CN105449665B - Time-lag power system stability method of discrimination based on SOD PS - Google Patents
Time-lag power system stability method of discrimination based on SOD PS Download PDFInfo
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Abstract
The invention discloses the time-lag power system stability method of discrimination based on SOD PS, comprise the following steps：Step (1)：Set up time-lag power system model；According to the relation between the characteristic value and time-lag power system solution to model operator characteristic value of time-lag power system model, the characteristic value for calculating time-lag power system model is changed into the characteristic value for calculating Solution operator；So as to be converted into the eigenvalue problem for the modulus value maximum for calculating Solution operator the problem of will determine that time-lag power system stability；Step (2)：Discretization is carried out to Solution operator using pseudo- spectral method, the discretization matrix of Solution operator is obtained；Step (3)：The maximum characteristic value μ of the discretization matrix norm value of the Solution operator obtained using sequential method or subspace method come calculation procedure (2)；Step (4)：The stability of time-lag power system is judged according to characteristic value μ size.This method has amount of calculation small, differentiates fast and accurately feature.
Description
Technical field
The present invention relates to a kind of time-lag power system stability method of discrimination based on SOD-PS.
Background technology
Extensive interconnecting electric power is given in the appearance of WAMS (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, by introducing the wide area feedback signal of effectively reflection inter-area oscillation mode, results in preferable damping control performance, its
To solve the problems, such as the inter-area low-frequency oscillation in interconnected network, and then improving the ability to transmit electricity of system to provide new control hand
Section, with good and be widely applied prospect.
Wide area signal is in the WAMS communications being made up of different communication medium (such as optical fiber, telephone wire, digital microwave, satellite)
When transmitting and handle in network, there is the communication delay changed between tens to hundreds of milliseconds.Time lag is to cause system control law to lose
A kind of major incentive of effect, operation conditions deterioration and system unstability【[1]Wu H X,Tsakalis K S,Heydt G
T.Evaluation of time delay effects to wide-area power system stabilizer
design.IEEE Trans.Power Syst.,2004,19(4):1935-1941.】.Therefore, entered using wide area measurement information
During row power system closed-loop control, it is necessary to the influence of meter and time lag.
Invention【[2] ox is newborn, Ye Hua, Wang Chunyi, wait based on time-lag power system characteristic value approximate Pad é calculate with
Convenient stable criterion .201210271783.8:[P].】Time Delay is approached using Pade approximation polynomials, and then calculates system
The critical eigenvalue of the system rightmost side, and judge the time lag stability of system.
Invention【[3] extensive time-lag power system characteristic value calculating sides of Ye Hua, Wang Yanyan, the Liu Yutian based on EIGD
Method .201510055743.3. China, 201510055743.3 [P]】Propose it is a kind of based on display IGD (Explicit IGD,
EIGD extensive time-lag power system characteristic value) is calculated.The critical eigenvalue of the system rightmost side obtained using calculating, can be with
Stability of the judgement system under fixed time lag.These time lag Convenient stable criterions, be required to by Multiple-Scan [0.1,
2.5] in Hz low-frequency oscillations frequency range, close to the critical eigenvalue of the imaginary axis, the time lag stability of system could be judged.
The content of the invention
The purpose of the present invention is exactly that based on SOD-PS, (puppet composes discretization Solution operator in order to solve the above problems there is provided one kind
(Pseudospectral discretization of solution operator, SOD-PS)) time-lag power system it is steady
Qualitative discrimination method, this method only needs to calculate a maximum characteristic value of Solution operator discretization matrix norm value or a pair of conjugation
Characteristic value, it is possible to judge stability of the system under fixed time lag.It is small with amount of calculation, the characteristics of differentiating accurate.
To achieve these goals, the present invention is adopted the following technical scheme that：
Power System Delay Convenient stable criterion based on SOD-PS, comprises the following steps：
Step (1)：Set up time-lag power system model；Characteristic value and time lag power train according to time-lag power system model
Relation between solution to model operator characteristic value of uniting, calculating Solution operator is changed into by the characteristic value for calculating time-lag power system model
Characteristic value；So as to be converted into the characteristic value for the modulus value maximum for calculating Solution operator the problem of will determine that time-lag power system stability
Problem；
Step (2)：Discretization is carried out to Solution operator using pseudo- spectral method, the discretization matrix of Solution operator is obtained；
Step (3)：The discretization matrix norm of the Solution operator obtained using sequential method or subspace method come calculation procedure (2)
It is worth maximum characteristic value μ；
Step (4)：The stability of time-lag power system is judged according to characteristic value μ size.
