CN108242808A - Time-lag power system stability method of discrimination based on IGD-LMS - Google Patents
Time-lag power system stability method of discrimination based on IGD-LMS 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
- H02J3/00—Circuit arrangements for ac mains or ac distribution networks
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
- H02J3/00—Circuit arrangements for ac mains or ac distribution networks
- H02J3/24—Arrangements for preventing or reducing oscillations of power in networks
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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
- H02J2203/00—Indexing scheme relating to details of circuit arrangements for AC mains or AC distribution networks
- H02J2203/20—Simulating, e g planning, reliability check, modelling or computer assisted design [CAD]
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Abstract
The invention discloses the time-lag power system stability method of discrimination based on IGD LMS, including：Establish time-lag power system model；Discretization is carried out to infinitesimal generator using linear multistep method LMS, obtains the discretization matrix of infinitesimal generator；Infinite dimensional eigenvalue problem is converted into the eigenvalue problem of finite dimension；The approximation of the characteristic value of the discretization inverse of a matrix matrix modulus value maximum of infinitesimal generator being calculated using implicit Arnoldi algorithm；Sparse realization is carried out using displacement inverse transformation and the Kronecker property accumulated in calculating process；According to spectrum mapping relations, the approximation of the characteristic value of infinitesimal generator discretization matrix modulus value maximum is converted into the approximate eigenvalue of time-lag power system model；It is modified using Newton iteration method pairing approximation characteristic value, obtains the accurate profile value of time-lag power system；The stability of time-lag power system is judged according to the size of accurate profile value.
Description
Technical field
The present invention relates to linear multi step discretization (the Infinitesimal Generator based on infinitesimal generator
Discretization Method with Linear Multistep, IGD-LMS) time-lag power system stability differentiate
Method.
Background technology
With the rise of global energy internet, the scale of interconnected electric power system gradually increases, section low-frequency oscillation problem
More significantly.Traditional solution is installation power system stabilizer, PSS (Power System Stabilizer, PSS), due to
Its feedback control signal can effectively solve interregional low-frequency oscillation problem, but be unable to effective damping interconnection electricity from locality
The inter-area oscillations of Force system.
Extensive interconnecting electric power is given in the appearance of Wide Area Measurement System (Wide-Area Measurement System, WAMS)
The development of system stability analysis and control brings new opportunity.Interconnected network low-frequency oscillation based on the WAMS Wide-area Measurement Informations provided
Control effectively reflects the wide area feedback signal of inter-area oscillation mode by introducing, can obtain preferable damping control performance,
To solve the problems, such as the inter-area low-frequency oscillation in interconnected network, and then the ability to transmit electricity for improving system provides new control hand
Section has well and is widely applied prospect.
Wide area signal is in the WAMS communications being made of different communication medium (such as optical fiber, telephone wire, digital microwave, satellite)
When transmitting and handle in network, there are the communication delays changed between tens to hundreds of milliseconds.Time lag is that system control law is caused to lose
Effect, operation conditions deteriorate and a kind of major incentive of system unstability.Therefore, electric system closed loop is carried out using wide area measurement information
During control, it is necessary to the influence of meter and time lag.
In modern power systems, concern is primarily with electromechanical oscillations problems for small interference stability.Using state-space model as
The eigenvalue Method on basis is to study the powerful tool of electromechanical oscillations.At present, many calculating have been proposed in researcher
The effective ways of large-scale electrical power system Critical eigenvalues mainly include the selection mode analytic approach based on reduced order system,
AESOPS algorithms and S- matrix methods, the sequential methods such as power method, Newton method, Rayleigh Rayleigh quotient iterations and simultaneous iterative,
The subspace iteration methods such as Arnoldi algorithm, the double iterative decomposed again and Jacobi-Davidson methods.These methods exist
The openness of augmented state matrix is all utilized during calculating section characteristic value, most methods are all by carrying out spectrum change to original system
The distribution so as to change characteristic spectrum is changed, then asks for the characteristic value of system, then the key feature of original system is obtained by inverse transformation
Value.But method mentioned above does not consider the influence of time lag.Chinese invention patent is based on the approximate time lag power trains of Pad é
Characteristic value of uniting calculates and Convenient stable criterion .201210271783.8:[P] approaches time lag ring using Pade approximation polynomials
Section, and then the critical eigenvalue of the computing system rightmost side, and judge the time lag stability of system.Chinese invention patent is based on EIGD
Extensive time-lag power system feature value calculating method .201510055743.3. [P] propose it is a kind of based on display IGD
The extensive time-lag power system characteristic value of (Explicit IGD, EIGD) calculates.The calculated system rightmost side
Critical eigenvalue, it can be determined that stability of the system under fixed time lag.These time lag Convenient stable criterions, are required to pass through
Multiple-Scan [0.1,2.5] Hz low-frequency oscillations frequency range is interior, critical eigenvalue close to the imaginary axis, could judge the time lag of system
Stability.
