CN108808702A - Time-lag power system electromechanic oscillation mode computational methods based on low order IGD-LMS algorithms - Google Patents
Time-lag power system electromechanic oscillation mode computational methods based on low order IGD-LMS algorithms Download PDFInfo
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
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Abstract
The invention discloses the time-lag power system electromechanic oscillation mode computational methods based on low order IGD-LMS algorithms, including：It establishes the dynamic model of time-lag power system and linearly turns to one group of differential equations with delay；By the state variable rearrangement of time-lag power system in differential equations with delay, it is divided into the state variable unrelated with time lag and state variable related with time lag；The characteristic value of time-lag power system is calculated and is converted into the characteristic value for calculating infinitesimal generator；Discretization is carried out to infinitesimal generator, infinitesimal generator low order discretization matrix is obtained and then obtains infinitesimal generator low order discretization inverse matrix；Sparse eigenvalue is carried out for infinitesimal generator low order discretization inverse matrix and calculates acquisition approximate eigenvalue, and pairing approximation characteristic value is modified, and obtains the accurate profile value of time-lag power system, and accurate profile value is the electromechanic oscillation mode of corresponding time-lag power system.
Description
Technical field
The present invention relates to technical field of power system computation, more particularly to the time lag electricity based on low order IGD-LMS algorithms
Force system electromechanic oscillation mode computational methods, IGD-LMS are " Infinitesimal Generator Discretization
The english abbreviation of With Linear Multistep ", Chinese meaning are：Infinitesimal generator linear multi step discretization.
Background technology
The trend that modern power systems develop to extensive interconnected network makes section low-frequency oscillation be increasingly becoming restriction electricity
The bottleneck of Force system small signal stability.For interconnecting transmission system, traditional power system stabilizer, PSS (Power on a large scale
System Stabilizer, PSS) local low frequency oscillation can be effectively treated, but wide area measurement information is not accounted for, it can not
Effectively weaken section low-frequency oscillation.
Wide Area Measurement System (Wide-Area based on synchronized phasor unit (Phasor Measurement Unit, PMU)
Measurement System, WAMS) can real-time remote acquire dynamic parameter, be extensive interconnected electric power system state
New information platform is built in perception.Interconnected network low-frequency oscillation control based on the WAMS Wide-area Measurement Informations provided, has by introducing
The wide area feedback signal of effect reflection inter-area oscillation mode, can obtain preferable damping control performance, be wide area protection and coordination
Control provides new realization rate.
Time delay is the inherent characteristic of information system.Wide area measurement system signal exists in acquisition, routing, transmission and processing procedure
Tens of to hundreds of milliseconds of communication delay.Electric system is therefore as the electric system (delay of time lag information physics fusion
Cyber-physical power system, DCPPS).Time lag is that the failure of system control law, operation conditions is caused to deteriorate and be
A kind of major incentive for unstability of uniting.Therefore, when carrying out electric system closed-loop control using wide area measurement information, it is necessary to meter and time lag
Influence.
Consider that the power system small signal stability analysis method of time-delay has two major classes, time domain method and frequency domain method.When
Domain method is that the stability analysis and controller design method using relatively broad time-lag power system utilize in principle
Krasovskii and Razumikhin theorems propose criterion, with the stability of decision-making system.Frequency domain method, it is intended to by calculating electric power
The characteristic value of system, stability of the analysis electric system near operating point.According to different, the frequency domain to Time Delay processing mode
Method can be further divided into transformation method, Matrix Pencil and spectrum 3 class of discretization method.It is well known that time lag system belongs to typical
Infinite dimensional system.Exponential term in time-lag power system characteristic equation shows it as transcendental equation and has infinite multiple characteristic values.
In order to avoid the difficulty of direct solution, transformation method utilizes such as Rekasius transformation (also referred to as bilinear transformation or feature
Root clustering procedure), Lambert-W functions, Pad é approximations exponential term is transformed to rational polynominal or Lambert functions.
In recent years, numerical analysis and calculate art of mathematics research and propose based on spectrum discretization time lag system part spy
Value indicative computational methods, can be carried out effectively feature calculation, obtain low-frequency oscillation of electric power system pattern, be used for small signal stability
Analysis.Chinese invention patent is based on the approximate time-lag power system characteristic value calculating of Pad é and Convenient stable criterion
.201210271783.8:[P] approaches Time Delay using Pad é approximation polynomials, and then the key of the computing system rightmost side is special
Value indicative, and judge the time lag stability of system.Extensive time-lag power system characteristic value meter of the Chinese invention patent based on EIGD
Calculation method .201510055743.3 [P] proposes a kind of based on display IGD (Explicit Infinitesimal
Generator Discretization, EIGD) extensive time-lag power system characteristic value calculate.Calculated system
The crucial Oscillatory mode shape of system, it can be determined that stability of the system under fixed time lag.
