CN107462809A - Phase-model transformation matrix design method for three-phase power circuit fault diagnosis - Google Patents

Phase-model transformation matrix design method for three-phase power circuit fault diagnosis Download PDF

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CN107462809A
CN107462809A CN201710588448.3A CN201710588448A CN107462809A CN 107462809 A CN107462809 A CN 107462809A CN 201710588448 A CN201710588448 A CN 201710588448A CN 107462809 A CN107462809 A CN 107462809A
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msub
mtd
mover
phase
mtr
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牛耕
孔力
裴玮
齐智平
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Institute of Electrical Engineering of CAS
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Institute of Electrical Engineering of CAS
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R31/00—Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
    • G01R31/08—Locating faults in cables, transmission lines, or networks
    • G01R31/088—Aspects of digital computing

Abstract

A kind of phase-model transformation matrix design method for three-phase power circuit fault diagnosis, it is first depending on design criteria and the parameter of phase-model transformation matrix is designed and chosen.Then Gram Schmidt orthogonalization process is carried out with the phase-model transformation matrix chosen to completing parameter designing, obtains the final design result of required phase-model transformation matrix.

Description

Phase-model transformation matrix design method for three-phase power circuit fault diagnosis
Technical field
The present invention relates to a kind of phase-model transformation matrix design method for three-phase power circuit fault diagnosis.
Background technology
At present, three-phase alternating current transmission of electricity is most important power transmission mode in power system.Power transmission line is in power system For electrical energy transportation and the critical elements of distribution.Most power circuits are often directly exposed among external environment condition for a long time, and this makes Obtain the frequency to be broken down on power circuit and probability is higher.The failure of power circuit influences whether the safety of whole power system Stable operation, such as processing will likely cause serious consequence not in time, bring huge economic loss.
The three-phase of each voltage class is defeated, is each equipped with corresponding route protection in distribution system, can occur in circuit After failure rapid disengagement failure and by Fault Isolation outside runtime, so as to ensure the safety of power system.In addition, pin Corresponding Fault Locating Method is additionally provided with to defeated, distribution line, for searching section or specific orientation where trouble point, so as to Quick investigation and repairing in time in line fault.It is the most key in the error protection and Fault Locating Method of three-phase line Link be line fault quick diagnosis.However, existing three-phase line protection and Fault Locating Method generally take split-phase The mode of configuration, for the fault diagnosis of circuit, this configuration mode causes the diagnosis of line fault to need to integrate circuit The failure determination result of each phase, its major defect are that fault diagnosis flow scheme is complicated, amount of calculation is larger, rapidity is bad.Meanwhile Each alternate coupling effect of circuit will adversely affect to malfunction analysis and problem shpoting on circuit.
In order to simplify the fault diagnosis flow scheme in the error protection and Fault Locating Method of three-phase line, the fast of method is improved Speed, and the influence of each Coupling Between Phases effect of circuit is eliminated, nowadays in the error protection and Fault Locating Method of many circuits In research, phase-model transformation technology has obtained more application, and conventional more traditional transformation matrix has：Symmetrical component transformation square Battle array, Clarke transformation matrixs, Karenbauer transformation matrixs, Wedpohl transformation matrixs etc..Wherein, symmetrical component transformation conduct The important tools of analysis of electric power system fault, it is theoretical more perfect, is widely used in the stable state phasor analysis of electric power system fault, Can reflect all types failure on circuit merely with its positive-sequence component, but in its transformation matrix containing complex operation because Son, calculating is complex, and rapidity is bad, and this limits its and led in the error protection of circuit with fault location to a certain extent Application in domain.Different from symmetrical component transformation, the conversion such as Clarke conversion, Karenbauer conversion and Wedpohl conversion The transformation matrix of method is real matrix, and this causes these transform methods to possess some superiority on quick calculate, but its is main Shortcoming is all types failure that can not be reflected on circuit using single modulus, it is necessary to the side for taking two modulus to cooperate Formula.Therefore, a kind of real number phase-model transformation matrix pair that can use all types failure on single modulus reflection circuit is found In the rapid failure diagnosis of three-phase line, error protection and fault location important in inhibiting.