The step of step (4) is：
If characteristic value μ modulus value is more than 1, time-lag power system is in small interference unsure state；
If characteristic value μ modulus value is equal to 1, time-lag power system is in the state of neutrality；
If characteristic value μ modulus value is less than 1, time-lag power system is in the state of asymptotically stability.
The time-lag power system model of the step (1) is as follows：
In formula：For the state variable vector of power system, n is system state variables sum.T is current time.
0<τ1<τ2<…<τi…<τmFor the time lag constant of Time Delay, maximum of which time lag is τm。
For systematic observation matrix.Δ x (t) is the increment of t system state variables, Δ x (t- τi) it is t- τiMoment system state variables
Increment,For the increment of t system state variables derivative.Δ x (0) is (i.e. initial for the initial value of system state variables
Condition), and be abbreviated as
The characteristic equation of linearized system that formula (1.1) is represented is：
In formula：λ is characterized value, and v is characterized the corresponding right characteristic vector of value.
The Solution operator is defined as follows：
Solution operator T (h) is defined as initial conditionIt is transferred to h (transfer step-length, 0≤h≤τm) time lag electric power after the moment
The linear operator of Solutions of Systems segmentation.
Wherein, s is integration variable, and θ is variable,WithThe shape of respectively 0 and h+ θ moment time-lag power systems
State..
Relation between the characteristic value and Solution operator characteristic value of the time-lag power system model：
Have such as between spectral mapping theorem, Solution operator T (h) characteristic value μ and the eigenvalue λ of time-lag power system
Lower relation：
In formula：σ (T (h)) represents the spectrum of Solution operator.
The step of step (2), is as follows：
Corresponding with Solution operator T (h), standard base form discretization matrix TM,NIt is expressed as follows：
TM,N=ΤM+Τ′M,N*(INn-UN)-1*UM,N (1.5)
In formula：
In formula (1.5), M and N are given positive integer, INnFor Nn ranks unit matrix (diagonal entry is 1, other elements be 0),
The representing matrix inversion operation of subscript -1.
In formula (1.6), Q is given positive integer,1M×11 M dimensional vectors, I are all for element(Q-2)MFor (Q-2) M rank unit matrix, InFor n rank unit matrix, 0(M+1)×MFor (M+1) × M rank null matrix, For Kronecker product operation.Matrix ΤMFor height sparse matrix, and with time-lag power system shape
State matrixIt is unrelated.
In formula (1.7),0QM×NFor QM × N rank null matrix,
In formula (1.8),I=0 ..., m, EiElement completely by drawing
Ge Lang coefficients are determined.
In formula (1.9),I=0 ..., m, FiElement determined completely by Lagrange coefficient
It is fixed.
The step of step (3), is as follows：
Assuming that in kth time iteration, it is necessary to calculate TM,NWith vectorProduct,
Comprise the following steps that：
Step (3-1)：By vector vkBy the matrix of row boil down to oneI=
1,…,QM+1.Correspondingly, have：vk=vec (Vk), wherein, vec () is by computing that matrix compression is column vector.
Step (3-2)：Calculate
Step (3-3)：Calculate
Step (3-4)：Calculate wk=ΤM*vk+Τ′M,N*qk。
The step of step (3-2), is as follows：
Formula (1.8) is substituted into, can be obtained：
In formula：Subscript T representing matrix transposition.K is kth time iteration；M is the number of time lag.
As the above analysis, p is calculatedk, first have to calculateI=0 ..., m, are then asked again
With the column vector that finally recompression is tieed up for a Nn.
It is worth noting that,It can be realized, so as to reduce amount of calculation, improved with sparse
Computational efficiency.
The step of step (3-3), is as follows：
After formula (1.9) is substituted into, it is known thatExpression is not shown.Thus, this
In q is calculated using iterative algorithmk=(INn-UN)-1pk.In solution procedure, it is related to matrix-vector multiplication computing b=(INn-UN)
Y, wherein
First, vectorial y is pressed into the matrix of row boil down to oneI=1 ..., N.
And then, b=(I can be obtainedNn-UN) the sparse of y realize that step is as follows：
It is worth noting that,It can be realized with sparse, so as to reduce amount of calculation, improve meter
Calculate efficiency.