However, the discretization matrix dimension that these time lag Eigenvalues analysis methods generate is larger, time and calculation amount are calculated
It is larger, it can not efficiently carry out characteristic value calculating.
Invention content
In order to solve the deficiencies in the prior art, the present invention provides the time-lag power systems based on low order IGD-LMS algorithms
Electromechanic oscillation mode computational methods, to carry out the stability analysis of extensive time-lag power system.Low order IGD-LMS algorithms are logical
It crosses and eliminates the item unrelated with time lag to obtain the discretization matrix of low order, efficiently realize IGD-LMS methods, solve original spectrum
The intrinsic matrix dimension of discretization method larger the problem of limiting its efficiency, to for calculating extensive time-lag power system
Electromechanic oscillation mode.
Time-lag power system electromechanic oscillation mode computational methods based on low order IGD-LMS algorithms, including：
It establishes the dynamic model of time-lag power system and linearly turns to one group of differential equations with delay；
By the state variable rearrangement of time-lag power system in differential equations with delay, it is divided into the state unrelated with time lag
Variable and state variable related with time lag, when carrying out discretization to the time-lag power system state of last time, reject with
The discretization of the unrelated state variable of time lag, the dimension reduction of corresponding discretization matrix；
Differential equations with delay is converted into ODE using infinitesimal generator, thus by the spy of time-lag power system
Sign equation equivalence is converted to general characteristics equation, and general characteristics equation is described by infinitesimal generator, and then by time lag electric power
The characteristic value of system calculates the characteristic value for being converted into and calculating infinitesimal generator；
Due to the characteristic value of infinitesimal generator have it is infinite multiple, need to infinitesimal generator carry out discretization, first,
Delay interval is divided into discrete function space by multiple discrete points on delay interval, to turn to continuous function is discrete
Piecemeal vector, then the functional value at Approximation Discrete point, arranges and the state equation rewritten at these discrete points obtains infinitesimal
First low order discretization matrix is generated, to convert infinite dimensional eigenvalue problem to the eigenvalue problem of finite dimension；
Displacement-inverse transformation pretreatment is carried out for infinitesimal generator low order discretization matrix, obtains infinitesimal generator
Low order discretization approximation inverse matrix, by the corresponding feature near the complex plane imaginary axis of time-lag power system electromechanic oscillation mode
Value is converted to the larger characteristic value of modulus value；
Sparse eigenvalue, which is carried out, for infinitesimal generator low order discretization approximation inverse matrix calculates acquisition approximate eigenvalue,
During sparse realization, matrix inversion-vector product in iterative solution calculating；
Pairing approximation characteristic value is modified, and obtains the accurate profile value of time-lag power system, accurate profile value i.e. to it is corresponding when
The electromechanic oscillation mode of stagnant electric system.
Further preferred technical solution, the dynamic model for establishing time-lag power system simultaneously linearly turn to one group of time lag
The differential equation, specially：Time-lag power system model is subjected to Taylor expansion at equalization point, the system after being linearized is dynamic
States model describes the dynamic model of time-lag power system by one group of differential equations with delay.
Further preferred technical solution, it is described that time-lag power system model is indicated by differential equations with delay, specially：
In formula：x∈Rn×1For the state variable vector of electric system；N is system state variables sum；When t is current operation
It carves；M is time lag number；τ1,…,τmFor time lag constant, τmaxIndicate maximum time lag；Δ x (t) is t moment system state variables
Increment；Δx(t-τi) it is t- τiThe increment of moment system state variables；For leading for t moment system state variables increment
Number；Δ x (0) is initial value, that is, original state of system state variables, and is abbreviated as WithDense systematic observation matrix and sparse hangover state matrix are indicated respectively.
System state variables x (t) is divided into and time lag outlier by further preferred technical solutionWith with
Time lag continuous itemAnd meet n1+n2=n, then the system state equation rewriting in formula (1) are reassembled as：
In formula (2)WithBe byWithRewriting is reassembled as related and unrelated with time lag
The matrix that part obtains, as follows：
In formula： WithIt is indicated respectively according to shape
The matrix in block form that the division of state variable is formed after being recombinated to matrix.The characteristic equation of linearized system that formula (2) indicates is：
In formula：λ is characterized value, and v is characterized the corresponding right feature vector of value.
