CN113488991A - Electromechanical transient small interference stability analysis method based on minimum characteristic trajectory method - Google Patents

Electromechanical transient small interference stability analysis method based on minimum characteristic trajectory method Download PDF

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CN113488991A
CN113488991A CN202110667904.XA CN202110667904A CN113488991A CN 113488991 A CN113488991 A CN 113488991A CN 202110667904 A CN202110667904 A CN 202110667904A CN 113488991 A CN113488991 A CN 113488991A
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黄伟
黄润
甘德强
张�杰
吴琛
张丹
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Yunnan Power Grid Co Ltd
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Abstract

The invention relates to an electromechanical transient small interference stability analysis method based on a minimum characteristic trajectory method, which is used for providing a frequency domain stability margin of small interference stability of a power system so as to maximize system performance parameters and ensure that the system meets stable operation requirements, and belongs to the technical field of power grid safety. The method comprises the following steps: selecting a rotor loop as a forward channel link to obtain a compact Heffron-Phillips model; applying a characteristic trajectory method to give a frequency domain stability margin of the system; writing the frequency domain stability margin into a loop addition form; simplifying the frequency domain stability margin by using a similar matrix method; analyzing the influence of a prime motor-speed regulator link; analyzing the influence of a PSS link of a power system stabilizer; and guiding the parameter setting of the controller according to the frequency domain stability margin.

Description

Electromechanical transient small interference stability analysis method based on minimum characteristic trajectory method
Technical Field
The invention belongs to the technical field of power grid safety, relates to an electromechanical transient small interference stability analysis method, and particularly relates to an electromechanical transient small interference stability analysis method of a multi-input-multi-output system based on a minimum characteristic trajectory method.
Background
Given a power grid, the electromechanical transient small disturbance stability analysis needs to obtain answers to the following questions: the role of controllers in the system, such as the additional excitation control (PSS) and speed regulators; and obtaining a controller-filter parameter setting method of expected gain and phase, and ensuring the robustness of control performance.
In our country, the structural form of the electric power system generally experiences the times of 'pure alternating current, alternating current and direct current + wind and light', and when the power grid has major structural changes, such as large-scale penetration of wind power and photovoltaic proportion currently in progress, the answer to the problems still hopes to be obtained. Obviously, these problems are more difficult because the mathematical model of the grid is more complex and the synchronization properties of the traditional grid have changed from quality to quality. Because of this, the study of the stability of the small interference is more extensive and profound than before.
In the field of stability analysis of power systems, small interference stability occupies an important position. At present, the problem of setting parameters of a controller with the PSS design as the core is well solved. Nevertheless, advances in the field of small disturbance stability analysis of power systems remain unsatisfactory. The existing methods almost provide sufficient conditions for stability, and are equivalent to each other, but the specific application is not easy.
In general, an engineering practical method should have a strict mathematical basis and an analytic stability margin expression, and can analyze the influence of any loop on the stability and the influence of any parameter on the stability, so as to obtain the answer to the above-mentioned problem.
The information disclosed in this background section is only for enhancement of understanding of the general background of the invention and should not be taken as an acknowledgement or any form of suggestion that this information forms the prior art already known to a person skilled in the art.
Disclosure of Invention
The invention aims to solve the defects of the prior art and provides an electromechanical transient small interference stability analysis method based on a minimum characteristic trajectory method.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
the electromechanical transient small interference stability analysis method based on the minimum characteristic trajectory method comprises the following steps:
1) selecting a rotor loop as a forward channel link to obtain a compact Heffron-Phillips model;
2) applying a characteristic trajectory method to give a frequency domain stability margin of the system;
3) writing the frequency domain stability margin into a loop addition form;
4) simplifying the frequency domain stability margin by using a similar matrix method;
5) analyzing the influence of a prime motor-speed regulator link;
6) analyzing the influence of a PSS link of a power system stabilizer;
7) and selecting the parameters of the controller according to the frequency domain stability margin.
