CN104505830B - Time-lag power system stability analysis method and device - Google Patents
Time-lag power system stability analysis method and device Download PDFInfo
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Abstract
A kind of time-lag power system stability analysis method and device, the method for analyzing stability include：A, collection time lag system network architecture parameters, generator frequency, generator rotor angle in system；B, using gathered data fault chains are constructed, and fault chains are combined with Markov process set up time lag Markov jump system state equation；C, according to time lag Markov jump system state equation, construct the time lag stability criterion for improving free-form curve and surface respectively based on Markov process transition rates and newton Leibnitz's formula, on this basis, by time lag stability criterion equivalence transformation, the stable upper limit of generalized eigenvalue method solving system time lag is utilized；D, the stable upper limit of the output system time lags.The stable upper limit of time lag can be effectively solved using the time-lag power system stability analysis method and device of the present invention, the stability of a system is improved.
Description
Technical Field
The invention relates to the technical field of power system analysis and control, in particular to a stability analysis technology of a time-lag power system.
Background
Many power systems in engineering may be described by differential equations in which state variables evolve over time. The phenomenon that time lag exists among state variables of a considerable part of power systems, namely the evolution trend of the system depends on the current state of the system and the state of a certain time or a plurality of times in the past, and the power systems are called time-lag power systems. In recent years, time-lag power systems have been the subject of intensive research in many fields, and especially in the field of power systems, research on the characteristics of time-lag power systems has attracted extensive attention. On the other hand, stability is the most basic quality of an electric power system, and for a time-lag electric power system, a characteristic equation is a transcendental equation containing an exponential function, and in principle, the characteristic equation has infinite roots, so that the distribution situation of the characteristic roots is quite complex.
The existence of the time lag makes the stability analysis and control of the power system become more complex, and becomes the root cause of system instability and performance deterioration. Therefore, it is urgently required to analyze the time lag stability of the system in depth so as to improve the stability of the system.
In the aspect of time lag stability analysis, there are many theoretical achievements, which are mainly divided into two categories: 1) frequency domain method. The method for solving the time lag stability upper limit of the system by utilizing Rekasius transformation is provided, but the method needs to search key characteristic values of the system in a time lag space, and the calculated amount is large; the characteristic equation of the time-delay system is converted into a pure virtual characteristic root of a polynomial equation solving system on a virtual axis, any intermediate variable substitution is not needed, and the time-delay stability upper limit of the single-time-delay power system can be effectively solved, but the method is hardly suitable for the calculation of a large-scale system. This solution is very difficult when there is uncertainty in the system and the time lag varies with time. Therefore, the method for researching the time lag stability of the system by adopting the frequency domain method has strong limitation. 2) Time domain methods. Respectively combining a Finsler guiding theory, a Park inequality, a Moon inequality and a Fridman generalized model transformation method with a Lyapunov stability theory, and analyzing a time lag related stability problem of the system; a free-weight matrix (FWM) method is provided, and the idea of time-lag system stability analysis is further widened. The time domain method can effectively deal with the time-varying time lag problem, but all have different degrees of conservation.
Disclosure of Invention
In view of this, the present invention aims to provide a time lag power system stability analysis method and apparatus, which solve the problem that the traditional time lag stability analysis method is difficult to analyze the cascading failure of the power system.
In order to achieve the purpose, the invention adopts the following technical scheme.
A time-lapse power system stability analysis method, the method comprising the steps of:
A. collecting time-lag system network structure parameters, generator frequency and power angle in the system;
B. constructing an accident chain by using the acquired data, and combining the accident chain with a Markov process to establish a state equation of a time-lag Markov jump system;
C. respectively constructing time lag stability criteria for improving a free weight matrix based on a Markov process transfer rate matrix and a Newton-Lebulitz formula according to a time lag Markov jump system state equation, equivalently transforming the time lag stability criteria on the basis, and solving a system time lag stability upper limit by utilizing a generalized characteristic value method;