Invention content
In order to solve the deficiencies in the prior art, the present invention provides the time-lag power system stabilities based on IGD-LMS to sentence
Other method；
To achieve these goals, the present invention adopts the following technical scheme that：
Time-lag power system stability method of discrimination based on IGD-LMS, including：
Step (1)：Establish time-lag power system model；According to the characteristic value of time-lag power system model and time lag power train
Spectrum mapping relations between the infinitesimal generator characteristic value of model of uniting convert the characteristic value for calculating time-lag power system model
Into the characteristic value for calculating infinitesimal generator；It is infinitely small that the problem of so as to will determine that time-lag power system stability, is converted into calculating
Generate the eigenvalue problem of the modulus value maximum of member；
Step (2)：Discretization is carried out to infinitesimal generator using linear multistep method LMS, obtains infinitesimal generator
Discretization matrix；Infinite dimensional eigenvalue problem is converted into the eigenvalue problem of finite dimension；
Step (3)：The discretization matrix of infinitesimal generator obtained using implicit Arnoldi algorithm calculating step (2)
Inverse matrix modulus value maximum characteristic value approximation；The property accumulated in calculating process using displacement-inverse transformation and Kronecker
Carry out sparse realization；
Step (4)：According to spectrum mapping relations, by the approximation of the characteristic value of infinitesimal generator discretization matrix modulus value maximum
Value is converted into the approximate eigenvalue of time-lag power system model；
Step (5)：It is modified using Newton iteration method pairing approximation characteristic value, obtains the accurate profile of time-lag power system
Value；
Step (6)：The stability of time-lag power system is judged according to the size of accurate profile value.
The time-lag power system model of the step (1) is：
In formula：For system mode.τi>0 is time lag constant.
Assuming thatWherein τmaxFor maximum time lag.
It is dense matrix for systematic observation matrix；It is height sparse matrix for system time lags state matrix.
IfTo be defined on the n dimensional linear vector spaces of complex field, if state spaceIt is
By delay interval [- τmax, 0] to n tie up real number spaceBanach (Banach) space that the continuous function of mapping is formed, and
Possesses supremum norm
Infinitesimal generator It can be defined as：
In formula：It is a linear functional：
The step of step (2) is：
According to the definition of infinitesimal generator, its approximate matrix namely infinitely small generation can be obtained by following methods
The discretization matrix of member.
Step (21)：Given positive integer N, utilizes section [- τmax, 0] on the different discrete points of N+1 form set omegasN,
ΩN={ θi, i=0,1..., N }, and then continuous state space X is converted into separate manufacturing firms
Step (22)：Given continuous functionIf its discrete approximation is
IfIts discrete approximation isSuitable mathematical method is selected to calculateIt is accurate
Derivative(i.e.) approximation ψ.Specifically, in discrete point θiAt (i=1 ..., N), ψ is utilizediTo approachThis point
Functional value：
In discrete point θ0At=0, obtained using the condition of splicing (1.5)Accurate derivativeAnd it is approximatelyI.e.
Step (23)：It willApproximation ψiIt is expressed asLinear combination, obtain：
In formula：ajFor constant, dijFor constant.
It is write formula (1.6) as matrix equation, is obtained：In fact, this is the discrete form of abstract Cauchy's equation.
The coefficient of equationAs infinitesimal generatorDiscretization matrix.
For single time lag system, have：θ0=0, θN=-τmax.At this point, matrixFirst block row can be simplified.
So far, the method for discussing infinitesimal generator discretization.
In order to make it easy to understand, (the Linear Multi-Step with based on backward difference linear multistep method are provided first
Backward Difference Formula, LMS-BDF) single time lag system infinitesimal generator discretization method, and then
This method is extended into Systems with Multiple Time-Delays.
The form for giving fixed step size h, k step BDF methods is as follows：
In formula：αl(l=0 ..., k) and βkCoefficient for linear k footworks.