Beneficial effects of the present invention：
This method only needs to calculate a maximum characteristic value of Solution operator discretization matrix norm value or a pair of conjugate characters
Value, it is possible to judge stability of the system under fixed time lag.It is small with amount of calculation, differentiate fast and accurately feature.
Brief description of the drawings
Fig. 1 is flow chart of the method for the present invention；
Fig. 2 (a) is time lag system characteristic value；
Fig. 2 (b) is Solution operator characteristic value；
Fig. 3 is the district system of four machine two；
Fig. 4 is matrix TM,NSpectrum, i.e. σ (TM,N)；
Fig. 5 is matrixSpectrum, i.e.,
Embodiment
The invention will be further described with embodiment below in conjunction with the accompanying drawings.
As shown in figure 1, the Power System Delay Convenient stable criterion based on SOD-PS, comprises the following steps：
Step (1)：Set up time-lag power system model；
Step (2)：By puppet spectrum discretization, Solution operator T (h) finite dimension discretization matrix T is obtainedM,N；
Step (3)：Obtained using sequential method or subspace method (such as implicit restarted Arnoldi algorithm) come calculation procedure (2)
The characteristic value μ of the Solution operator discretization matrix norm value maximum arrived；
Step (4)：
If characteristic value μ modulus value is more than 1, time-lag power system is in small interference unsure state；
If characteristic value μ modulus value is equal to 1, time-lag power system is in the state of neutrality；
If characteristic value μ modulus value is less than 1, time-lag power system is in the state of asymptotically stability.
1. time-lag power system model
Consider after wide-area communication time-delay, one group of time-delayed differential equations that power system can be as follows are described：
In formula：For the state variable vector of power system, n is system state variables sum.T is current time.0
<τ1<τ2<…<τi…<τmFor the time lag constant of Time Delay, maximum of which time lag is τm。For
Systematic observation matrix.
The characteristic equation of linearized system that formula (1) is represented is：
In formula：λ and v are respectively characteristic value and corresponding right characteristic vector.
2. Solution operator
Solution operator is defined as initial conditionIt is transferred to h (transfer step-length, 0≤h≤τm) Solutions of Systems is segmented after the moment
Linear operator.
There is such as ShiShimonoseki between spectral mapping theorem, Solution operator T (h) characteristic value μ and the eigenvalue λ of time lag system
System.Its graph-based, such as Fig. 2 (a) and Fig. 2 (b) are shown.
In formula：σ (T (h)) represents the spectrum of Solution operator.
Time lag system is mapped in Fig. 2 (b) within unit circle positioned at the characteristic value of left half complex plane in Fig. 2 (a), and when
The characteristic value that stagnant system is located at right half complex plane is mapped as the characteristic value that Solution operator modulus value is more than 1, and outside unit circle.
Therefore, the characteristic value of Solution operator is utilized, it is possible to judge the stability of former time lag system.
If Solution operator at least has the characteristic value that a modulus value is more than 1, it is unstable for may determine that former time lag system
,
If the modulus value of all characteristic values of Solution operator is respectively less than 1, former time lag system is asymptotically stability.
Solution operator T (h) is Infinite Dimension Linear operator.In order to calculate the characteristic value of Solution operator and judge time lag system according to this
Stability, carries out discretization to T (h) using pseudo- spectral method (Pesudospectral, PS) first, obtains one and Solution operator pair
Approximate matrix answer, finite dimension, and then calculate the characteristic value of approximate matrix and judge the stability of former time lag system.
3. the Solution operator discretization based on puppet spectrum
Corresponding with Solution operator T (h), standard base form discretization matrix TM,NIt can be expressed as follows：
TM,N=ΤM+Τ′M,N*(INn-UN)-1*UM,N (1.5)
In formula：
In formula (1.6), Q, M and N are given positive integer,1M×1The M for being all 1 for element ties up row
Vector, I(Q-2)MFor (Q-2) M rank unit matrix, 0(M+1)×MFor (M+1) × M rank null matrix, For Kronecker product operation.Matrix ΤMFor height sparse matrix, and and systematic observation matrix It is unrelated.
In formula (1.7),0QM×NFor QM × N rank null matrix,
InFor n rank unit matrix.
In formula (1.8),I=0 ..., m, EiElement completely by drawing
Ge Lang coefficients are determined.
In formula (1.9),I=0 ..., m, FiElement determined completely by Lagrange coefficient
It is fixed.