D. and outputting the upper limit of the system time lag stability.
The method comprises the following steps of respectively constructing time lag stability criteria for improving a free weight matrix according to a time lag Markov jump system state equation and based on a Markov process transfer rate matrix and a Newton-Lebulitz formula, equivalently transforming the time lag stability criteria on the basis, and solving a system time lag stability upper limit by using a generalized characteristic value method, wherein the steps comprise:
c1, constructing a Lyapunov-Classy-Fuji functional considering Markov jump and solving a derivative function of the Lyapunov-Classy-Fuji functional along the system, respectively constructing free weight terms by utilizing a Markov process transfer rate matrix and a Newton-Lebunitz formula, introducing the free weight terms into a weak infinite small operator of the Lyapunov-Classy-Fuji functional, and decomposing the whole time-varying interval into two subintervals to obtain a time-varying system stability criterion;
and C2, equivalently transforming the time lag system stability criterion in the step C1 into a standard form which accords with the generalized eigenvalue method solution, and solving the time lag stability upper limit.
Specifically, in step C1, the markov process transfer rate matrix and the newton-raphicz equation are used to construct the free weight terms:
the equation for the modified free weight matrix constructed by the newton-lebeniz equation is:
wherein,
x(t)∈Rnis a state vector for a time-skewed power system,
the first derivative of the state vector x (t) with respect to time,
htin order to be a time lag,
in order to be the upper limit of the time-lag stability,
mu is the maximum rate of change of the time lag,
w, U and V are the pending matrices,
n, L and M are modified free weight matrices,
πijan element of a markov transition probability matrix for a time-lapse power system refers to a transition probability density for a system mode in the i state at time t and in the j state at time t + delta,
wherein the time lag htWith its first derivativeThe conditions are satisfied:
in addition, the solving of the time lag stability upper limit is to solve an optimization problem:
min d，
the constraint conditions are as follows:
and
wherein d is an optimization objective, and
in order to be the upper limit of the time-lag stability,
Aiis a state matrix of a time-lag power system under an operating condition i, and Ai∈Rn×n，
AdiIs a time lag matrix of a time lag power system under an operating condition i, and Adi∈Rn×n，
T1、T2、T3Is an additional matrix, and
P、Q、Ri、K、Zi、Ui、Vi、Wiare all undetermined matrices, and P ═ PT＞0，Q＝QT≥0，K＝KT＞0， And
Ω2＝[N+M L-N -L-M]，
πijan element of a markov transition probability matrix for a time-lapse power system refers to a transition rate at which a system mode is in an i state at time t and in a j state at time t + Δ, Δ being the amount of change at time t,
mu is the maximum rate of change of the time lag,
s is a set of finite modes and,
n, L and M are modified free weight matrices.
A time-lag electric power system stability analysis device comprises a data acquisition module, a time-lag Markov jump system generation module, a time-lag upper limit solving module and a result output module;
the data acquisition module is used for acquiring network structure parameters, the frequency and the power angle of a generator in the system and sending the acquired data to the time-lag Markov jump system generation module;
the Markov jump system generation module constructs an accident chain by using the acquired data, and combines the accident chain with a Markov process to establish a time-lag Markov jump system state equation;
the time lag upper limit solving module is used for respectively constructing time lag stability criteria of an improved free weight matrix based on a Markov process transfer rate matrix and a Newton-Lebulitz formula according to a time lag Markov jump system state equation, equivalently transforming the time lag stability criteria on the basis, and solving the time lag stability upper limit of the system by utilizing a generalized characteristic value method;
and the result output module is used for outputting the time lag stability upper limit of the system.
The time-lag upper limit solving module comprises a time-lag system stability criterion determining unit and a standard solving unit, wherein:
a time-lag system stability criterion determining unit, which is used for constructing a Lyapunov-Classy Function considering Markov jump and solving a derivative function of the Lyapunov-Classy Function along the system, respectively constructing free weight terms by utilizing a Markov process transfer rate matrix and a Newton-Labrinitz formula, introducing the free weight terms into a weak infinite operator of the Lyapunov-Classy Function, and decomposing the whole time-varying time-lag interval into two sub-intervals to obtain a time-lag system stability criterion;
and the standard solving unit is used for equivalently transforming the time-lag system stability criterion obtained by the time-lag system stability criterion determining unit into a standard form which accords with the generalized characteristic value method for solving, and solving the time-lag stability upper limit.