Further preferred technical solution, infinitesimal generatorDefinition：
In formula：DomainFor the dense subspace of X；
Utilize infinitesimal generatorIt is following formula that formula (2) can be changed：
In formula：u(t):[0, ∞) → X is continuously differentiable function, it is the solution of formula (7), and u (t)=Δ x (t+ θ), θ ∈
[-τmax,0]。
Relationship between the characteristic value and infinitesimal generator characteristic value of the time-lag power system model is：
In formula：λ is the characteristic value of time-lag power system, and σ () indicates that the spectrum of infinitesimal generator, the formula (8) illustrate to ask
The characteristic value for solving time-lag power system is to calculate infinitesimal generatorCharacteristic value.
Further preferred technical solution, due to solvingCharacteristic value be Infinite-dimensional problem, need use linear multi step
Discretization method pairDiscretization is carried out, finite dimension approximate matrix is obtainedAnd then its characteristic value is calculated, as time lag electric power
The approximation of system accurate profile value.
Further preferred technical solution establishes discrete point set when carrying out discretization to infinitesimal generator, will be infinite
Small generation member carries out discretization at each discrete point：
In section [- τmax, 0] on use disperse segmentaly thought, establish discrete point set omegaN：
In formula：θj,iIndicate each discrete point,For section [- τi,-τi-1] between
Away from for hiNiThe set that a discrete point is constituted.
Further preferred technical solution, in order to ensure the availability of LMS methods, the discrete points N on subintervali(i=
1 ..., m) have to be larger than the step number k, i.e. N of linear multistep method BDF methodsi>k。
Further preferred technical solution, the iterative formula based on linear multistep method, and according to the basic think of of low order algorithm
Road carries out discretization to infinitesimal generator, only retains discretization related with time lag part, realize the drop to discretization matrix
Rank obtains the discretization matrix of low order corresponding with infinitesimal generatorOrder of matrix number is (n+Nn2), it indicates as follows：
In formula：N is given positive integer,Indicate Kronecker product operation,It is n2Rank unit matrix,
It is Nn2×n1Null matrix,It is matrixFirst piece of row；SubmatrixIt is that height is sparse
Matrix, it is unrelated with systematic observation matrix.
∑NIt can indicate as follows：
In formula：For the dense state matrix after recombination,For hangover state matrix after recombination
Rear n2Row,For unit vector；
Further preferred technical solution, using displacement-inverse transformation technology by time-lag power system electromechanic oscillation mode pair
The smaller characteristic value of the modulus value answered is converted to the larger characteristic value of dominant eigenvalue i.e. modulus value, the specific steps are：
First, displacement point s is given, the eigenvalue λ of time-lag power system is substituted with λ '+s, then the spy after displacement can be obtained
Equation is levied, i.e.,：
In formula：
After displacement operation,Partitioned organization do not change, for different infinitesimal generator discretization schemes,
It willIn expression formulaWithIt is replaced, you can obtain infinite after displacement operation
The small discretization matrix for generating memberThe infinitesimal generator discretization approximate matrix that IGD-LMS methods obtainIt is mapped
ForIn turn, their inverse matrix is expressed as：
In formula (18)：
In formula (19)：By the state matrix after recombinatingIt is obtained through shift transformation,ForIt is obtained after shift transformation
MatrixRear n2Row；
In formula,For matrixRear n2Arrange n2The matrix that row is constituted.Further preferred technical solution utilizes sparse calculation
Method is soughtModulus value the best part characteristic value, in IRA algorithms, the maximum operation of calculation amount is to utilizeWith to
It measures product and forms Krylov subspace；If k-th of Krylov vector isThen+1 Krylov vectors q of kthk+1
It can calculate as follows：
Due to matrixWithout special logical construction,Without explicit expression form, for it is extensive when
Stagnant electric system is calculated using direct inversion techniqueInverse matrix when, it is very high to request memory, and cannot make full use of
The sparse characteristic of system augmented state matrix；
In order to avoid direct solutionQ is calculated using alternative mannerk+1, then, formula (21) is converted to：
In formula：It is q after the l times iterationk+1Approximation；
The advantage of iterative solution is during solving system of linear equations, not increasing any element, maintain's
Sparse characteristic is calculated using induction dimension reduction methodIt is as follows：
First, willIn rear (N+1) n2Row element is rearranged according to the direction of row, obtains matrixI.e.Operator vec () is indicated each of matrix
Leie is arranged in a vector.