2. The electromechanical transient small interference stability analysis method based on the minimum feature trajectory method according to claim 1, wherein the specific method of the step 1) is as follows:
extracting a rotor loop in a Heffron-Phillips model, and enabling a rotor channel (sM + K)D)-1The other links are combined together to be used as a feedback channel to obtain a compact Heffron-Phillips model;
defined in the compact form of the Heffron-Phillips model described above:
GQ(s)＝GQ1(s)+GQ2(s), formula (3);
wherein G isQ1(s)＝-K2[(K3+sT′d0)+GEX(s)K6]-1(GEX(s)K5+K4) Formula (1);
Figure BDA0003117651620000021
in the formula, matrix K2、K3、K4、K5、K6Is a linearized model coefficient matrix; diagonal matrix T'd0The transient time constant of the d axis of each generator is contained; the diagonal matrix M contains a generator rotor motion inertia constant; diagonal matrix KDThe damping coefficient of the rotor motion is contained; diagonal matrix GEXIs an excitation system transfer function matrix; diagonal matrix HPSS(s) is the PSS transfer function matrix; omega0For system synchronous speed, s is the generalized frequency.
Further, preferably, the applying the feature trajectory method in step 2) gives a frequency domain stability margin of the system, specifically:
vis (sM + K)D)-1And (3) transmitting a matrix for a forward channel, and taking other links as feedback channels to obtain a return difference matrix I + L of the closed-loop system:
Figure BDA0003117651620000022
wherein, the matrix K1Is a linearized model coefficient matrix; diagonal matrix GMIs a governor-prime mover system transfer function matrix;
according to the nature of the matrix eigenvalues:
λ (I + L) ═ I + λ (L), formula (5);
taking a characteristic track lambdamin(I + L) distance | λ from-1 Point-1(L) | is the margin of stability.
Further, it is preferable that the frequency domain stability margin in step 3) is written in a loop addition form, specifically:
substituting s ═ j ω into the expression for L according to equation (4)
Figure BDA0003117651620000023
Wherein ω represents an angular frequency;
in formula (6), the matrices M and KDAre all diagonal matrices, therefore
Figure BDA0003117651620000026
Bringing formula (7) into formula (6)
Figure BDA0003117651620000024
The following matrix is introduced:
Figure BDA0003117651620000025
LGQ1＝(jωM+KD)-1GQ1ω0/(j ω), formula (10);
LGQ2＝(jωM+KD)-1GQ2ω0/(j ω), formula (11);
LGM＝(jωM+KD)-1GMformula (12);
equation (9) represents the damping component, equation (10) represents the excitation component, equation (11) represents the PSS component, and equation (12) represents the prime mover-governor component.
Further, preferably, the step 4) of simplifying the frequency domain stability margin by using a similarity matrix method specifically includes:
in equation (4), feature decomposition is performed on the related term of K1:
Figure BDA0003117651620000031
wherein Λ represents a characteristic value matrix, and U represents a characteristic vector matrix; the above equation holds if the matrix K1 can be diagonalized; then:
I+L＝UΛU-1+ΔL＝U(Λ+U-1ΔLU)U-1formula (14);
based on the definition of the similarity matrix
λ(I+L)＝λ(Λ+U-1ΔLU)＝λ(SM) Formula (15);
wherein
SM＝Λ+U-1Δ LU, equation (16);
the stability margin is equivalent to the minimum eigenvalue for solving the similarity matrix
λ-1(L)＝λmin(SM)-1＝λmin(Λ+U-1Δ LU) -1, formula (17);
the eigenvalues are approximated with minimum modulo diagonal bins to simplify the frequency domain stability margin:
λ-1(L)＝SM11-1, formula (18);
equation (18) is the simplified frequency domain stability margin, where SM11Is the minimum modulo diagonal of the similarity matrix.
Further, preferably, the step 5) analyzes the influence of the prime mover-speed regulator link, specifically:
since the transfer matrix component of the prime mover-governor is as shown in equation (12), the similarity matrix S isMThe primary motor-governor having a component of
U-1LGMU＝U-1(jωM+KD)-1GMU, formula (19);
suppose the power grid has m units, V ═ U-1)TIs a left eigenvector matrix, then
Figure BDA0003117651620000032
Due to SM＝Λ+U-1Δ LU, Λ is a diagonal matrix, consider
Figure BDA0003117651620000033
Smallest modular diagonal element in
Figure BDA0003117651620000034
Component(s) of
Figure BDA0003117651620000041
Wherein u is1,v1Respectively representing the right and left eigenvectors, symbols
Figure BDA0003117651620000049
Represents the Hadamard product of the matrix;
handle LGMFormula (21):
Figure BDA0003117651620000042
wherein, gMiRepresenting a diagonal matrix GMThe ith diagonal element of (1)DiRepresenting a diagonal matrix KDThe ith diagonal element of (1)iDenotes the ith diagonal element, u, of the diagonal matrix Mi1、vi1Represents u1、v1The ith element of (1);
in the ultra-low mode frequency band, due to v1≈Mu1Right eigenvector u1Each element is equal, assuming K D0, then
Figure BDA0003117651620000043
Equation (23) shows the association of the common unified frequency model in the literature with the (more general) model in this document, we can just follow the individual units gMThe phase of (j omega) can be judged to be stableSexual contribution, still very simple: gMThe (j ω) hysteresis deteriorates the stability when it exceeds 90 degrees, and improves the stability when it falls below 90 degrees.