Song state soldier et al. exists《Automation of Electric Systems》Written by the 14th phases of volume 31 in 2007《A kind of new phase-model transformation square Battle array》In propose a kind of new phase-model transformation matrix, the matrix is a real number matrix, can utilize single modulus reflection three All types failure on phase line, significantly reduce the flow complexity of fault diagnosis.S.Song et al. exists《IEEE TRANSACTIONS ON POWER DELIVERY》Written by the 4th phases of volume 30 in 2015《A Novel Busbar Protection Method Based on Polarity Comparison of Superimposed Current》Middle proposition It is a kind of from different above, the new real number phase mould of all types failure on single modulus reflection three-phase line can be utilized Transformation matrix.But the research of two papers of the above fails to provide the specific design method of such phase-model transformation matrix.It is public Temporarily the file of the relevant design method without such phase-model transformation matrix is reported in the Chinese invention patent opened.
At present, for three-phase line, although experts and scholars both domestic and external have pointed out several new phase-model transformation matrixes, adopt With such phase-model transformation matrix, all types failure on single modulus reflection circuit can be utilized.But for this kind of new The parameter designing of phase-model transformation matrix, specific a design method and criterion are lacked in existing research.It is in view of this kind of new Fast failure protection and fast failure positioning of the phase-model transformation matrix for three-phase line play the role of positive with significantly anticipating Justice, therefore in this field there is an urgent need to a kind of method and criterion being designed to such phase-model transformation matrix, to specification The parameter designing of this kind of phase-model transformation matrix and selection.
The content of the invention
It is an object of the invention to overcome shortcoming and deficiency present in existing research, propose that one kind is applied to three-phase power The phase-model transformation matrix design method of Circuit fault diagnosis.
The phase-model transformation matrix of the present invention is a real number matrix, is reflected merely with one of modulus can The various faults occurred on three-phase line, such as single-line to ground fault, two-phase grounding fault, two-phase phase fault, three-phase shortcircuit Situation, realize the fault diagnosis on the circuit.
The technical solution used in the present invention is：
The inventive method is first depending on design criteria and the parameter of phase-model transformation matrix is designed and chosen；Then, it is right Complete parameter designing and carry out Gram-Schmidt orthogonalization process with the phase-model transformation matrix chosen, obtain required phase-model transformation The final design result of matrix.
The inventive method comprises the following steps that：
(1) parameter of phase-model transformation matrix is designed and chosen according to design criteria；
The parameter designing of described phase-model transformation matrix is with choosing method：For one section of both-end without branch, line parameter circuit value Along thread path to the three-phase power circuit being evenly distributed and three-phase uniformly replaces, it is known that the series impedance under the circuit unit length Matrix Z is respectively with shunt admittance matrix Y：
Wherein, z is series connection self-impedance of each phase of the circuit under unit length；Z ' is that each phase of the circuit is grown in unit Series connection mutual impedance under degree；Y is in parallel self-admittance of each phase of the circuit under unit length；Y ' is each phase of the circuit in list Transadmittance in parallel under bit length.Due to coupling effect in circuit often be present, therefore in general, z, z ', y and y ' Value often not be 0, make they be all non-zero plural number.
Phase-model transformation is the decoupling method commonly used in power system, can effectively eliminate the coupling between each phase of the circuit Effect.The three-phase decoupling of line parameter circuit value can not only be realized by phase-model transformation, moreover it is possible to realize node voltage on circuit, electric current Three-phase decouples.Make the node voltage matrix on the circuit at a certain node and node current matrix is respectively U and I, their table It is respectively up to formula：
Wherein, UA、UBAnd UCA, B and C phase voltage respectively at the node；IA、IBAnd ICA, B respectively at the node And C phase currents.
Correspondingly, the node mode voltage after phase-model transformation decouples at the node and the node mould current matrix can be made to be respectivelyWithTheir expression formula is respectively：
Wherein,Andα, β and γ mode voltage respectively at the node；AndThe respectively node α, β and γ mould electric current at place.
The phase-model transformation matrix that the circuit can be made is Φ, then the matrix is a three rank real number square formations, its expression formula For：
Wherein,For i-th of column vector in matrix Φ,The element arranged for the i-th row jth in matrix Φ, i=1,2,3, j =1,2,3.
The phase-model transformation relational expression of line node voltage and node current based on matrix Φ is：
Understand that matrix Φ allows for meeting relational expression through analysis：
Φ-1K Φ=λ (1)
Wherein, matrix K is the parameter matrix of the circuit, and hasλ is the characteristic value of matrix K Matrix, the matrix are a diagonal matrix, i.e. λ=diag (λ1,λ2,λ3), λiFor the element on matrix λ inner opposite angle lines；Matrix Φ-1 For matrix Φ inverse matrix；I=1,2,3.