By adopting the method and the device, the improved free weight matrix is respectively constructed by utilizing the Markov process transfer rate matrix and the Newton-Lei-Bluetz formula to establish the time lag Markov variable system stability criterion, and the upper limit of the time lag of the system is solved by utilizing the generalized characteristic value method. The result shows that the method can reasonably reveal the time lag stability of the fault power system and effectively solve the time lag stability upper limit.
Drawings
Fig. 1 is a configuration diagram of a time lag power system stability analysis device in an embodiment of the present invention.
Fig. 2 is a view of a scene topology structure applied to an embodiment of the present invention.
Fig. 3 is a graph illustrating dynamic response of partial power generator to power angle under different delay conditions according to an embodiment of the present invention.
FIG. 4 is a graph illustrating the dynamic response of the power generator to the power angle under different delay conditions according to an embodiment of the present invention
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings.
Detailed exemplary embodiments are disclosed below. However, specific structural and functional details disclosed herein are merely for purposes of describing example embodiments.
It should be understood, however, that the intention is not to limit the invention to the particular exemplary embodiments disclosed, but to cover all modifications, equivalents, and alternatives falling within the scope of the disclosure. Like reference numerals refer to like elements throughout the description of the figures.
It will also be understood that the term "and/or" as used herein includes any and all combinations of one or more of the associated listed items. It will be further understood that when an element or unit is referred to as being "connected" or "coupled" to another element or unit, it can be directly connected or coupled to the other element or unit or intervening elements or units may also be present. Moreover, other words used to describe the relationship between components or elements should be understood in the same manner (e.g., "between" versus "directly between," "adjacent" versus "directly adjacent," etc.).
To describe the technical solution of the present invention, first, the technical principle of the present invention is explained.
The basic principle of the time-lag power system stability analysis method and the time-lag power system stability analysis device provided by the invention is as follows:
firstly, collecting time-lag system network structure parameters, generator frequency and power angle in the system.
When the power system is normally operated, the line has a certain initial load, but when a certain line is cut off and is stopped, the load on the line is transferred to other lines, and the lines are possibly overloaded or protected and malacted successively, so that cascading failure occursAs an intermediate link prediction index, the index determines the next-level fault line of the accident chain.
Suppose that line i fails in the system, SiFor the flow through line i, α, β, and γ are calculated as follows:
in the formula,representing the flow of the line i before the accident,representing the flow of the line j before the accident,representing the power flow of line j after an accident on line i, SjmaxRepresenting the maximum power flow that line j is allowed to flow through.
Defining intermediate link prediction indexComprises the following steps:
in the formula (4), the reaction mixture is,the larger the value of (a), the more the non-faulty line j is affected by the fault of the line i, indicating that the line j has a higher possibility of becoming a faulty line of the next stage. Therefore, the probability of state transition caused by power flow transition can be calculated by the following formula:
as can be seen from the above description, the development process of the accident chain can be described as a series of evolutionary processes with conditional probability events. In each link of accident chain development, each level of generation is only related to the previous level, but not to the accident before the system, and the conditional probability distribution of the future state is only dependent on the current state, so that the development process of the accident chain belongs to the Markov process, and the Markov theory can be used for realizing the development processThe accident chain with randomness and correlation features is described: let rtR (t) is a system mode, which is a homogeneous markov process that takes values from a finite set S {1, 2. For accident chains L ═ L1L2L3...LnAn intermediate link LiModality r as a stochastic Markov processiAnd r (i), namely the finite mode set S corresponds to the number of intermediate links of the accident chain L, and then the Markov transition probability matrix pi is calculated by using the formula (5) to establish the Markov process based on the accident chain.