Single time lag situation
Given positive integer N, the collection that the N+1 discrete point that spacing is h on section [- τ, 0] is formed are combined into ΩN.So as to continuously
State space X is converted into discrete space
In θ0At=0, functionDerivativeApproximationIt can be obtained by splicing condition (1.3).
In discrete point θjPlace, functionDerivativeApproximationBy infinitesimal generatorDefinition (1.2)
It obtains.
The expression of formula (1.11), and two kinds of situations can be divided into and derived.
First, in discrete point θjAt (j=k ..., N), ψjIt can be arranged to obtain by (1.8) to BDF methods.
Then, in discrete point θjAt (j=1 ..., k-1), ψ is calculated using formula (1.13) " startup " methodj.At this point,
Assuming that ψjWith the similar form of formula (1.12), i.e.,：
In formula：γjl(j=1 ..., k-1；L=0,1 ..., k) it is unknowm coefficient.
Determining γ is given belowjlMethod.
Near, it will be in formula (1.13) right endIt is launched into about step
Long h, cut-off error arePower series.
Formula (1.14) is substituted into formula (1.13), and enables the coefficient of the same power item of both members h equal, is obtained：
In formula：J=1 ..., k-1.
For the j, unknowm coefficient γ of some settingjlBy solving (k+1) corresponding with formula (1.15) × (k+1) rank
System of linear equations obtains；By coefficient gammajlIt is write as (k-1) × (k+1) rank matrixes, is obtained：
For example, for BDF methods, as k=3 and k=5, matrix Γ can be respectively obtained by calculating3And Γ5。
Simultaneous formula (1.10), formula (1.12) and formula (1.13), the relational expression being derived by between Φ and Ψ：
In formula：The discretization matrix of infinitesimal generator is tieed up for (N+1) n × (N+1) n.
Multiple time delay situation
Infinitesimal generator BDF discretization methods in the case of single time lag are expanded to containing m time lag τi(i=1 ...,
M) system.
First, in section [- τmax, 0] on establish discrete point set omegaN：
In formula： For section [- τi,-τi-1] on spacing be hiNiThe set that a discrete point is formed is
Ensure the availability of linear multistep method, the discrete points N on subintervaliHave to be larger than step number k, i.e. Ni>k。
According to ΩN, continuous space X is converted into discrete spaceAnd have
In θ0At=0, functionThe approximation ψ of derivative0, obtained by formula (1.3).
For i-th of time lag subinterval [- τi,-τi-1] on discrete point θj,i(j=1 ..., Ni), functionDerivative it is near
Like value ψj,i, by estimating infinitesimal generatorDefinition (1.2) obtain.
In subinterval [- τi,-τi-1] go forward k-1 discrete point θj,iAt (j=1 ..., k-1), using similar to single time lag
In the case of " startup " method design factor γ for usingj,l；In remaining Ni- k+1 discrete point θj,i(j=k ..., Ni) at, directly
It connects and utilizes BDF method design factors γj,l。
Then, formula (1.24) is embodied as：
It enables
Then formula (1.25) is write as matrix form：
In formula：
For all time lag subinterval [- τi,-τi-1] discrete point θ on (i=1 ..., m)j,i(j=1 ..., Ni；I=
1 ..., m), have：
In formula：
It enables
Simultaneous formula (1.23) and formula (1.29), can be derived by the relational expression between Φ and Ψ in the case of multiple time delay：
In formula：The discretization matrix of infinitesimal generator, first block are tieed up for (N+1) n × (N+1) n
RowIt can be write as unit vectorAnd systematic observation matrix's
The sum of Kronecker products.
The step of step (3) is：
First, the eigenvalue λ of time-lag power system is substituted with λ '+s, then can obtain the characteristic equation after displacement, i.e.,：
In formula：
After displacement operation, infinitesimal generator discretization approximate matrix that IGD-LMS methods obtainIt is mapped asAnd then inverse matrix is represented by：
In formula：
Then, it is dilute using implicitly Arnoldi (Implicitly Restarted Arnoldi, IRA) algorithm etc. is restarted
Thin algorithm is asked forModulus value the best part characteristic value.
In IRA algorithms, the operation of calculation amount maximum is to utilizeKrylov subspace is formed with vector product.If
K-th of Krylov vector beThen+1 Krylov vectors q of kthk+1It 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 technique (such as LU is decomposed and Gaussian elimination method)Inverse matrix when, on the one hand internally
It is very high to deposit requirement, and memory overflow problem may be caused；On the other hand, it is impossible to make full use of the sparse of system augmented state matrix
Characteristic.