, can be by the off-diagonal in the parameter matrix of the circuit based on matrix Φ from formula (1) through analysis Coupling parameter is eliminated, and the parameter matrix K of the circuit is turned to the eigenvalue matrix λ of a diagonal matrix, i.e. matrix K, realizes this Each phase parameter decoupling of circuit.Therefore, matrix Φ can be referred to as to the right eigenvectors matrix of matrix K；Accordingly, can be by matrix Φ-1The referred to as left eigenvector matrix of matrix K.Further, can incite somebody to actionIn referred to as matrix Φ ith feature vector, i=1,2, 3.Formula (1) and then it can turn to：
Wherein, i=1,2,3, E are three rank unit matrixs.
The characteristic value of matrix K can be solved based on following formula：
Det (K- λ E)=0 (3)
Can be in the hope of the characteristic value of matrix K according to formula (3)：
Formula (4) is substituted into formula (2), it can be deduced that：
An invertible matrix is necessary in view of matrix Φ, order matrix Φ inverse matrix is Φ-1, its expression formula is：
Wherein,For matrix Φ-1The element of interior i-th row jth row, i=1,2,3, j=1,2,3.
Understood through analysis, to ensure matrix Φ inverse matrix Φ-1Existence, it is necessary to have following formula establishment：
It is more complicated in view of matrix Φ inversion operation, it is unfavorable for quickly calculating, is simplification matrix Φ fortune of inverting Calculate, canonical orthogonal processing can be carried out to matrix Φ, obtain a new phase-model transformation matrixMatrixWith following Property：
Canonical orthogonal processing can be carried out to matrix Φ using Gram-Schmidt orthogonalization methods, and then obtain matrixGram-Schmidt orthogonalization methods are computed matrix Φ being changed into an orthogonal matrix Γ first, then by matrix Γ unit standardization, finally gives matrixMatrix Γ andExpression formula be respectively：
Wherein,For i-th of column vector in matrix Γ, tijThe element arranged for the i-th row jth in matrix Γ；For matrix I-th interior of column vector,For matrixThe element of interior i-th row jth row；I=1,2,3, j=1,2,3.And there is following computing to close It is formula：
In formula (9) and (10), there are i=1,2,3.
From formula (4), there are two identical characteristic values in matrix K.In matrix Φ, corresponding characteristic value not phase Two same characteristic vectors are mutually orthogonal；Two equal characteristic vector linear independences of corresponding characteristic value., can be with based on this Draw：
Formula (11) is substituted into formula (9) to obtain：
Wherein, have：
Simultaneous formula (5) and formula (12) can prove：
t13+t23+t33=0 (13)
Based on formula (10), (12) and (13), matrix can be further obtainedThe relational expression of interior each element：
Formula (12) and formula (14) analysis are understood, reversible matrixCan be as the phase-model transformation square of the circuit Battle array.After Gram-Schmidt orthogonalization process, matrixIn the direction of the 1st and the 2nd characteristic vector will not become Change, still the direction with characteristic vector corresponding in matrix Φ is consistent, but the direction of wherein the 3rd characteristic vector will Change, it is no longer identical with the direction of characteristic vector corresponding in matrix Φ.
To matrixKnowable to interior the 1st and the 2nd characteristic vector are analyzed：
(1) for matrixThe 1st interior characteristic vector, the modulus corresponding to it can not reflect what is occurred on the circuit Two-phase phase fault and three phase short circuit fault, therefore can not realize the circuit merely with the modulus corresponding to this feature vector The all types failure of upper generation, comprising single-phase grounding fault, two-phase short circuit and ground fault, two-phase phase fault, The fault diagnosis of three phase short circuit fault；
(2) for matrixThe 2nd interior characteristic vector, can merely with the modulus corresponding to this feature vector for guarantee Reflect all types failure occurred on the circuit, include single-phase grounding fault, two-phase short circuit and ground fault, two-phase phase Between short trouble, three phase short circuit fault, it is also necessary to supplement following constraints：
Understood according to formula (10) and formula (12), due to matrixThe direction of the 2nd interior characteristic vector and matrix Φ In the 2nd characteristic vector be consistent, therefore, understood according to formula (15), for the 2nd characteristic vector in matrix Φ, Need to meet following constraints：
To sum up, simultaneous formula (5), (7) and (16), each element in the phase-model transformation matrix Φ of the circuit can be exported Between the relational expression of required satisfaction be：
Formula (17) is design criteria of the parameter designing with choosing method institute foundation of the phase-model transformation matrix.Foundation The formula can be designed and choose to the parameter of the phase-model transformation matrix Φ suitable for three-phase power circuit fault diagnosis, and profit It can reflect all types failure on the circuit with the modulus corresponding to wherein the 2nd characteristic vector, it is short comprising single-phase earthing Road failure, two-phase short circuit and ground fault, two-phase phase fault, three phase short circuit fault.