Further, preferably, the step 6) analyzes the influence of the PSS link of the power system stabilizer, specifically:
similar to the formula (21),
Figure BDA0003117651620000044
smallest modular diagonal element in
Figure BDA0003117651620000045
As follows
Figure BDA0003117651620000046
The transfer matrix component of the PSS is shown in formula (11);
defining forward channel transfer matrix through which PSS passes
HPVr＝K2[K3+jωT′d0+GEXK6]-1GEXFormula (25);
the transfer matrix component of the PSS may be further written as
Figure BDA0003117651620000047
Let fijRepresentative matrix (j ω M + K)D)-1HPVrElement at position (i, j), the contribution of PSS to stability margin is obtained:
Figure BDA0003117651620000048
the contribution of PSS to stability margin has the form of a dot product:
Figure BDA0003117651620000051
wherein, vector sum (F)uv) Is a matrix FuvColumn (c) and, diag (H)PSS) Are vectors of the same dimension, where the elements represent the PSS frequency response for each unit;
assuming that the grid has n units, then
Figure BDA0003117651620000052
According to the above formula, the parameter of the PSS should satisfy the following phase condition, so that the minimum feature trajectory moves vertically downward and is away from the-1 point on the complex plane, that is, the phase parameter of the PSS is set by using the following formula:
Figure BDA0003117651620000053
further, preferably, the step 7) selects the controller parameter according to the frequency domain stability margin, and the specific method includes: selecting PID parameters of the speed regulator according to the formula (23) obtained in the step 5), and selecting parameters of the power system stabilizer PSS according to the formula (30) obtained in the step 6).
The method provided by the invention can provide a frequency domain stability index based on a minimum characteristic trajectory method only by simplifying a forward channel function of the rotor loop, can convert a complicated multi-input multi-output system matrix stability analysis problem into a simple and feasible single variable function, is a powerful tool for obtaining a simple frequency domain stability margin, and is never applied to solving the electromechanical transient small interference stability problem.
Compared with the prior art, the invention has the beneficial effects that:
the method provided by the invention can provide the frequency domain stability margin of the multi-input-multi-output power system, the mathematical relation of the stability margin is simple, the simple analysis of each controller link loop is realized, the influence of a prime mover-speed regulator link, a power system stabilizer PSS link and the like on the small interference stability can be clearly seen, and the electromechanical transient small interference stability problem of the multi-input-multi-output power system can be converted into a simple and feasible single variable function analysis problem. Compared with the existing damping torque method and the existing number-keeping method, the method has the advantages of simpler analysis and more rigorous theoretical basis, and can realize quantitative analysis on a prime motor-speed regulator link and a power system stabilizer PSS link.
Drawings
FIG. 1 is a flow chart of an electromechanical transient small disturbance stability analysis method based on a minimum characteristic trajectory method;
FIG. 2 is a Heffron-Phillips model of a conventional small interference stability analysis provided by the present invention;
FIG. 3 is a compact form of the transformed Heffron-Phillips model provided by the present invention;
FIG. 4 is a vector diagram of the contribution of the prime mover-governor loop of each unit of the 11-node system to the damping of the zone mode, provided by the application example of the present invention;
FIG. 5 shows the PSS compensation angle of the 11-node 6-machine system area mode obtained by the characteristic trace method provided by the application example of the present invention;
FIG. 6 shows the PSS compensation angle of the 11-node 6-machine system area mode obtained by the ideal phase-frequency characteristic method according to the embodiment of the present invention;
fig. 7 shows the PSS compensation angle of the 11-node 6-machine system area pattern obtained by the residue method according to the embodiment of the present invention.
Detailed Description
The present invention will be described in further detail with reference to examples.
It will be appreciated by those skilled in the art that the following examples are illustrative of the invention only and should not be taken as limiting the scope of the invention. The examples do not specify particular techniques or conditions, and are performed according to the techniques or conditions described in the literature in the art or according to the product specifications. The materials or equipment used are not indicated by manufacturers, and all are conventional products available by purchase.
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail with reference to the following embodiments and accompanying drawings. The exemplary embodiments and descriptions of the present invention are provided to explain the present invention, but not to limit the present invention.