Then matrix Q is taken1Rear N row, be expressed as matrixIn turn, the left end of formula (22)
It is calculated using the property of Kronecker product：A, B and C is enabled to meet AXB=C, then Then, formula (22) is converted to：
In formula (23), the maximum operation of calculation amount is matrix-vector productUsing the power method of sparse realization
It is calculated, reduces computation burden, improves computational efficiency.
Further preferred technical solution, if what IRA algorithms were calculatedCharacteristic value be λ ", thenIt is close
It is like characteristic value：
" the vector that the preceding n element of corresponding Krylov vectors is formed with λThe corresponding features of accurate profile value λ to
The approximation for measuring v, withWithFor initial value accurate profile value λ and corresponding feature vector are obtained using Newton method by iteration
v。
Compared with prior art, the beneficial effects of the invention are as follows：
The first, low order IGD-LMS algorithms proposed by the present invention are for calculating the corresponding pass of real system electromechanic oscillation mode
When key characteristic value, the scale of real system and the influence of communication delay have been fully considered.
The second, low order IGD-LMS algorithms proposed by the present invention are gone on the basis of IGD-LMS algorithms using the thinking of depression of order
Except the state variable unrelated with time lag, the dimension of discretization matrix is greatly reduced, dimension is dropped into approximation and no time lag system
The dimension of state matrix is essentially identical, reduces and calculates time and calculation amount.
It is using displacement inverse transformation technology that part modulus value is smaller in third, low order IGD-LMS algorithms proposed by the present invention
Characteristic value is converted to the larger characteristic value of modulus value, preferentially calculates the low frequency oscillation mode of the real system of needs.
4th, low order IGD-LMS algorithms proposed by the present invention carry out sparse eigenvalue calculating using IRA algorithms, and use
Iterative algorithm carries out the product calculation of matrix inverse vector, using the sparse characteristic of systematic observation matrix and discretization matrix, reduces
Calculation amount.
5th, method of the invention is that the time-lag power system electromechanic oscillation mode based on low order IGD-LMS algorithms corresponds to
Feature value calculating method, core innovative point is following various technologies being combined together, and utilizes the thought of low order algorithm, will
Infinite dimensional time-lag power system is converted to the discretization matrix of finite dimension, and displacement inverse transformation technology is used to it, then leads to
It crosses sparse features value calculating method and calculates its characteristic value, finally verify to obtain the accurate profile of time-lag power system by newton
Value.
Description of the drawings
The accompanying drawings which form a part of this application are used for providing further understanding of the present application, and the application's shows
Meaning property embodiment and its explanation do not constitute the improper restriction to the application for explaining the application.
Fig. 1 is the discrete point set omega in IGD-LMS methodsN；
Fig. 2 (a)-Fig. 2 (b) is the schematic diagram of displacement inverse transformation preconditioning technique；
Fig. 3 is the flow chart based on low order IGD-LMS algorithm time-lag power system electromechanic oscillation mode computational methods.
Specific implementation mode
It is noted that following detailed description is all illustrative, it is intended to provide further instruction to the application.Unless another
It indicates, all technical and scientific terms used herein has usual with the application person of an ordinary skill in the technical field
The identical meanings of understanding.
It should be noted that term used herein above is merely to describe specific implementation mode, and be not intended to restricted root
According to the illustrative embodiments of the application.As used herein, unless the context clearly indicates otherwise, otherwise singulative
It is also intended to include plural form, additionally, it should be understood that, when in the present specification using term "comprising" and/or " packet
Include " when, indicate existing characteristics, step, operation, device, component and/or combination thereof.
In a kind of typical embodiment of the application, as shown in Figure 1：Time lag electric power based on low order IGD-LMS algorithms
System electromechanical oscillations mode computation method will first establish discrete point set in discretization infinitesimal generator.Namely based on m
Time lag subinterval, with step-length h on each time lag subintervaliEstablish NiA discrete point, and then convert continuous space to discrete sky
Between, it lays a good foundation for subsequent discretization scheme.