In order to avoid direct solutionHere q is calculated using alternative mannerk+1.Then, formula (1.40) is converted to：
In formula：It is q after the l times iterationk+1Approximation.
The advantage of iterative solution is during system of linear equations is solved, not increasing any element, maintain's
Sparse characteristic.
It is calculated using induction dimension reduction methodIt is as follows：
First, willIn element rearranged according to the direction of row, obtain matrix
I.e.And then the property accumulated using Kronecker, the left end of formula (1.41) can be calculated as：
In formula:
In formula (1.42), the operation of calculation amount maximum is matrix-vector productThe power method of sparse realization can be used
It is calculated, reduces computation burden, improves computational efficiency.It is as follows：
Given convergence precision ε1, then solveThe conditions of convergence of IDR (s) algorithms be：
The step of step (4) is：If IRA algorithms are calculatedCharacteristic value for λ ", thenApproximation
Characteristic value, the i.e. approximate eigenvalue of time lag electric power system model are：
The step of step (5) is：The vector that the preceding n element of formula (1.44) Krylov vectors corresponding with λ " is formedIt is the good approximation of the corresponding feature vector v of accurate profile value λ.WithWithIt, can be by repeatedly using Newton method for initial value
In generation, obtains accurate profile value λ and corresponding feature vector v.
The step of step (6) is：The stability of time-lag power system is judged according to the size of accurate profile value.
If the damping ratio there are some characteristic value is less than 0, time-lag power system is in small interference unsure state；
If the minimum damping ratio of all characteristic values is equal to 0, time-lag power system is in the state of neutrality；
If the damping ratio of all characteristic values is all higher than 0, time-lag power system is in the state of asymptotically stability.
Compared with prior art, the beneficial effects of the invention are as follows：
Firstth, IGD-LMS algorithms proposed by the present invention are corresponding crucial special for calculating real system electromechanic oscillation mode
During value indicative, the scale of real system and the influence of communication delay have been fully considered.
Secondth, IGD-LMS algorithms proposed by the present invention calculate the partial feature value near designated displacement point, it is possible to obtain
The corresponding characteristic value of time-lag power system electromechanic oscillation mode, computational accuracy are more accurate.
Third, IGD-LMS algorithms proposed by the present invention have expanded extensive time lag using linear multistep method discretization scheme
The computational methods of electric system critical eigenvalue, and its obtained critical eigenvalue contributes to setting for wide area damping control
Meter.
4th, the present invention is based on the time-lag power system key feature value calculating methods of IGD-LMS so that follow conventional lines electricity
The Eigenvalues analysis theoretical method and frame of Force system small signal stability, analyze time-lag power system small signal stability and
Design wide area damping control is possibly realized.This is theoretical for the analysis on Small Disturbance Stability of perfect and abundant feature based value,
Have great importance and be worth.
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 form the improper restriction to the application for explaining the application.
Fig. 1 is the flow chart of the present invention；
Fig. 2 is the discrete point set omega in IGD-LMS methods in the case of single time lagN；
Fig. 3 is the discrete point set omega in IGD-LMS methods in the case of multiple time delayN。
Specific embodiment
It is noted that following detailed description is all illustrative, it is intended to provide further instruction to the application.It is 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 embodiment, 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.
As shown in Figure 1, the time-lag power system stability method of discrimination based on IGD-LMS, including：
Step (1)：Establish time-lag power system model；According to the characteristic value of time-lag power system model and time lag power train
Spectrum mapping relations between the infinitesimal generator characteristic value of model of uniting convert the characteristic value for calculating time-lag power system model
Into the characteristic value for calculating infinitesimal generator；It is infinitely small that the problem of so as to will determine that time-lag power system stability, is converted into calculating
Generate the eigenvalue problem of the modulus value maximum of member；
Step (2)：Discretization is carried out to infinitesimal generator using linear multistep method LMS, obtains infinitesimal generator
Discretization matrix；Infinite dimensional eigenvalue problem is converted into the eigenvalue problem of finite dimension；
Step (3)：The discretization matrix of infinitesimal generator obtained using implicit Arnoldi algorithm calculating step (2)
Inverse matrix modulus value maximum characteristic value approximation；The property accumulated in calculating process using displacement-inverse transformation and Kronecker
Carry out sparse realization；
Step (4)：According to spectrum mapping relations, by the approximation of the characteristic value of infinitesimal generator discretization matrix modulus value maximum
Value is converted into the approximate eigenvalue of time-lag power system model；
Step (5)：It is modified using Newton iteration method pairing approximation characteristic value, obtains the accurate profile of time-lag power system
Value；
Step (6)：The stability of time-lag power system is judged according to the size of accurate profile value.