4. sparse realize
Matrix TM,NExponent number be (QM+1) n.For large-scale electrical power system, matrix TM,NExponent number will be very huge.Cause
This, in the corresponding discretization matrix T of application Solution operatorM,NWhen judging the time lag stability of large-scale electrical power system, it is necessary to use and change
Several maximum characteristic values of its modulus value are calculated for feature value-based algorithm (sequential method or subspace method).
Assuming that in kth time iteration, it is necessary to calculate TM,NWith vectorProduct,
Comprise the following steps that：
(1) by vector vkBy the matrix of row boil down to oneI=1 ...,
QM+1.Correspondingly, have：vk=vec (Vk), wherein vec () is by computing that matrix compression is column vector.
(2) calculateFormula (1.8) is substituted into, can be obtained：
As the above analysis, p is calculatedk, first have to calculateI=0 ..., m, are then asked again
With the column vector that finally recompression is tieed up for a Nn.It is worth noting that,Can be with sparse reality
It is existing, so as to reduce amount of calculation, improve computational efficiency.
(3) calculate
After formula (1.9) is substituted into, it is known thatExpression is not shown.Thus, this
In q is calculated using iterative algorithmk=(INn-UN)-1*pk.In solution procedure, it is related to matrix-vector multiplication computing b=(INn-
UN) * y, wherein
First, vectorial y is pressed into the matrix of row boil down to oneI=1 ..., N.
And then, b=(I can be obtainedNn-UN) the sparse of * y realize that step is as follows：
It is worth noting that,It can be realized with sparse, so as to reduce amount of calculation, improve meter
Calculate efficiency.
(4) w is calculatedk=ΤM*vk+Τ′M,N*qk。
It is proposed by the present invention based on the big of display infinitesimal generator discretization to verify using four Ji Liang areas example systems
The validity of the sparse features value calculating method of scale time-lag power system.All analyses are in Matlab and in Inter
Carried out on Core i5 4 × 3.4GHz 8GB RAM personal computers.
Four Ji Liang areas example systems are as shown in Figure 3.All generators use high-gain Thyristor Excitation System, and install with
Rotating speed deviation is the power system stabilizer, PSS (PSS) of input.On this basis, in G1Upper installing is with G1And G3Relative rotation speed deviation
Δω13For the wide area PSS of feedback signal, the damping with further raising to inter-area low-frequency oscillation pattern.Assuming that wide area PSS
Feedback Delays and input delay are respectively τ1=150ms and τ2=100ms.The dimension difference of system state variables and algebraic variable
For n=56 and l=22.
(1) approximate matrix TM,NApproximation capability analysis
First, by Solution operator discretization Spectral radius σ (TM,N) it is converted into the approximate eigenvalue of time-lag power systemThen verified using newton, obtain the accurate characteristic value of time-lag power systemFinally willIt is converted into σ
(T), and by comparing σ (TM,N) and σ (T), to verify discretization matrix TM,NApproach Solution operator T accuracy.
If the Feedback Delays of wide area damping control are τ=0.2045s, step-length h=0.05s is taken, thenTake
M=N=20, TM,NDimension be (QM+1) n=5656.For anti-leak-stopping root, T is calculated using QR algorithmsM,NAll Eigenvalues σ
(TM,N), wherein partial feature value of the modulus value more than 1e-3 is as shown in Figure 4.And then, it is converted to the characteristic value of time lag systemAnd newton verification is carried out, as shown in Figure 5.By the accurate characteristic value of time-lag power systemThe time lag being converted to
The accurate profile value σ (T) of power system Solution operator, is added in Fig. 4.
As shown in Figure 4, SOD-PS methods can accurately calculate the partial feature value that Solution operator modulus value is more than 0.45.Wherein,
The maximum characteristic value of modulus value is μ=0.87495+0.48415i, and corresponding modulus value is 0.99997.Thus, it is possible to which decision-making system is faced
Boundary is stable, and the time lag upper limit that system can be born is τm=0.2045s.Now, the characteristic value of time lag system corresponding with μ, be
Close to low frequency oscillation mode λ=- 0.0005+10.1082i of zero damping.
Above-mentioned analysis shows：The approximate matrix for the Solution operator that SOD-PS methods are obtained, can relatively accurately approach resolving
Submodule value the best part characteristic value, is capable of the time lag stability of judgement system exactly.