As shown in Fig. 2 (a)-Fig. 2 (b)：The corresponding characteristic value of electromechanic oscillation mode of electric system, is usually located at complex plane
Near the imaginary axis.In order to preferentially calculate these critical eigenvalues, need to infinitesimal generator discretization matrixCarry out displacement-
Inverse transformation.In figure, λ ' indicates infinitesimal generator discretization matrix after displacement sCharacteristic value.By further inversion operation
Afterwards,Partial key eigenvalue λ near displacement point s translates intoModulus value the best part eigenvalue λ ", and
Have
Fig. 3 is the flow chart based on low order IGD-LMS algorithm time-lag power system electromechanic oscillation mode computational methods, including
Following steps：
S1：Establish time-lag power system model；Then recombination time lag electricity is rewritten according to the correlation of state variable and time lag
Force system state equation.
S2：It is infinite according to calculating is converted into the problem of composing mapping principle, time-lag power system electromechanic oscillation mode will be calculated
The small eigenvalue problem for generating member.
S3：Based on the theory of linear multi step discretization scheme, discrete point set is initially set up, by infinitesimal generator every
At a discrete point carry out discretization, in discretization process, extract with the relevant part of time lag, to obtain infinitely small generation
The low order discretization matrix of member.
S4：Using the inverse preconditioning technique of displacement-, electric system electromechanical oscillations pattern is converted into the larger feature of modulus value
Value preferentially calculates the critical eigenvalue that these real parts are smaller, damping is weaker.
S5：Sparse eigenvalue calculating is carried out using such as IRA scheduling algorithms；During sparse realization, square is iteratively solved
Inverse-vector product the operation of battle array.
S6：Eigenvalue λ is obtained from step S5 " later, to be verified by newton, finally obtain time-lag power system model
Accurate profile value λ.
Time-lag power system model in the step S1 is as follows：
In formula：For the state variable vector of electric system, n is system state variables sum.T is current time.0
<τ1<τ2<…<τi…<τmFor the time lag constant of Time Delay, τmaxIndicate maximum time lag.For systematic observation matrix,
For dense matrix.It is sparse matrix for system time lags state matrix.Δ x (t) is t moment system shape
The increment of state variable, Δ x (t- τi) it is t- τiThe increment of moment system state variables,It is led for t moment system state variables
Several increments.Δ x (0) is the initial value (i.e. primary condition) of system state variables, and is abbreviated as
System state variables x (t) is divided into and time lag outlierWith with time lag continuous itemAnd meet n1+n2=n, then the system state equation in formula (1), which can be rewritten, is reassembled as：
In formula (2)WithBe byWithRewriting is reassembled as related and unrelated with time lag
The matrix that part obtains, as follows：
In formula： WithIt is indicated respectively according to shape
The matrix in block form that the division of state variable is formed after being recombinated to matrix.The characteristic equation of linearized system that formula (2) indicates is：
In formula：λ is characterized value, and v is characterized the corresponding right feature vector of value.
In the step S2, infinitesimal generatorIt can define：
In formula：DomainFor the dense subspace of X.
Utilize infinitesimal generatorIt is following formula that formula (2) can be changed：
In formula：u(t):[0, ∞) → X is continuously differentiable function, it is the solution of formula (7).And u (t)=Δ x (t+ θ), θ ∈
[-τmax,0]。
Relationship between the characteristic value and infinitesimal generator characteristic value of the time-lag power system model is：
In formula：λ is the characteristic value of electric system.σ () indicates the spectrum of infinitesimal generator.The formula illustrates to solve power train
The characteristic value of system is to calculateCharacteristic value.
Due to solvingCharacteristic value be Infinite-dimensional problem, need use linear multi step discretization method pairIt carries out discrete
Change, obtains finite dimension approximate matrixAnd then its characteristic value is calculated, the approximation as electric system accurate profile value.
The step S3：First, in section [- τmax, 0] on use disperse segmentaly thought, establish discrete point set
ΩN：
In formula：θj,iIndicate each discrete point,For section [- τi,-τi-1] between
Away from for hiNiThe set that a discrete point is constituted.It is worth noting that, in order to ensure the availability of LMS methods, on subinterval from
Dissipate points Ni(i=1 ..., m) has to be larger than the step number k of linear multistep method BDF methods, i.e. Ni>k。
Iterative formula based on linear multistep method, and according to the basic ideas of low order algorithm, infinitesimal generator is carried out
Discretization only retains discretization related with time lag part, and realization obtains and infinitesimal generator the depression of order of discretization matrix
Corresponding low order discretization matrixOrder of matrix number is (n+Nn2), it indicates as follows：
In formula：N is given positive integer,Indicate Kronecker product operation,It is n2Rank unit matrix,
It is Nn2×n1Null matrix.It is matrixFirst piece of row；SubmatrixIt is that height is sparse
Matrix, it is unrelated with systematic observation matrix.∑NIt can indicate as follows：
In formula：For the dense state matrix after recombination,For hangover state matrix after recombination
Rear n2Row,For unit vector.