(2) Gram-Schmidt orthogonalization process is carried out with the phase-model transformation matrix chosen to completing parameter designing, obtained The final design result of required phase-model transformation matrix；
The Gram-Schmidt orthogonalization process method includes three steps altogether：
Step 1, the phase-model transformation matrix that will be completed parameter designing in step (1) and choose are designated as matrix Φ, its expression formula For：
Step 2, according to below equation matrix Φ is calculated：
Matrix Φ is changed into an orthogonal matrix Γ by above-mentioned calculating process, and its expression formula is：
Wherein,For i-th of column vector in matrix Γ, tijThe element arranged for the i-th row jth in matrix Γ, i=1,2,3, J=1,2,3；
Step 3, according to below equation matrix Γ is calculated：
Above-mentioned calculating process standardizes matrix Γ units, finally gives and exports required phase-model transformation matrixIts Expression formula is：
Wherein,For matrixI-th interior of column vector,For matrixThe element of interior i-th row jth row；I=1,2,3, j =1,2,3.
The present invention proposes one kind for the phase-model transformation matrix design method of three-phase power circuit fault diagnosis and can be used for The design method of three-phase power circuit fault diagnosis, error protection and the phase-model transformation matrix in FLT field, with Prior art is compared, and the beneficial effect that this method can be generated is：
First, the present invention has filled up reflects on three-phase power circuit that what is occurred owns that can utilize single modulus both at home and abroad The failure of type, include single-phase grounding fault, two-phase short circuit and ground fault, two-phase phase fault, three-phase shortcircuit event Blank on this field of the design method of the phase-model transformation real matrix of barrier；
Second, the phase-model transformation matrix of the three-phase power circuit obtained based on the inventive method, can not only utilize its 2nd Modulus corresponding to individual characteristic vector can reflect all types of failures occurred on three-phase power transmission line, comprising single-phase Ground short circuit failure, two-phase short circuit and ground fault, two-phase phase fault, three phase short circuit fault, also transported with matrix inversion Calculate the advantages of simple, amount of calculation is small.
Brief description of the drawings
Fig. 1 is the Method And Principle flow chart of the present invention；
Fig. 2 is the schematic diagram of the specific embodiment of the present invention；
Fig. 3 is front and rear, the of the invention phase-model transformation square that broken down on a certain bar three-phase line in low-voltage network The simulation waveform of the mould electric current on circuit in battle array corresponding to the 2nd characteristic vector.
Embodiment
The present invention will be further described with reference to the accompanying drawings and detailed description.
Fig. 1 is the Method And Principle flow chart of the present invention.As shown in figure 1, step 001 is first carried out, according to design criteria pair The parameter of phase-model transformation matrix is designed and chosen.Then, step 002 is performed, to the phase mould completed parameter designing with chosen Transformation matrix carries out Gram-Schmidt orthogonalization process.Finally, step 003 is performed, the final of output phase-model transformation matrix is set Count result.
Fig. 2 is the schematic diagram of the specific embodiment of the present invention.As shown in Fig. 2 step 101 is first carried out, to phase mould The parameter of transformation matrix is designed selection.Then, step 102 is performed, is by phase-model transformation matrix design：
Then, step 103 is performed, Gram-Schmidt orthogonalization process is carried out to phase-model transformation matrix Φ.Finally, perform Step 104, final phase-model transformation matrix is obtainedFor：
According to the specific embodiment of the present invention, it is analyzed and understood, it is final obtained by phase-model transformation matrix's Inverse matrixFor：
As can be seen here, the phase-model transformation matrix obtained using present invention design has the advantages of matrix inversion operation is simple, Amount of calculation caused by matrix inversion can effectively be reduced.
