 94％ 91％ 74％
Acc completion of the second group 99％ 99％ 94％
Third Acc deletion 98％ 91％ 83％
Acc completion of the third group 100％ 99％ 96％
By combining the tables, the two defects obviously improve certain accuracy rate when the invention is applied to matrix completion or mutual identification.
In summary, through simulation analysis of a certain actual distribution area, it is verified that the accuracy of the current data of the low-voltage distribution area and the identification accuracy of the phase-to-phase relationship of the low-voltage distribution area can be effectively improved by using the matrix completion-based low-voltage distribution network phase-to-phase relationship identification method provided by this embodiment.
The above embodiments are preferred embodiments of the present invention, but the present invention is not limited to the above embodiments, and any other modifications, substitutions, combinations, and simplifications which do not depart from the spirit and principle of the present invention should be construed as equivalents and are intended to be included in the scope of the present invention.

Claims (10)

1. The method for identifying the phase-to-household relationship of the low-voltage distribution network based on matrix completion is characterized by comprising the following steps of:
s1, collecting current time sequence data of a target platform area low-voltage bus and all electric meters in multiple time periods to form an original current matrix;
s2, decomposing, optimizing and combining the original current matrix with the data missing condition by using a singular value threshold decomposition method to form a current completion matrix;
s3, calculating the relative error of the completed matrix relative to the original current matrix by a method of solving the ratio of the norm of the difference between the current completed matrix and the original current matrix to the norm of the original current matrix;
and S4, identifying the corresponding correlation in the transformer area by adopting a mathematical optimization method.
2. The method for identifying the phase relationship of the low-voltage distribution network based on the matrix completion in the step S1 is characterized in that, according to the current time sequence data of the target area low-voltage bus and all the electric meters in a plurality of time periods, an original current matrix with the abscissa as the user and the ordinate as the time is formed on the basis of the current data of the electric meters in the actual area.
3. The method for identifying the household relationship of the low-voltage distribution network based on the matrix completion as claimed in claim 1, wherein the step S2 specifically comprises the following steps:
s2.1, converting the problem of complementing the original current matrix into an optimization target problem with constraint solving:
Figure FDA0003277659180000011
in the formula, M is the original current matrix, X is the current completion matrix, | | X | | | is the nuclear norm of the current completion matrix X, min | | | X | | | means that the nuclear norm of the current completion matrix X is minimized, M | | | X | | | means that M is the current completion matrix XijIs an element of the ith row and jth column of the original current matrix, XijThe current is added to the elements in the ith row and the jth column of the current matrix, W is the known element set after the data missing condition of the original current matrix M occurs, i,j refers to the abscissa and ordinate, s.t.x, of the known element, respectivelyij＝Mij(i, j) epsilon W represents a constraint condition when a function of the X nuclear norm of the 0 minimized current completion matrix is solved, wherein the constraint condition is that elements of the original current matrix which are not subjected to the missing condition are kept unchanged;
defining a projection quantity PWComprises the following steps:
Figure FDA0003277659180000021
solving the optimization objective problem with constraints is represented as:
Figure FDA0003277659180000022
s2.2, introducing an intermediate matrix, and generating an intermediate matrix Y from the initial by using a singular value threshold decomposition algorithm00, the current completion matrix X0Starting at 0, τ is a set singular value threshold, τ is selected as a singular value median method, that is, the singular value σ of the original current matrix M isrAnd (3) arranging the values from large to small, wherein r is the number of singular values of the original current matrix M, and then:
τ＝median(σ1，σ2，...，σr)； (3-4)
the median function represents taking the median value in the array;
definition k0Are integers that satisfy the following formula:
Figure FDA0003277659180000023
obtaining an intermediate matrix Y generated by the kth iteration by combining the singular value threshold value decomposition (SVT) principlekThe method comprises the following steps:
Yk＝kδPW(M) k＝1，...，k0； (3-6)
wherein δ is a set iteration step; then get the firstIntermediate matrix Y generated by k iterationskIs an intermediate matrix Y;
for the intermediate matrix Y with the rank R belongs to Rn1×n2Singular value decomposition is carried out, the number s of singular values is specified, and then the previous s maximum singular values and corresponding singular vectors are calculated; therefore, Y is needed at the k-th iterationk-1Number of singular values of skLet sk＝rk-1+1，rk-1Is Xk-1Number of non-zero singular values in the previous iteration, rk-1＝rank(Xk-1) Then calculate Yk-1S beforekSingular value, skThe predefined integer l needs to be added repeatedly until skAmong the singular values, there are singular values less than τ as follows:
Y＝USV*，S＝diag({σi}1≤i≤r)； (3-7)
wherein σiIs a singular value of the intermediate matrix Y and σiBeing positive, S is the singular value σ of the intermediate matrix YiDiagonal matrix being diagonal, U and V being singular values σ of the intermediate matrix YiCorresponding left singular vectors and right singular vectors, Vx is the transpose of V; the transposition of Y and Y are subjected to matrix multiplication to obtain an n multiplied by n square matrix YTY; pair matrix YTAnd performing characteristic decomposition on Y to obtain a characteristic value and a characteristic vector which satisfy the following formula:
(YTY)vg＝λgvg； (3-8)
obtain matrix YTN eigenvalues λ of YgAnd corresponding n feature vectors vgG takes the value of 1-n; will YTAll the eigenvectors of Y are expanded into an n multiplied by n matrix V, and the matrix V is the right singular vector of Y;
the transpose of Y and Y is used for matrix multiplication to obtain a square matrix YY of m x mTOpposite square matrix YYTAnd performing characteristic decomposition to obtain characteristic values and characteristic vectors which satisfy the following formula:
(YYT)ul＝λlul； (3-9)
thereby obtaining a matrix YYTM eigenvalues λ oflAnd correspondingm eigenvectors ulL is 1-m; general matrix YYTAll the eigenvectors are expanded into a matrix U with the size of m multiplied by m, and the matrix U is a left singular vector of the middle matrix Y;
s2.3, introducing a soft threshold operator D for each tau being more than or equal to 0τThe definition is as follows:
Dτ(X)＝UDτ(S)V*，Dτ(S)＝diag({(σi-τ)+}) ；(3-10)
Dτ(X) according to a set threshold value, taking a value larger than the threshold value for an element in the current completion matrix X, and then forming a matrix; wherein (sigma)i-τ)+Is (sigma)i- τ) is a positive fraction, i.e. (σ)i-τ)+＝max(0，(σi- τ)); the current completion matrix X thus generated for the kth iterationkConverging to a solution of minimizing X, and combining with the step S2.2 to obtain U, S, V; based on this, a current completion matrix X is obtained.