7.1.1 time lag system equation
The state equation of time lag system is：
In formula：For system mode.τi>0 (i=1,2 ..., m) it is time lag constant.Without loss of generality, it is assumed that 0
=τ0<τ1<…<τi Wherein τmaxFor maximum time lag.It is dense matrix for systematic observation matrix；It is height sparse matrix for system time lags state matrix.
IfTo be defined on the n dimensional linear vector spaces of complex field, if state spaceIt is
By section [- τmax, 0] to n tie up real number spaceBanach (Banach) space that the continuous function of mapping is formed, and possess
Supremum norm
Infinitesimal generator It can be defined as：
In formula：It is a linear functional：
The linear multi step discretization (IGD-LMS) of 7.2 infinitesimal generators
In order to make it easy to understand, this section provides (the Linear Multi-Step based on backward difference linear multistep method first
With Backward Difference Formula, LMS-BDF) single time lag system infinitesimal generator discretization side
Method, and then this method is extended into Systems with Multiple Time-Delays.
The general type for giving fixed step size h, k step BDF methods is as follows：
In formula：αl(l=0 ..., k) and βkCoefficient for linear k footworks.
7.2.1 single time lag situation
1. discrete point set
Given positive integer N, the collection that the N+1 discrete point that spacing is h on section [- τ, 0] is formed are combined into ΩN.So as to continuously
State space X is converted into discrete spaceAs shown in Figure 2.
2. in θ0Function at=0The estimated value ψ of derivative0
In θ0At=0, functionDerivativeApproximationIt can be obtained by splicing condition (1.48).
3. in θjFunction at (j=k ..., N)The estimated value ψ of derivativej
In discrete point θjAt (j=1 ..., N), functionDerivativeApproximationIt can be by infinitesimal generatorDefinition (1.47) obtain.
The expression of formula (1.53), and two kinds of situations can be divided into and derived.First, in discrete point θj(j=
K ..., N) at, ψjIt can be arranged to obtain by the general type (1.50) to BDF methods.
4. in θjFunction at (j=1 ..., k-1)The estimated value ψ of derivativej
Then, in discrete point θjAt (j=1 ..., k-1), " startup " method using formula (1.55) is needed to calculate ψj。
At this time, it is assumed that ψjWith the similar form of formula (1.54), i.e.,：
In formula：γjl(j=1 ..., k-1；L=0,1 ..., k) it is unknown, to be asked coefficient.Determining γ is given belowjlSide
Method.
Near, it will be in formula (1.55) right endIt is launched into about step
Long h, cut-off error arePower series.
Formula (1.56) is substituted into formula (1.55), and enables the coefficient of the same power item of both members h equal, is obtained：
In formula：J=1 ..., k-1.
For some specific j, unknowm coefficient γjl(l=0,1 ..., k) it can be right with formula (1.57) by solving one
(k+1) answered × (k+1) rank system of linear equations obtains.By coefficient gammajl(j=1 ..., k-1；L=0,1 ..., k) it is write as (k-1)
× (k+1) rank matrixes, obtain：
For example, for BDF methods, as k=3 and k=5, matrix Γ can be respectively obtained by calculating3And Γ5。
5. the BDF discretization matrixes of infinitesimal generator
Simultaneous formula (1.52), formula (1.54) and formula (1.55), the relational expression that can be derived by between Φ and Ψ：
In formula：The discretization matrix of infinitesimal generator is tieed up for (N+1) n × (N+1) n.
7.2.2 multiple time delay situation
This section expands to infinitesimal generator BDF discretization methods in the case of single time lag containing m time lag τi(i=
1 ..., m) system.
1. discrete point set
First, in section [- τmax, 0] on establish discrete point set omegaN：
In formula： For section [- τi,-τi-1] on spacing be hiNiA discrete point structure
Into set, as shown in Figure 3.It is worth noting that, in order to ensure the availability of linear multistep method, the discrete points on subinterval
Ni(i=1 ..., m) has to be larger than step number k, i.e. Ni>k。
According to ΩN, continuous space X is converted into discrete spaceAnd have
2. in θ0Function at=0The estimated value ψ of derivative0
In θ0At=0, functionThe approximation ψ of derivative0, can be obtained by splicing condition (1.48).