Fig. 3 is front and rear, the of the invention phase-model transformation square that broken down on a certain bar three-phase line in low-voltage network The simulation waveform of the mould electric current on circuit in battle array corresponding to the 2nd characteristic vector.As shown in figure 3, in low-voltage network A three-phase line carry out fault simulation, the fault moment of setting is 0.5 second, fault type have single-phase grounding fault, Two-phase short circuit and ground fault, two-phase phase fault and three phase short circuit fault, the simulation waveform in the case of all kinds of failures It is divided into：The simulation waveform 201 of A phase ground short circuit failures, the simulation waveform 202 of B phase ground short circuit failures, C phases are grounded short The simulation waveform 203 of road failure, the simulation waveform 204 of AB phase ground short circuit failures, the emulation of AC phase ground short circuit failures Oscillogram 205, the simulation waveform 206 of BC phase ground short circuit failures, the simulation waveform 207 of AB phase faults, AC phases Between short trouble simulation waveform 208, the simulation waveform 209 of BC phase faults, the emulation ripple of ABC phase short troubles Shape Figure 21 0.It is right with the 2nd characteristic vector institute in the phase-model transformation matrix of the present invention on the circuit in the case of 10 kinds of failures more than The simulation waveform for the circuit mould electric current answered is visible, and phase-model transformation matrix of the invention is merely with corresponding to its 2nd characteristic vector Modulus can just reflect all types of failures occurred on three-phase line, connect comprising single-phase grounding fault, two Ground short circuit failure, two-phase phase fault, three phase short circuit fault.

Claims (4)

  1. A kind of 1. phase-model transformation matrix design method for three-phase power circuit fault diagnosis, it is characterised in that：The design Method is first depending on design criteria and the parameter of phase-model transformation matrix is designed and chosen；Then, to complete parameter designing with The phase-model transformation matrix of selection carries out Gram-Schmidt orthogonalization process, obtains the final design of required phase-model transformation matrix As a result.
  2. 2. according to the phase-model transformation matrix design method described in claim 1, it is characterised in that：It is described to phase-model transformation matrix Parameter be designed with choose method be：For one section of both-end without branch, line parameter circuit value along thread path to being evenly distributed and The three-phase power circuit that three-phase uniformly replaces, the phase-model transformation matrix that can make the circuit are Φ, then the matrix is one three Rank real number square formation, its expression formula are：
    Wherein,For i-th of column vector in matrix Φ,For in matrix Φ the i-th row jth arrange element, i=1,2,3, j=1, 2、3；The design of each element in matrix Φ with select the criterion of institute's foundation for：
  3. 3. according to the phase-model transformation matrix design method described in claim 2, it is characterised in that：Described phase-model transformation matrix Φ, the phase-model transformation relational expression of line node voltage and node current based on matrix Φ are：
    <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>U</mi> <mo>=</mo> <mi>&amp;Phi;</mi> <mo>&amp;CenterDot;</mo> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>I</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>&amp;Phi;</mi> <mi>T</mi> </msup> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>&amp;CenterDot;</mo> <mover> <mi>I</mi> <mo>&amp;OverBar;</mo> </mover> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>&amp;DoubleLeftRightArrow;</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mo>=</mo> <msup> <mi>&amp;Phi;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>&amp;CenterDot;</mo> <mi>U</mi> </mtd> </mtr> <mtr> <mtd> <mover> <mi>I</mi> <mo>&amp;OverBar;</mo> </mover> <mo>=</mo> <msup> <mi>&amp;Phi;</mi> <mi>T</mi> </msup> <mo>&amp;CenterDot;</mo> <mi>I</mi> </mtd> </mtr> </mtable> </mfenced> </mrow>
    Wherein, U and I is respectively the node voltage matrix and node current matrix on the circuit at a certain node,WithRespectively Node mode voltage and node mould current matrix after phase-model transformation decouples at the node, and have：
    <mrow> <mi>U</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>U</mi> <mi>A</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mi>B</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mi>C</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mi>I</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>I</mi> <mi>A</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mi>B</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mi>C</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mi>&amp;alpha;</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mi>&amp;beta;</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>U</mi> <mo>&amp;OverBar;</mo> </mover> <mi>&amp;gamma;</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mover> <mi>I</mi> <mo>&amp;OverBar;</mo> </mover> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mover> <mi>I</mi> <mo>&amp;OverBar;</mo> </mover> <mi>&amp;alpha;</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>I</mi> <mo>&amp;OverBar;</mo> </mover> <mi>&amp;beta;</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>I</mi> <mo>&amp;OverBar;</mo> </mover> <mi>&amp;gamma;</mi> </msub> </mtd> </mtr> </mtable> </mfenced> </mrow>
    Wherein, UA、UBAnd UCA, B and C phase voltage respectively at the node；IA、IBAnd ICA, B and C respectively at the node Phase current；Andα, β and γ mode voltage respectively at the node；Andα, β respectively at the node And γ mould electric currents.