(2) analysis of the accuracy that delay margin is calculated
Below by the contrast with calculating the maximum time lag that the district system of four machine two can be born using LMI methods, explanation
The accuracy of SOD-PS methods.
(open loop) POWER SYSTEM STATE order of matrix number not comprising wide area damping control is 52 ranks.In order to utilize
LMI (Linear in the robust control tool box (Robust Control Toolbox) that Matlab is provided
Matrix Inequality, LMI) solver, calculate the time lag for obtaining (closed loop) power system comprising wide area damping control
Stability margin【[5] Yao W, Jiang L, Wu Q H, wait .Delay-dependent stability analysis of the
power system with a wide-area damping controller embedded.IEEE Trans.Power
Syst.,2011,26(1):233-240. [6] Wu M, He Y, She J-H, wait .Delay-dependent criteria for
robust stability of time-varying delay systems.Automatica,2004,40(8):1435-
1439.】, it is necessary to depression of order is carried out to split loop system.It is 7 ranks by open cycle system depression of order using the Schur depressions of order, it transmits letter
Number is as follows：
The frequency characteristic of open cycle system is it can be found that in 0.2-2.5Hz low-frequency oscillation frequency models before and after contrast depression of order
In enclosing, reduced-order model and full rank model frequency response difference are very small.And then, the gevp letters provided using robust control tool box
Number, the delay margin for trying to achieve system is 187.9ms.By approximate matrix TM, the analysis of N approximation capability analysis is understood, profit
It is 204.5ms that obtained system time lags stability margin is calculated with SOD-PS methods.It is hereby understood that there is larger guarantor in LMI methods
Keeping property, is (204.5-187.9)/204.5*100%=8.12%.
Although above-mentioned the embodiment of the present invention is described with reference to accompanying drawing, not to present invention protection model
The limitation enclosed, one of ordinary skill in the art should be understood that on the basis of technical scheme those skilled in the art are not
Need to pay various modifications or deform still within protection scope of the present invention that creative work can make.
Claims (9)
1. the time-lag power system stability method of discrimination based on SOD-PS, it is characterized in that, comprise the following steps：
Step (1)：Set up time-lag power system model；Characteristic value and time-lag power system mould according to time-lag power system model
Relation between the Solution operator characteristic value of type, the characteristic value for calculating time-lag power system model is changed into the spy for calculating Solution operator
Value indicative；So as to which the characteristic value that the modulus value maximum for calculating Solution operator is converted into the problem of will determine that time-lag power system stability is asked
Topic；
Step (2)：Discretization is carried out to Solution operator using pseudo- spectral method, the discretization matrix of Solution operator is obtained；
Step (3)：The discretization matrix norm value of the Solution operator obtained using sequential method or subspace method come calculation procedure (2) is most
Big characteristic value μ；
Step (4)：The stability of time-lag power system is judged according to characteristic value μ size.
2. the time-lag power system stability method of discrimination as claimed in claim 1 based on SOD-PS, it is characterized in that, the step
Suddenly the step of (4) are：
If characteristic value μ modulus value is more than 1, time-lag power system is in small interference unsure state；
If characteristic value μ modulus value is equal to 1, time-lag power system is in the state of neutrality；
If characteristic value μ modulus value is less than 1, time-lag power system is in the state of asymptotically stability.
3. the time-lag power system stability method of discrimination as claimed in claim 1 based on SOD-PS, it is characterized in that,
The time-lag power system model of the step (1) is as follows：
In formula：For the state variable vector of power system, n is system state variables sum；T is current time；0<τ1<τ2
<…<τi…<τmFor the time lag constant of Time Delay, wherein, m is the number of time lag, and maximum time lag constant is τm；i
=0,1 ..., m is systematic observation matrix；Δ x (t) is the increment of t system state variables, Δ x (t- τi) it is t- τiMoment be
The increment for state variable of uniting,For the increment of t system state variables derivative；Δ x (0) is the first of system state variables
Initial value, and be abbreviated as
The characteristic equation of linearized system that formula (1.1) is represented is：
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In formula：λ is characterized value, and v is characterized the corresponding right characteristic vector of value.
4. the time-lag power system stability method of discrimination as claimed in claim 3 based on SOD-PS, it is characterized in that,
The Solution operator is defined as follows：
Solution operator T (h) is defined as initial conditionH is transferred to, h is transfer step-length, 0≤h≤τmTime lag power train after moment
The linear operator of system solution segmentation；
Wherein, s is integration variable, and θ is variable,WithThe state of respectively 0 and θ+h moment time-lag power systems.