3. i-th of subinterval [- τi,-τi-1] on discrete point θj,i(j=1 ..., Ni；I=1 ..., m) at functionDerivative
Estimated value ψj,i
For i-th of time lag subinterval [- τi,-τi-1] on discrete point θj,i(j=1 ..., Ni), functionDerivative it is near
Like value ψj,i, can be by estimating infinitesimal generatorDefinition (1.47) obtain.
Specifically, in subinterval [- τi,-τi-1] go forward k-1 discrete point θj,iAt (j=1 ..., k-1), using similar to
" startup " the method design factor γ used in the case of single time lagj,l；In remaining Ni- k+1 discrete point θj,i(j=k ..., Ni)
Place directly utilizes BDF method design factors γj,l.Then, formula (1.61) may particularly denote for：
It enables
Then formula (1.62) can be write as matrix form：
In formula：
For all time lag subinterval [- τi,-τi-1] discrete point θ on (i=1 ..., m)j,i(j=1 ..., Ni；I=
1 ..., m), have：
In formula：
4. infinitesimal generatorBDF discretization matrixes
It enables
Simultaneous formula (1.60) and formula (1.66), can be derived by the relational expression between Φ and Ψ in the case of multiple time delay：
In formula：The discretization matrix of infinitesimal generator is tieed up for (N+1) n × (N+1) n.Its first block
RowIt can be write as unit vectorAnd systematic observation matrix's
The sum of Kronecker products.
The foregoing is merely the preferred embodiments of the application, are not limited to the application, for the skill of this field
For art personnel, the application can have various modifications and variations.It is all within spirit herein and principle, made any repair
Change, equivalent replacement, improvement etc., should be included within the protection domain of the application.
Claims (8)
1. the time-lag power system stability method of discrimination based on IGD-LMS, it is characterized in that, including：
Step (1)：Establish time-lag power system model；According to the characteristic value of time-lag power system model and time-lag power system mould
The characteristic value for calculating time-lag power system model is converted to meter by the spectrum mapping relations between the infinitesimal generator characteristic value of type
Calculate the characteristic value of infinitesimal generator；The problem of so as to will determine that time-lag power system stability, is converted into the infinitely small generation of calculating
The eigenvalue problem of the modulus value maximum of member；
Step (2)：Discretization is carried out to infinitesimal generator using linear multistep method LMS, obtains the discrete of infinitesimal generator
Change matrix；Infinite dimensional eigenvalue problem is converted into the eigenvalue problem of finite dimension；
Step (3)：The discretization inverse of a matrix of infinitesimal generator obtained using implicit Arnoldi algorithm calculating step (2)
The approximation of the characteristic value of matrix modulus value maximum；It is carried out in calculating process using displacement-inverse transformation and the Kronecker property accumulated
Sparse realization；
Step (4)：According to spectrum mapping relations, the approximation of the characteristic value of infinitesimal generator discretization matrix modulus value maximum is turned
Turn to the approximate eigenvalue of time-lag power system model；
Step (5)：It is modified using Newton iteration method pairing approximation characteristic value, obtains the accurate profile value of time-lag power system；
Step (6)：The stability of time-lag power system is judged according to the size of accurate profile value.
2. the time-lag power system stability method of discrimination based on IGD-LMS as described in claim 1, it is characterized in that, it is described
The time-lag power system model of step (1) is：
In formula：For system mode；τi>0 is time lag constant；
Assuming thatWherein τmaxFor maximum time lag；
For systematic observation matrix,For dense matrix；For system time lags state matrix,For height sparse matrix；
IfTo be defined on the n dimensional linear vector spaces of complex field, if state space X:=C is by delay interval [- τmax,0]
Real number space is tieed up to nThe Banach Banach spaces that the continuous function of mapping is formed, and possess supremum norm
3. the time-lag power system stability method of discrimination based on IGD-LMS as claimed in claim 2, it is characterized in that, it is infinite
Small generation memberIt is defined as：
In formula：It is a linear functional：
The step of step (2) is：
According to the definition of infinitesimal generator, the discretization matrix of infinitesimal generator is obtained；
Step (21)：Given positive integer N, utilizes section [- τmax, 0] on the different discrete points of N+1 form set omegasN, ΩN=
{θi, i=0,1..., N }, and then continuous state space X is converted into separate manufacturing firms
Step (22)：Given continuous functionIf its discrete approximation is
IfIts discrete approximation isIt calculatesAccurate derivativeApproximation ψ；
Specifically, in discrete point θiPlace, utilizes ψiTo approachIn the functional value of the point：
In discrete point θ0At=0, obtained using the condition of splicing (1.5)Accurate derivativeAnd it is approximatelyI.e.