  4. 4. according to the phase-model transformation matrix design method described in claim 1, it is characterised in that：It is described to completing parameter designing The method that Gram-Schmidt orthogonalization process is carried out with the phase-model transformation matrix of selection is as follows：
    Step 1, the phase-model transformation matrix that will be completed parameter designing and choose are designated as matrix Φ, and its expression formula is：
    Step 2, for matrix Φ, calculated according to below equation：
    <mrow> <msub> <mover> <mi>&amp;tau;</mi> <mo>&amp;RightArrow;</mo> </mover> <mi>i</mi> </msub> <mo>=</mo> <msub> <mover> <mi>&amp;phi;</mi> <mo>&amp;RightArrow;</mo> </mover> <mi>i</mi> </msub> <mo>-</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <mo>{</mo> <mfrac> <mrow> <mo>&amp;lsqb;</mo> <msub> <mover> <mi>&amp;phi;</mi> <mo>&amp;RightArrow;</mo> </mover> <mi>i</mi> </msub> <mo>&amp;CenterDot;</mo> <msub> <mover> <mi>&amp;phi;</mi> <mo>&amp;RightArrow;</mo> </mover> <mi>k</mi> </msub> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mo>&amp;lsqb;</mo> <msub> <mover> <mi>&amp;phi;</mi> <mo>&amp;RightArrow;</mo> </mover> <mi>k</mi> </msub> <mo>&amp;CenterDot;</mo> <msub> <mover> <mi>&amp;phi;</mi> <mo>&amp;RightArrow;</mo> </mover> <mi>k</mi> </msub> <mo>&amp;rsqb;</mo> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <msub> <mover> <mi>&amp;phi;</mi> <mo>&amp;RightArrow;</mo> </mover> <mi>k</mi> </msub> <mo>}</mo> </mrow>
    Matrix Φ is changed into an orthogonal matrix Γ by above-mentioned calculating process, and its expression formula is：
    <mrow> <mi>&amp;Gamma;</mi> <mo>=</mo> <mo>&amp;lsqb;</mo> <mtable> <mtr> <mtd> <msub> <mover> <mi>&amp;tau;</mi> <mo>&amp;RightArrow;</mo> </mover> <mn>1</mn> </msub> </mtd> <mtd> <msub> <mover> <mi>&amp;tau;</mi> <mo>&amp;RightArrow;</mo> </mover> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mover> <mi>&amp;tau;</mi> <mo>&amp;RightArrow;</mo> </mover> <mn>3</mn> </msub> </mtd> </mtr> </mtable> <mo>&amp;rsqb;</mo> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>t</mi> <mn>11</mn> </msub> </mtd> <mtd> <msub> <mi>t</mi> <mn>12</mn> </msub> </mtd> <mtd> <msub> <mi>t</mi> <mn>13</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>t</mi> <mn>21</mn> </msub> </mtd> <mtd> <msub> <mi>t</mi> <mn>22</mn> </msub> </mtd> <mtd> <msub> <mi>t</mi> <mn>23</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>t</mi> <mn>31</mn> </msub> </mtd> <mtd> <msub> <mi>t</mi> <mn>32</mn> </msub> </mtd> <mtd> <msub> <mi>t</mi> <mn>33</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </mrow>
    Wherein,For i-th of column vector in matrix Γ, tijFor in matrix Γ the i-th row jth arrange element, i=1,2,3, j=1, 2、3；
    Step 3, for matrix Γ, calculated according to below equation：
    <mrow> <msub> <mover> <mover> <mi>&amp;phi;</mi> <mo>^</mo> </mover> <mo>&amp;RightArrow;</mo> </mover> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <msub> <mover> <mi>&amp;tau;</mi> <mo>&amp;RightArrow;</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>|</mo> <msub> <mover> <mi>&amp;tau;</mi> <mo>&amp;RightArrow;</mo> </mover> <mi>i</mi> </msub> <mo>|</mo> </mrow> </mfrac> </mrow>
    Above-mentioned calculating process standardizes matrix Γ units, finally gives and exports required phase-model transformation matrixIt is expressed Formula is：
    Wherein,For matrixI-th interior of column vector,For matrixThe element of interior i-th row jth row；I=1,2,3, j=1, 2、3。
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