In the step S4, further preferred technical solution, using displacement-inverse transformation technology by time-lag power system machine
The smaller characteristic value of the corresponding modulus value of electric oscillation pattern is converted into the larger characteristic value of modulus value.
First, displacement point s is given, the eigenvalue λ of time-lag power system is substituted with λ '+s, then the spy after displacement can be obtained
Equation is levied, i.e.,：
In formula：
After displacement operation,Partitioned organization do not change.For different infinitesimal generator discretization schemes,
It willIn expression formulaWithIt is replaced, you can obtain infinite after displacement operation
The small discretization matrix for generating memberThe infinitesimal generator discretization approximate matrix that IGD-LMS methods obtainIt is mapped
ForIn turn, their inverse matrix is represented by：
In formula (18)：
In formula (19)：By the state matrix after recombinatingIt is obtained through shift transformation,ForIt is obtained after shift transformation
MatrixRear n2Row；
In formula,For matrixRear n2Arrange n2The matrix that row is constituted.
The step S5：Common QR algorithms are only applicable to medium-scale electric system, utilize the Corresponding Sparse Algorithms such as IRA
It seeksModulus value the best part characteristic value.In IRA algorithms, the maximum operation of calculation amount is to utilizeWith vector
Product forms Krylov subspace.If k-th of Krylov vector isThen+1 Krylov vectors q of kthk+1It can
It calculates as follows：
Due to matrixWithout special logical construction,Without explicit expression form.For it is extensive when
Stagnant electric system is calculated using direct inversion techniqueInverse matrix when, it is very high to request memory, and cannot make full use of
The sparse characteristic of system augmented state matrix.
In order to avoid direct solutionUse alternative manner to calculate q herek+1.Then, formula (21) is converted to：
In formula：It is q after the l times iterationk+1Approximation.
The advantage of iterative solution is during solving system of linear equations, not increasing any element, maintain's
Sparse characteristic.Here it is calculated using induction dimensionality reduction (Induced Dimension Reduction, IDR (s)) methodSpecifically
Steps are as follows.
First, willIn rear (N+1) n2Row element is rearranged according to the direction of row, obtains matrixI.e.Operator vec () is indicated each of matrix
Leie is arranged in a vector.
Then matrix Q is taken1Rear N row, matrix can be expressed asIn turn, formula (22)
Left end can utilize the property of Kronecker product to be calculated：A, B and C is enabled to meet AXB=C, then Then, formula (22) can be exchanged into：
In formula (23), the maximum operation of calculation amount is matrix-vector productThe power of sparse realization can be used
Method is calculated, and is reduced computation burden, is improved computational efficiency.
In the step S6, if what IRA algorithms were calculatedCharacteristic value be λ ", thenApproximate eigenvalue
For：
" the vector that the preceding n element of corresponding Krylov vectors is formed with λThe corresponding features of accurate profile value λ to
Measure the good approximation of v.WithWithFor initial value, accurate profile value λ and corresponding can be obtained by iteration using Newton method
Feature vector v.
The foregoing is merely the preferred embodiments of the application, are not intended to limit this application, for the skill of this field
For art personnel, the application can have various modifications and variations.Within the spirit and principles of this application, any made by repair
Change, equivalent replacement, improvement etc., should be included within the protection domain of the application.