4. The method for identifying the phase relationship of the low-voltage distribution network based on matrix completion of claim 1, wherein in step S3, the relative error of the current completion matrix with respect to the original current matrix is calculated by calculating the ratio of the norm of the difference between the current completion matrix and the original current matrix to the norm of the original current matrix, which is specifically as follows:
Figure FDA0003277659180000041
wherein M is an original current matrix, X represents a current complement matrix, | | | | | YFThe F-norm of the matrix is represented and r represents the relative error of the current completion matrix with respect to the original current matrix.
5. The method for identifying the phase-to-household relationship of the low-voltage distribution network based on the matrix completion of the claim 1, wherein in the step S4, the basic principle of the mathematical optimization method is the current conservation law in the circuit.
6. The matrix completion based low-voltage distribution network household relationship identification method according to claim 5, wherein the mathematical optimization method in step S4 is specifically as follows: at any instant, the sum of the currents flowing to a node is constantly equal to the sum of the currents flowing from the node;
the mathematical optimization method introduces a binary variable of 0-1 to represent the attribution relationship of the electric meter and the phase line, and the objective function is that the sum of the total current of each phase and the electric meter power of the phase obtained according to the binary variable is basically equal at each moment:
Figure FDA0003277659180000042
wherein the content of the first and second substances,
Figure FDA0003277659180000043
indicating the phase sequence of the low-voltage side of the home distribution transformer
Figure FDA0003277659180000044
The current error at the time t above,
Figure FDA0003277659180000045
distributing phase sequence to low-voltage side for attribution
Figure FDA0003277659180000046
Collecting all the electric meters;
Figure FDA0003277659180000047
distributing phase sequence to low-voltage side for attribution
Figure FDA0003277659180000048
The current value of the mth meter at the time t.
7. The matrix completion-based low-voltage distribution network household relationship identification method according to claim 6Characterized in that, in step S4, after a binary variable of 0-1 is introduced, if the meter g belongs to the second meter
Figure FDA0003277659180000049
Phase, then
Figure FDA00032776591800000410
Otherwise
Figure FDA00032776591800000411
Formula (5-1) to:
Figure FDA00032776591800000412
wherein the content of the first and second substances,
Figure FDA0003277659180000051
is the current value of the g-th ammeter at the moment t,
Figure FDA0003277659180000052
is to indicate the g-th electric meter and the
Figure FDA0003277659180000053
The variable 0-1, G ═ 1, 2, of the phase attribution, G, represents the total number of meters.
8. The method for identifying the household relationship of the low-voltage distribution network based on the matrix completion of claim 7, wherein in step S4, the order is
Figure FDA0003277659180000054
Figure FDA0003277659180000055
Wherein, T is 1, 2.. times.t; t represents the number of time instants;
Figure FDA0003277659180000056
refer to the phase sequence
Figure FDA0003277659180000057
The affiliation of the G-th meter in (c),
Figure FDA0003277659180000058
is shown as
Figure FDA0003277659180000059
The current error of the phase at the time T,
Figure FDA00032776591800000510
phase sequence at time T
Figure FDA00032776591800000511
A terminal current; the phase sequence definition ammeter current matrix Q, the ammeter phase sequence matrix Z, the current error matrix xi and the phase sequence current I are as follows:
Figure FDA00032776591800000512
wherein, A, C is phase sequence
Figure FDA00032776591800000513
And A is less than C.
9. The matrix completion-based low-voltage distribution network phase-to-household relationship identification method according to any one of claims 1 to 8, wherein in the step S4, the 'phase-to-household' relationship identification problem is converted into a 0-1 variable solving problem about the current completion matrix X, and the concrete model for solving the current completion matrix X is the minimum value of two-norm squares of the difference between the terminal current matrix and the matrix obtained by multiplying the current matrix of the ammeter at each moment and the phase sequence attribution relationship matrix point of the ammeter at each moment.
10. The method for identifying the phase-to-household relationship of the low-voltage distribution network based on matrix completion of claim 9, wherein in step S4, the specific model for solving the current completion matrix X is as follows:
Figure FDA00032776591800000514
wherein Z represents the affiliation of the electricity meter and the phase sequence.
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