Step (23)：It willApproximation ψiIt is expressed asLinear combination, obtain：
In formula：ajFor constant, dijFor constant；
It is write formula (1.6) as matrix equation, is obtained：In fact, this is the discrete form of abstract Cauchy's equation；Equation
CoefficientAs infinitesimal generatorDiscretization matrix；
For single time lag system, have：θ0=0, θN=-τmax；At this point, matrixFirst block row be simplified；
So far, the method for discussing infinitesimal generator discretization.
4. the time-lag power system stability method of discrimination based on IGD-LMS as claimed in claim 2, it is characterized in that,
In order to make it easy to understand, the infinite your pupil of single time lag system of the LMS-BDF based on backward difference linear multistep method is provided first
The discretization method of Cheng Yuan, and then this method is extended into Systems with Multiple Time-Delays；
The form for giving fixed step size h, k step BDF methods is as follows：
In formula：αlAnd βkCoefficient for linear k footworks；
Single time lag situation
Given positive integer N, the collection that the N+1 discrete point that spacing is h on section [- τ, 0] is formed are combined into ΩN；So as to continuous state
Space X is converted into discrete space
In θ0At=0, functionDerivativeApproximationIt can be obtained by splicing condition (1.3)；
In discrete point θjPlace, functionDerivativeApproximationBy infinitesimal generatorDefinition (1.2) obtain；
The expression of formula (1.11), and two kinds of situations can be divided into and derived；
First, in discrete point θjPlace, ψjIt can be arranged to obtain by (1.8) to BDF methods；
Then, in discrete point θjPlace, wherein, j=1 ..., k-1 calculate ψ using formula (1.13) " startup " methodj；At this point,
Assuming that ψjWith the similar form of formula (1.12), i.e.,：
In formula：γjlFor unknowm coefficient, j=1 ..., k-1；L=0,1 ..., k；
Determining γ is given belowjlMethod：
It nearby will be in formula (1.13) right endIt is launched into and is about step-length h, cut-off errorPower series；
Formula (1.14) is substituted into formula (1.13), and enables the coefficient of the same power item of both members h equal, is obtained：
For the j, unknowm coefficient γ of some settingjlIt is linear by solving (k+1) corresponding with formula (1.15) × (k+1) rank
Equation group obtains；By coefficient gammajlIt is write as (k-1) × (k+1) rank matrixes, is obtained：
Simultaneous formula (1.10), formula (1.12) and formula (1.13), the relational expression being derived by between Φ and Ψ：
In formula：The discretization matrix of infinitesimal generator is tieed up for (N+1) n × (N+1) n；
Multiple time delay situation
Infinitesimal generator BDF discretization methods in the case of single time lag are expanded to containing m time lag τiSystem；
First, in section [- τmax, 0] on establish discrete point set omegaN：
In formula：For section [- τi,-τi-1] on spacing be hiNiThe set that a discrete point is formed, in order to protect
The availability of linear multistep method is demonstrate,proved, the discrete points N on subintervaliHave to be larger than step number k, i.e. Ni>k；
According to ΩN, continuous space X is converted into discrete spaceAnd have
In θ0At=0, functionThe approximation ψ of derivative0, obtained by formula (1.3)；
For i-th of time lag subinterval [- τi,-τi-1] on discrete point θj,i, wherein j=1 ..., Ni, functionThe approximation of derivative
Value ψj,i, by estimating infinitesimal generatorDefinition (1.2) obtain；
In subinterval [- τi,-τi-1] go forward k-1 discrete point θj,iPlace, wherein j=1 ..., k-1, using similar to single time lag feelings
" startup " the method design factor γ used under conditionj,l；In remaining Ni- k+1 discrete point θj,iPlace, wherein j=k ..., Ni, directly
It connects and utilizes BDF method design factors γj,l；
Then, formula (1.24) is embodied as：
It enables
Then formula (1.25) is write as matrix form：
In formula：
For all time lag subinterval [- τi,-τi-1] on discrete point θj,i, have：
In formula：
It enables
Simultaneous formula (1.23) and formula (1.29), can be derived by the relational expression between Φ and Ψ in the case of multiple time delay：
In formula：The discretization matrix of infinitesimal generator, first block row are tieed up for (N+1) n × (N+1) nWrite as unit vectorAnd systematic observation matrixKronecker product the sum of, i=0,1 ...,
m；
5. the time-lag power system stability method of discrimination based on IGD-LMS as claimed in claim 4, it is characterized in that,