Combining the accident chain with the Markov process to obtain a time-lag Markov jump system model:
in the formula (6), time lag htThe conditions are satisfied:
wherein, x (t) ∈ RnAnd z (t) ∈ RnRespectively, a state vector and an output vector of the system, Ai、Adi、Bi、BdiFor a known matrix of appropriate dimensions, htFor the system time lag, h is the time lag stability upper limit, and μ is the maximum rate of change of the time lag, then the state transition probability of the homogeneous markov process is:
wherein
πijIs that the system modality is in the i state at time t and at time t + ΔA transition rate in the j state, andΔ is the amount of change at time t, and o (Δ) is a higher order infinitesimal amount of change Δ at time t.
The following form of lyapunov-krassofski functional is thus constructed:
p, Q, R thereini、K、ZiAre all undetermined matrices, and P ═ PT＞0，Q＝QT≥0，K＝KT＞0，
Consider that the relation is satisfied between the elements in the markov transfer rate matrix pi for a time-lapse system characterized by equation (6):then for any suitable dimension of the matrixComprises the following steps:
wherein U isi、Vi、WiAre all undetermined matrices, and and
meanwhile, according to the newton-lebeniz formula, for any dimension-adaptive matrix N, L and M, the following formula holds:
wherein,for V (x)tT) taking a random process { xt,rtT is not less than 0Then there are:
in order to effectively reduce the conservatism of the method, a free weight term (formula (11) -formula (13)) constructed by a Markov transfer rate matrix pi and a free weight term (formula (14) -formula (16)) constructed by a Newton-Lernitz formula are introduced to the right side of formula (17), and the whole time-varying time-lag interval [0, h ] is decomposed into two sub-intervals [0, h ], [ h, h ] (h > 0) and is sorted into:
in formula 18:
wherein N, L and M are modified free weight matrices.
When phi + phivAnd (3) being less than or equal to 0, based on the Schur supplement theorem, the time-lag system stability criterion represented by the formula (6) can be obtained as follows:
given scalar quantityAnd μ, if present, P ═ PT＞0，Q＝QT≥0，K＝KT＞0， And(i＝1,2…S)，N＝[N1N2N3]T，L＝[L1L2L3]Tand M ═ M1M2M3]TIf the following expression (23) is satisfied, the Markov jump lag system (6) satisfying the conditional expression (7) is randomly stabilized:
wherein:
Ω2＝[N+M L-N -L-M]，
i＝{1,2，…，N}，
wherein omega and omega1、Ω2、Are all intermediate variables. A. theiIs a state matrix of a time-lag power system under an operating condition i, and Ai∈Rn×n，AdiIs a time lag matrix of a time lag power system under an operating condition i, and Adi∈Rn×n。
The matrix inequality represented by the formula (23) can only judge whether the system is stable, but cannot acquire information such as the time lag stability upper limit of the system, and the time lag stability upper limit is calculated by using a generalized characteristic value method in consideration of the fact that the solution of the time lag stability upper limit is a convex optimization problem with linear inequality constraint. Since the expression (23) is not in the form of a standard generalized eigenvalue and requires necessary processing, and it is noted that the elements of the four columns of the expression (23) except the first column all contain h, the matrix is usedLeft and right multipliers (23) to obtain:
then, orderAccording to Schur supplementary properties, it is possible to obtain:
order toAndobtaining:
in formula (25), reacting1,-T2,-T3Respectively replaceObtaining:
from equations (24) - (27), the problem of the time lag stability upper limit h can be transformed into the following optimization problem:
by solving the optimization problem (28) and using the equations (26) and (27) as constraints, finally, the time lag stability upper limit of the power system in the cascading failure situation can be deduced by using h to 1/d.