Claims (10)
1. the time-lag power system electromechanic oscillation mode computational methods based on low order IGD-LMS algorithms, characterized in that including：
It establishes the dynamic model of time-lag power system and linearly turns to one group of differential equations with delay；
By the state variable rearrangement of time-lag power system in differential equations with delay, it is divided into the state variable unrelated with time lag
State variable related with time lag, when the time-lag power system state to last time carries out discretization, rejecting and time lag
The discretization of unrelated state variable, the dimension reduction of corresponding discretization matrix；
Differential equations with delay is converted into ODE using infinitesimal generator, thus by the feature side of time-lag power system
Journey equivalence is converted to general characteristics equation, and general characteristics equation is described by infinitesimal generator, and then by time-lag power system
Characteristic value calculate be converted into calculate infinitesimal generator characteristic value；
Due to the characteristic value of infinitesimal generator have it is infinite multiple, need to infinitesimal generator carry out discretization pass through first
Delay interval is divided into discrete function space by multiple discrete points on delay interval, to turn to piecemeal by continuous function is discrete
Vector, the then functional value at Approximation Discrete point arrange and rewrite the state equation at these discrete points and obtain infinitely small generation
First low order discretization matrix, to convert infinite dimensional eigenvalue problem to the eigenvalue problem of finite dimension；
Displacement-inverse transformation pretreatment is carried out for infinitesimal generator low order discretization matrix, obtains infinitesimal generator low order
Discretization approximation inverse matrix turns the corresponding characteristic value near the complex plane imaginary axis of time-lag power system electromechanic oscillation mode
It is changed to dominant eigenvalue；
Sparse eigenvalue is carried out for infinitesimal generator low order discretization approximation inverse matrix and calculates acquisition approximate eigenvalue, dilute
It dredges during realizing, matrix inversion-vector product in iterative solution calculating；
Pairing approximation characteristic value is modified, and obtains the accurate profile value of time-lag power system, the i.e. corresponding time lag electricity of accurate profile value
The electromechanic oscillation mode of Force system.
2. the time-lag power system electromechanic oscillation mode computational methods based on low order IGD-LMS algorithms as described in claim 1,
It is characterized in that the dynamic model for establishing time-lag power system and linearly turning to one group of differential equations with delay, specially：By when
Stagnant electric power system model carries out Taylor expansion at equalization point, the system dynamic model after being linearized, i.e., by one group when
The dynamic model of stagnant differential equation time-lag power system.
3. the time-lag power system electromechanic oscillation mode computational methods based on low order IGD-LMS algorithms as claimed in claim 2,
It is characterized in that described indicate time-lag power system model by differential equations with delay, specially：
In formula：x∈Rn×1For the state variable vector of electric system；N is system state variables sum；T is the current time of running；m
For time lag number；τ1,…,τmFor time lag constant, τmaxIndicate maximum time lag；Δ x (t) is the increasing of t moment system state variables
Amount；Δx(t-τi) it is t- τiThe increment of moment system state variables；For the derivative of t moment system state variables increment；Δ
X (0) is initial value, that is, original state of system state variables, and is abbreviated as WithPoint
Dense systematic observation matrix and sparse hangover state matrix are not indicated；
System state variables x (t) is divided into and time lag outlierWith with time lag continuous item
And meet n1+n2=n, then the system state equation rewriting in formula (1) are reassembled as：
In formula (2)WithBe byWithRewriting be reassembled as with time lag in relation to and unrelated part
Obtained matrix, as follows：
In formula：WithIt is indicated respectively according to state
The matrix in block form that the division of variable is formed after being recombinated to matrix, the characteristic equation of linearized system that formula (2) indicates are：
In formula：λ is characterized value, and v is characterized the corresponding right feature vector of value.
4. the time-lag power system electromechanic oscillation mode computational methods based on low order IGD-LMS algorithms as claimed in claim 3,
It is characterized in that infinitesimal generatorDefinition：
In formula：DomainFor the dense subspace of X；
Utilize infinitesimal generatorIt is following formula that formula (2) can be changed：
In formula：u(t):[0, ∞) → X is continuously differentiable function, it is the solution of formula (7), and u (t)=Δ x (t+ θ), θ ∈ [-
τmax,0]；
Relationship between the characteristic value and infinitesimal generator characteristic value of the time-lag power system model is：
In formula：λ is the characteristic value of time-lag power system, and σ () indicates the spectrum of infinitesimal generator, when which illustrates to solve
The characteristic value of stagnant electric system is to calculate infinitesimal generatorCharacteristic value；
Due to solvingCharacteristic value be Infinite-dimensional problem, need use linear multi step discretization method pairDiscretization is carried out,
Obtain finite dimension approximate matrixAnd then its characteristic value is calculated, the approximation as time-lag power system accurate profile value.