The step of step (3) is：
First, the eigenvalue λ of time-lag power system is substituted with λ '+s, then can obtain the characteristic equation after displacement, i.e.,：
In formula：
After displacement operation, infinitesimal generator discretization approximate matrix that IGD-LMS methods obtainIt is mapped asInto
And inverse matrix is expressed as：
In formula：
Then, it is asked for using Corresponding Sparse Algorithms such as implicit restarted Arnoldi algorithmsModulus value the best part characteristic value；
In IRA algorithms, the operation of calculation amount maximum is to utilizeKrylov subspace is formed with vector product；If k-th
Krylov vectors areThen+1 Krylov vectors q of kthk+1It calculates as follows：
In order to avoid direct solutionHere q is calculated using alternative mannerk+1；Then, formula (1.40) is converted to：
In formula：It is q after the l times iterationk+1Approximation；
It is calculated using induction dimension reduction methodIt is as follows：
First, willIn element rearranged according to the direction of row, obtain matrixI.e.And then the property accumulated using Kronecker, the left end of formula (1.41) are calculated as：
In formula:
In formula (1.42), the operation of calculation amount maximum is matrix-vector productIt is counted using the power method of sparse realization
It calculates：
Given convergence precision ε1, then solveThe conditions of convergence of IDR (s) algorithms be：
6. the time-lag power system stability method of discrimination based on IGD-LMS as claimed in claim 5, it is characterized in that,
The step of step (4) is：If IRA algorithms are calculatedCharacteristic value for λ ", thenApproximation characteristic
Value, the i.e. approximate eigenvalue of time lag electric power system model are：
7. the time-lag power system stability method of discrimination based on IGD-LMS as claimed in claim 6, it is characterized in that,
The step of step (5) is：The vector that the preceding n element of formula (1.44) Krylov vectors corresponding with λ " is formedIt is
The approximation of the corresponding feature vector v of accurate profile value λ；WithWithFor initial value, can essence be obtained by iteration using Newton method
True eigenvalue λ and corresponding feature vector v.
8. the time-lag power system stability method of discrimination based on IGD-LMS as claimed in claim 7, it is characterized in that,
The step of step (6) is：The stability of time-lag power system is judged according to the size of accurate profile value；
If the damping ratio there are some characteristic value is less than 0, time-lag power system is in small interference unsure state；
If the minimum damping ratio of all characteristic values is equal to 0, time-lag power system is in the state of neutrality；
If the damping ratio of all characteristic values is all higher than 0, time-lag power system is in the state of asymptotically stability.
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN108808703A (en) * | 2018-07-13 | 2018-11-13 | 山东大学 | Time-lag power system analysis on Small Disturbance Stability method based on low order IGD-IRK |
| CN108808702A (en) * | 2018-07-13 | 2018-11-13 | 山东大学 | Time-lag power system electromechanic oscillation mode computational methods based on low order IGD-LMS algorithms |
| CN111614078A (en) * | 2020-04-28 | 2020-09-01 | 南方电网科学研究院有限责任公司 | Method, device, equipment and storage medium for analyzing small interference stability of power system |
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| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN108808703A (en) * | 2018-07-13 | 2018-11-13 | 山东大学 | Time-lag power system analysis on Small Disturbance Stability method based on low order IGD-IRK |
| CN108808702A (en) * | 2018-07-13 | 2018-11-13 | 山东大学 | Time-lag power system electromechanic oscillation mode computational methods based on low order IGD-LMS algorithms |
| CN108808703B (en) * | 2018-07-13 | 2020-07-31 | 山东大学 | Time-lag power system small-interference stability analysis method based on low-order IGD-IRK |
| CN111614078A (en) * | 2020-04-28 | 2020-09-01 | 南方电网科学研究院有限责任公司 | Method, device, equipment and storage medium for analyzing small interference stability of power system |
| CN111614078B (en) * | 2020-04-28 | 2021-12-14 | 南方电网科学研究院有限责任公司 | Method, device, equipment and storage medium for analyzing small interference stability of power system |
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