An embodiment of the present invention therefore includes a method for skew power system stability analysis, the method comprising the steps of:
A. collecting time-lag system network structure parameters, generator frequency and power angle in the system;
B. constructing an accident chain by using the acquired data, and combining the accident chain with a Markov process to establish a state equation of a time-lag Markov jump system;
C. respectively constructing time lag stability criteria for improving a free weight matrix based on a Markov process transfer rate matrix and a Newton-Lebulitz formula according to a time lag Markov jump system state equation, equivalently transforming the time lag stability criteria on the basis, and solving a system time lag stability upper limit by utilizing a generalized characteristic value method;
D. and outputting the upper limit of the system time lag stability.
The method comprises the following steps of constructing a time lag stability criterion based on an improved free weight matrix respectively constructed by a Markov process transfer rate matrix and a Newton-Lebulitz formula according to a time lag Markov jump system state equation, equivalently transforming the time lag stability criterion on the basis, and solving a time lag stability upper limit of the system by using a generalized characteristic value method, wherein the step comprises the following steps of:
c1, constructing a class of Lyapunov-Classy Functions considering Markov jump and solving a derivative function of the Lyapunov-Classy Functions along the system, respectively constructing free weight terms by utilizing a Markov process transfer rate matrix and a Newton-Lei-Bluez formula, introducing the free weight terms into a weak infinite small operator of the Lyapunov-Classy Functions, and decomposing the whole time-varying interval into two subintervals to obtain a time-varying system stability criterion;
and C2, equivalently transforming the time lag system stability criterion in the step C1 into a standard form which accords with the generalized eigenvalue method solution, and solving the time lag stability upper limit.
Specifically, in step C1, the markov process transfer rate matrix and the newton-raphicz equation are used to construct the free weight terms:
the equation for the modified free weight matrix constructed by the newton-lebeniz equation is:
wherein,
x(t)∈Rnis a state vector for a time-skewed power system,
the first derivative of the state vector x (t) with respect to time,
in order to be the upper limit of the time-lag stability,
mu is the maximum rate of change of the time lag,
w, U and V are the pending matrices,
n, L and M are modified free weight matrices,
πijan element of a markov transition probability matrix for a time-lapse power system refers to a transition rate at which a system mode is in the i state at time t, and in the j state at time t + delta,
wherein the time lag htWith its first derivativeThe conditions are satisfied:
in addition, the solving of the time lag stability upper limit is to solve an optimization problem:
min d，
the constraint conditions are as follows:
and
wherein d is an optimization objective, and
in order to be the upper limit of the time-lag stability,
x(t)∈Rnis a state vector for a time-skewed power system,
the first derivative of the state vector x (t) with respect to time,
Aiis a state matrix of a time-lag power system under an operating condition i, and Ai∈Rn×n，
AdiIs a time lag matrix of a time lag power system under an operating condition i, and Adi∈Rn×n，
T1、T2、T3Is an additional matrix, and
P、Q、Ri、K、Zi、Ui、Vi、Wiare all undetermined matrices, and P ═ PT＞0，Q＝QT≥0，K＝KT＞0， And
Ω2＝[N+M L-N -L-M]，
πijan element of a markov transition probability matrix for a time-lapse power system refers to a transition rate at which a system mode is in an i state at time t and in a j state at time t + Δ, Δ being the amount of change at time t,
mu is the maximum rate of change of the time lag,
s is a set of finite modes and,
n, L and M are modified free weight matrices.
In order to implement the method for analyzing the stability of the time-lag power system, the embodiment of the invention further comprises a device for analyzing the stability of the time-lag power system, as shown in fig. 1, the device for analyzing the stability of the time-lag power system comprises a data acquisition module, a time-lag markov jump system generation module, a time-lag upper limit solving module and a result output module;
the data acquisition module is used for acquiring network structure parameters, the frequency and the power angle of a generator in the system and sending the acquired data to the time-lag Markov jump system generation module;
the Markov jump system generation module constructs an accident chain by using the acquired data, and combines the accident chain with a Markov process to establish a time-lag Markov jump system state equation;
the time lag upper limit solving module is used for respectively constructing time lag stability criteria of an improved free weight matrix based on a Markov process transfer rate matrix and a Newton-Lebulitz formula according to a time lag Markov jump system state equation, equivalently transforming the time lag stability criteria on the basis, and solving the time lag stability upper limit of the system by utilizing a generalized characteristic value method;
and the result output module is used for outputting the time lag stability upper limit of the system.