5. the time-lag power system electromechanic oscillation mode computational methods based on low order IGD-LMS algorithms as claimed in claim 2,
It is characterized in that when carrying out discretization to infinitesimal generator, discrete point set is established, by infinitesimal generator in each discrete point
Place carries out discretization：
In section [- τmax, 0] on use disperse segmentaly thought, establish discrete point set omegaN：
In formula：θj,iIndicate each discrete point,For section [- τi,-τi-1] on spacing be
hiNiThe set that a discrete point is constituted；
In order to ensure the availability of LMS methods, the discrete points N on subintervali(i=1 ..., m) have to be larger than linear multistep method
The step number k of BDF methods, i.e. Ni>k。
6. the time-lag power system electromechanic oscillation mode computational methods based on low order IGD-LMS algorithms as claimed in claim 5,
It is characterized in that the iterative formula based on linear multistep method, and according to the basic ideas of low order algorithm, infinitesimal generator is carried out
Discretization only retains discretization related with time lag part, and realization obtains and infinitesimal generator the depression of order of discretization matrix
The discretization matrix of corresponding low orderOrder of matrix number is (n+Nn2), it indicates as follows：
In formula：N is given positive integer,Indicate Kronecker product operation,It is n2Rank unit matrix,It is Nn2
×n1Null matrix,It is matrixFirst piece of row；SubmatrixIt is the sparse square of height
Battle array, it is unrelated with systematic observation matrix；
∑NIt can indicate as follows：
In formula：For the dense state matrix after recombination,For hangover state matrix after recombinationRear n2
Row,For unit vector；
7. the time-lag power system electromechanic oscillation mode computational methods based on low order IGD-LMS algorithms as described in claim 1,
It is characterized in that converting the smaller characteristic value of the corresponding modulus value of time-lag power system electromechanic oscillation mode to the larger feature of modulus value
Value；The specific steps are：
First, displacement point s is given, the eigenvalue λ of time-lag power system is substituted with λ '+s, then the feature side after displacement can be obtained
Journey, i.e.,：
In formula：
After displacement operation,Partitioned organization do not change, will for different infinitesimal generator discretization schemes
In expression formulaWithIt is replaced, you can obtain the infinite your pupil after displacement operation
The discretization matrix of Cheng YuanThe infinitesimal generator discretization approximate matrix that IGD-LMS methods obtainIt is mapped asIn turn, their inverse matrix is expressed as：
In formula (18)：
In formula (19)：By the state matrix after recombinatingIt is obtained through shift transformation,ForThe square obtained after shift transformation
Battle arrayRear n2Row；
In formula,For matrixRear n2Arrange n2The matrix that row is constituted.
8. the time-lag power system electromechanic oscillation mode computational methods based on low order IGD-LMS algorithms as claimed in claim 7,
It is characterized in that being sought using Corresponding Sparse AlgorithmModulus value the best part characteristic value, in IRA algorithms, calculation amount is maximum
Operation is to utilizeKrylov subspace is formed with vector product；If k-th of Krylov vector isThen
+ 1 Krylov vectors q of kthk+1It can calculate as follows：
Due to matrixWithout special logical construction,Without explicit expression form, for extensive time lag electricity
Force system is calculated using direct inversion techniqueInverse matrix when, it is very high to request memory, and system cannot be made full use of
The sparse characteristic of augmented state matrix；
In order to avoid direct solutionQ is calculated using alternative mannerk+1, then, formula (21) is converted to：
In formula：It is q after the l times iterationk+1Approximation.
9. the time-lag power system electromechanic oscillation mode computational methods based on low order IGD-LMS algorithms as claimed in claim 8,
It is characterized in that the advantage of iterative solution is during solving system of linear equations, not increasing any element, maintain
Sparse characteristic, using induction dimension reduction method calculateIt is as follows：
First, willIn rear (N+1) n2Row element is rearranged according to the direction of row, obtains matrixI.e.Operator vec () is indicated each of matrix
Leie is arranged in a vector；
Then matrix Q is taken1Rear N row, be expressed as matrixIn turn, the left end of formula (22) utilizes
The property of Kronecker product is calculated：A, B and C is enabled to meet AXB=C, then In
It is that formula (22) is converted to：
In formula (23), the maximum operation of calculation amount is matrix-vector productIt is carried out using the power method of sparse realization
It calculates.
10. the time-lag power system electromechanic oscillation mode calculating side based on low order IGD-LMS algorithms as claimed in claim 8
Method, characterized in that set what IRA algorithms were calculatedCharacteristic value be λ ", thenApproximate eigenvalue be：
" the vector that the preceding n element of corresponding Krylov vectors is formed with λIt is the corresponding feature vector v of accurate profile value λ
Approximation, withWithFor initial value accurate profile value λ and corresponding feature vector v are obtained using Newton method by iteration.
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