5. the time-lag power system stability method of discrimination as claimed in claim 4 based on SOD-PS, it is characterized in that,
Relation between the characteristic value and Solution operator characteristic value of the time-lag power system model：
There is such as ShiShimonoseki between spectral mapping theorem, Solution operator T (h) characteristic value μ and the eigenvalue λ of time-lag power system
System：
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6. the time-lag power system stability method of discrimination as claimed in claim 4 based on SOD-PS, it is characterized in that,
The step of step (2), is as follows：
Corresponding with Solution operator T (h), standard base form discretization matrix TM,NIt is expressed as follows：
TM,N=ΤM+Τ′M,N*(INn-UN)-1*UM,N (1.5)
In formula：
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In formula (1.5), M and N are given positive integer, INnFor Nn rank unit matrix, the representing matrix inversion operation of subscript -1；
In formula (1.6), Q is given positive integer,1M×11 M dimensional vectors, I are all for element(Q-2)M
For (Q-2) M rank unit matrix, InFor n rank unit matrix, 0(M+1)×MFor (M+1) × M rank null matrix, For Kronecker product operation；Matrix ΤMFor height sparse matrix, and with time-lag power system state
MatrixI=0 ..., m is unrelated；
In formula (1.7),0QM×NFor QM × N rank null matrix,
In formula (1.8),I=0 ..., m, EiElement completely it is bright by glug
Day coefficient decision；
In formula (1.9),I=0 ..., m, FiElement determined completely by Lagrange coefficient.
7. the time-lag power system stability method of discrimination as claimed in claim 6 based on SOD-PS, it is characterized in that,
The step of step (3), is as follows：
Assuming that in kth time iteration, it is necessary to calculate TM,NWith vectorProduct,
Comprise the following steps that：
Step (3-1)：By vector vkBy the matrix of row boil down to oneI=1 ..., QM+
1；Correspondingly, have：vk=vec (Vk), wherein, vec () is by computing that matrix compression is column vector；
Step (3-2)：Calculate
Step (3-3)：Calculate
Step (3-4)：Calculate wk=ΤM*vk+Τ′M,N*qk。
8. the time-lag power system stability method of discrimination as claimed in claim 7 based on SOD-PS, it is characterized in that,
The step of step (3-2), is as follows：
Formula (1.8) is substituted into, can be obtained：
In formula：Subscript T representing matrix transposition；K is kth time iteration；
As the above analysis, p is calculatedk, first have to calculateI=0 ..., m, are then summed again, finally
Recompress as the column vector of a Nn dimension；
Realized by sparse.
9. the time-lag power system stability method of discrimination as claimed in claim 7 based on SOD-PS, it is characterized in that,
The step of step (3-3), is as follows：
After formula (1.9) is substituted into, it is known thatExpression is not shown；
Thus, q is calculated using iterative algorithm herek=(INn-UN)-1*pk；
In solution procedure, it is related to matrix-vector multiplication computing b=(INn-UN) * y, wherein
First, vectorial y is pressed into the matrix of row boil down to oneI=1 ..., N；
And then, b=(I can be obtainedNn-UN) the sparse of * y realize that step is as follows：
Realized by sparse.
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| CN105958476B (en) * | 2016-05-06 | 2018-03-20 | 河海大学 | Time-lag power system stability sex determination method based on Wirtinger inequality |
| CN106099921B (en) * | 2016-07-21 | 2018-09-28 | 天津大学 | A kind of Power System Delay stability margin fast solution method |
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| CN108321821B (en) * | 2018-02-12 | 2019-12-27 | 山东大学 | Time-lag power system stability discrimination method based on SOD-IRK |
| CN109685400B (en) * | 2018-02-24 | 2020-07-31 | 山东大学 | Time-lag power system stability discrimination method based on time integral IGD |
| CN108647906B (en) * | 2018-05-25 | 2020-05-08 | 山东大学 | Time-lag power system stability analysis method based on low-order EIGD |
| CN108808705B (en) * | 2018-07-13 | 2020-05-22 | 山东大学 | Time-lag power system electromechanical oscillation mode calculation method based on low-order SOD-PS-II-R algorithm |
| 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 |
| CN108923421B (en) * | 2018-07-13 | 2020-04-21 | 山东大学 | Time-lag power system key characteristic value calculation method based on low-order SOD-IRK algorithm |
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