The time-lag upper limit solving module comprises a time-lag system stability criterion determining unit and a standard solving unit, wherein: a time-lag system stability criterion determining unit, which is used for constructing a class of Lyapunov-Classy Functions considering Markov jump and solving the derivative function of the Lyapunov-Classy Functions along the system, respectively constructing free weight terms by utilizing a Markov process transfer rate matrix and a Newton-Laeviz formula, introducing the free weight terms into a weak infinite small operator of the Lyapunov-Classy Functions, and decomposing the whole time-varying time-lag interval into two sub-intervals to obtain a time-lag system stability criterion;
and the standard solving unit is used for equivalently transforming the time-lag system stability criterion obtained by the time-lag system stability criterion determining unit into a standard form which accords with the generalized characteristic value method for solving, and solving the time-lag stability upper limit.
The technical effects of the present invention are illustrated below by specific embodiments, and it should be understood by those skilled in the art that the specific embodiments are merely illustrative and are not meant to impose any limitation on the scope of the present invention.
Figure 2 shows a new england-new york interconnection system of IEEE16 machine 68 nodes. The system can be divided into 5 large areas: regions 1,2 and 3 are the equivalent systems, region 4 is the new york system and region 5 is the new england system, respectively. The generator adopts a 6-order detailed model, the excitation adopts IEEE-DC1 type excitation, and the load model adopts a WECC load model, 50% of constant active load, 50% of constant reactive impedance load and 20% of dynamic load.
Most of the faults of the power system are line faults, and the importance of the tie line between areas is considered, and the tie line faults are used as triggering links of an accident chain. In the regional links of the IEEE16 machine 68 node system, the power flows of the links 1-2 between the regions 4 and 5 and the links 46-49 between the regions 3 and 4 are 160.47MW and 208.47MW, respectively, and are close to the maximum transmission capacity of the line, so that the links 1-2 and the links 46-49 are selected as trigger links of different accident chains, respectively, and the accident chain generation is completed until the system generates the constant amplitude and amplified low frequency oscillation. According to the indexes mentioned in the foregoing of the present invention, the predicted accident chain is:
l1 ═ { line 1-2 → line 3-4 → line 2-3}
L2 ═ lines 46-49 → lines 32-33 → lines 31-38}
Markov process for establishing corresponding accident chains L1 and L2, system modality rtR (t) each being of a finite set Sm1,2,3, and m is 1,2, and the transition probability matrix is:
aiming at an accident chain L1, a state matrix A of a sixteen-machine system after Schur order reductioniAnd a time lag matrix Adi(i is 1,2, 3) is substituted for the formulas (27) and (28), and the time lag stability upper limit is determined by the LMI method。
To verify the feasibility and effectiveness of the method provided by the invention, damping controller design is carried out on the IEEE16 machine 68 node system based on the H2/H infinity control method, time lags are respectively set to be 0ms, 40ms, 82.9ms and 120ms, and relative power angle differential dynamic response curves between the generators 8-15 and the generators 1-16 are observed, as shown in FIG. 3. As can be seen from the figure, when the time lag is smaller than the time lag stability upper limit, the system is in a stable state in the accident chain development process, and can rapidly damp interval oscillation; when the time lag is equal to the time lag stability upper limit, the system is in a weak damping state at the initial link of the accident chain, but gradually recovers to a stable state along with the evolution of the accident chain; when the time lag is larger than the time lag stability upper limit, the system gradually develops from a weak damping state to an unstable state along with the generation of an accident chain.
The damping ratios of the relative power angle difference curves of the generators 8-15 and the generators 1-16 at the final link of the accident chain, which are obtained by utilizing the prony algorithm, are respectively listed in the table 1 and the table 2. It can be seen from the table that when the time lag reaches 82.9ms in the case of a cascading failure of the system, the damping ratios of the power-angle difference curves are 8.49% and 9.62%, respectively, and although there is still a certain damping, the damping ratio is already reduced to below 10%, and the control requirement cannot be met. Therefore, the accident chain is combined with the Markov process, the time-lag power system stability method is feasible, and the time-lag stability upper limit obtained by the time-lag power system stability method is closer to the actual value.
Damping ratio of G8 and G15 power angle difference of watch 116 machine system under each time lag time
Damping ratio of G1 and G16 power angle difference of table 216 under each time lag time
For the accident chain L2, according to the transition probability matrixAnd a known coefficient matrix ai,Adi(i ═ 1,2, 3), solving the time lag stability upper limit by the generalized eigenvalue method. To verify the effectiveness of the method, the time lag is set to 0ms, 50ms, 101.4ms and 150ms respectively, and the relative power angle differential dynamic response curves between the generators 4-13 and the generators 7-14 are observed, as shown in fig. 4.
As can be seen from fig. 4, when the time lag is smaller than the time lag stability upper limit, each link in the accident chain development process can oscillate in the damping interval within 20 s; when the time lag is increased from 0ms to 101.4ms, the damping effect is weakened, and the relative power angle difference curve of the generator has certain swing in each link; when the time lag is increased to 150ms, the relative power angle difference curve of the generator generates constant-amplitude and amplified oscillation, and the system is in a destabilization state.
The damping ratio of the relative power angle difference curves of the generators 4-13 and the generators 7-14 at the final link of the accident chain, which are obtained by utilizing the prony algorithm, are respectively shown in the table 3 and the table 4. As can be seen from the table, the damping ratio of the system is 17.59% and 15.75% respectively when the time lag is 0ms, and the damping ratio of the system is 11.81% and 11.54% respectively when the time lag is 40ms, and the system has larger damping and is in a stable state in both cases. However, when the time lag reaches 101.4ms, the damping ratio of the power-angle difference curve is reduced to below 10%, which is 5.16% and 5.92%, respectively, and the system is in a critical stable state. Therefore, the accident chain is combined with the Markov process, so that the time lag stability of the fault power system can be reasonably disclosed, and the time lag stability upper limit can be effectively solved.
Damping ratio of G4 and G13 power angle difference of table 316 machine system under each time lag time
TABLE 416 damping ratio of G7 and G14 power angle difference at each time lag time
It should be noted that the above-mentioned embodiments are only preferred embodiments of the present invention, and should not be construed as limiting the scope of the present invention, and any minor changes and modifications to the present invention are within the scope of the present invention without departing from the spirit of the present invention.
Claims (4)
1. A time-lapse power system stability analysis method, the method comprising the steps of:
A. collecting time-lag system network structure parameters, generator frequency and power angle in the system;
B. constructing an accident chain by using the acquired data, and combining the accident chain with a Markov process to establish a state equation of a time-lag Markov jump system;
C. respectively constructing time lag stability criteria for improving a free weight matrix based on a Markov process transfer rate matrix and a Newton-Lebulitz formula according to a time lag Markov jump system state equation, equivalently transforming the time lag stability criteria on the basis, and solving a system time lag stability upper limit by utilizing a generalized characteristic value method;
D. outputting the upper limit of the system time lag stability;
the construction method of the accident chain comprises the following steps:
α, β and gamma are respectively used as evaluation indexes of 3 aspects of the tidal current change rate and the overload margin of the line and the coupling degree between the fault line and the predicted line, and the indexes are usedAs an intermediate link prediction index, the index determines the next-level fault line of the accident chain;
suppose that line i fails in the system, SiFor the flow through line i, α, β, and γ are calculated as follows:
in the formula,representing the flow of the line i before the accident,representing the flow of the line j before the accident,representing the power flow of line j after an accident on line i, SjmaxRepresents the maximum power flow allowed to flow through line j;
defining intermediate link prediction indexComprises the following steps:
in the formula,the larger the value of (a), the more the non-faulty line j is affected by the fault of the line i, which indicates that the line j has a higher possibility of becoming a next-stage faulty line; therefore, the probability of state transition caused by power flow transition can be calculated by the following formula:
as can be seen from the above description, the development process of the accident chain can be described as a series of evolution processes with conditional probability events; in each link of accident chain development, each level of generation is only related to the previous level, but is not related to the accident before the system, and the conditional probability distribution of the future state of the accident chain is only dependent on the current state, so that the development process of the accident chain belongs to a Markov process, and the accident chain with randomness and relevance characteristics can be described by using a Markov theory: let rtR (t) is a system mode, which is a homogeneous markov process that takes values from a finite set S {1,2, …, S }; for accident chains L ═ L1L2L3…LnAn intermediate link LiModality r as a stochastic Markov processiAnd r (i), namely the finite mode set S corresponds to the number of intermediate links of the accident chain L, and then the Markov transition probability matrix pi is calculated by using the formula, so as to establish the Markov process based on the accident chain.
2. The method for analyzing the stability of the time-lag power system as recited in claim 1, wherein the step of constructing the time-lag stabilization criterion of the improved free-weight matrix based on the markov process transfer rate matrix and the newton-levenz formula according to the state equation of the time-lag markov jump system, equivalently transforming the time-lag stabilization criterion on the basis, and solving the upper limit of the time-lag stabilization of the system by using the generalized eigenvalue method comprises:
c1, constructing a Lyapunov-Classy-Fuji functional considering Markov jump and solving a derivative function of the Lyapunov-Classy-Fuji functional along the system, respectively constructing free weight terms by utilizing a Markov process transfer rate matrix and a Newton-Lebunitz formula, introducing the free weight terms into a weak infinite small operator of the Lyapunov-Classy-Fuji functional, and decomposing the whole time-varying interval into two subintervals to obtain a time-varying system stability criterion;
and C2, equivalently transforming the time lag system stability criterion in the step C1 into a standard form which accords with the generalized eigenvalue method solution, and solving the time lag stability upper limit.
3. The method for analyzing the stability of the time-lag power system as claimed in claim 2, wherein in step C1, the markov process transfer rate matrix and the newton-raphicz equation are used to construct the free weight terms respectively as follows:
the equation for the modified free weight matrix constructed by the newton-lebeniz equation is:
wherein,
x(t)∈Rnis a state vector for a time-skewed power system,
the first derivative of the state vector x (t) with respect to time,
htin order to be a time lag,
in order to be the upper limit of the time-lag stability,
mu is the maximum rate of change of the time lag,
w, U and V are the pending matrices,
n, L and M are modified free weight matrices,
πijan element of a markov transition probability matrix for a time-lapse power system refers to a transition probability density for a system mode in the i state at time t and in the j state at time t + delta,
wherein the time lag htWith its first derivativeThe conditions are satisfied:
4. the time lag power system stability analysis method of claim 2, wherein the solving of the upper time lag stability limit is an optimization problem:
min d，
the constraint conditions are as follows:
and
wherein d is an optimization objective, and
in order to be the upper limit of the time-lag stability,
Aiis a state matrix of a time-lag power system under an operating condition i, and Ai∈Rn×n，
AdiIs a time lag matrix of a time lag power system under an operating condition i, and Adi∈Rn×n，
T1、T2、T3Is an additional matrix, and
P、Q、Ri、K、Zi、Ui、Vi、Wiare all undetermined matrices, and P ═ PT>0，Q＝QT≥0，K＝KT>0，Zi＝Zi T≥0，Ui＝Ui T≥0，Vi＝Vi TNot less than 0 and Wi＝Wi T≥0，
Ω2＝[N+M L-N -L-M]，
πijAn element of a markov transition probability matrix for a time-lapse power system refers to a transition rate at which a system mode is in an i state at time t and in a j state at time t + Δ, Δ being the amount of change at time t,
mu is the maximum rate of change of the time lag,
s is a set of finite modes and,
n, L and M are modified free weight matrices.
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