A method for analyzing electromechanical transient small interference stability of a minimum characteristic trajectory method comprises the following steps:
1) selecting a rotor loop as a forward channel link to obtain a compact Heffron-Phillips model;
2) applying a characteristic trajectory method to give a frequency domain stability margin of the system;
3) writing the frequency domain stability margin into a loop addition form;
4) simplifying the frequency domain stability margin by using a similar matrix method;
5) analyzing the influence of a prime motor-speed regulator link;
6) analyzing the influence of a PSS link of a power system stabilizer;
7) and selecting the parameters of the controller according to the frequency domain stability margin.
Selecting a rotor loop as a forward channel link in the step 1) to obtain a compact Heffron-Phillips model;
as shown in FIG. 2, the column vector Δ P in the conventional Heffron-Phillips modelmOutputting mechanical power for the prime mover; column vector Δ Pe1And Δ Pe2For the electromagnetic power Δ P of the uniteA component of (a); column vector Δ PdIs the system power disturbance; the column vector delta omega is the rotating speed of each generator; the column vector delta is the angular displacement of each generator rotor relative to the synchronous reference shaft, namely the power angle; column vector Δ E'qThe quadrature axis transient electromotive force of each generator; column vector Δ E'fdOutputting a voltage for the automatic voltage regulator; column vector Δ uPSSOutputting the signal for the PSS; omega0Synchronizing the rotation speed of the system; the diagonal matrix M contains a generator rotor motion inertia constant; diagonal matrix KDThe damping coefficient of the rotor motion is contained; diagonal matrix T'd0The transient time constant of the d axis of each generator is contained; matrix K1、K2、K3、K4、K5、K6The method is a linear model coefficient matrix, and reflects network structure, element parameters, operation conditions and load characteristics; diagonal matrix GM(s) is a governor-prime mover system transfer function matrix; diagonal matrix GEX(s) is an excitation system transfer function matrix; diagonal matrix HPSS(s) is the PSS transfer function matrix; s is a generalized frequency.
Extracting the rotor loop, passing the rotor channel (sM + K)D)-1The other links are combined together to be used as a feedback channel, and a more compact Heffron-Phillips model is obtained as shown in figure 3.
Defined in the compact model
GQ1(s)＝-K2[(K3+sT′d0)+GEX(s)K6]-1(GEX(s)K5+K4) Formula (1);
Figure BDA0003117651620000061
GQ(s)＝GQ1(s)+GQ2(s), formula (3);
the frequency domain stability margin of the system given by the characteristic trajectory method in the step 2) is specifically as follows:
vis (sM + K)D)-1A forward channel transfer matrix is adopted, and the other links are feedback channels, so that a return difference matrix I + L of the closed-loop system is obtained
Figure BDA0003117651620000062
Substituting s to j ω in the above equation, ω represents the angular frequency. Defined according to the eigen-track method, the minimum eigen-track λmin[L(jω)]The distance from the origin is the stability margin. For simplicity of description, "j ω" is omitted hereinafter.
According to the nature of the matrix eigenvalues:
λ (I + L) ═ I + λ (L), formula (5);
following the convention of single input-single output (SISO) system stability analysis, taking the characteristic trajectory lambdamin(I + L) distance | λ from-1 Point-1(L) | is the margin of stability.
The step of writing the frequency domain stability margin into a loop addition form in the step 3) specifically comprises the following steps:
substituting s ═ j ω into the expression of L according to the expression of L obtained in step 2)
Figure BDA0003117651620000071
In formula (6), the matrices M and KDAre all diagonal matrices, therefore
Figure BDA0003117651620000075
Bringing formula (7) into formula (6)
Figure BDA0003117651620000072
The following matrix is introduced:
Figure BDA0003117651620000073
LGQ1＝(jωM+KD)-1GQ1ω0/(j ω), formula (10);
LGQ2＝(jωM+KD)-1GQ2ω0/(j ω), formula (11);
LGM＝(jωM+KD)-1GMformula (12);
equation (9) represents the damping component, equation (10) represents the excitation component, equation (11) represents the PSS component, and equation (12) represents the prime mover-governor component.
The step of simplifying the frequency domain stability margin by using a similar matrix method in the step 4) specifically comprises the following steps:
in equation (4), feature decomposition is performed on the related term of K1:
Figure BDA0003117651620000074
wherein Λ represents a characteristic value matrix, and U represents a characteristic vector matrix; the above equation holds if the matrix K1 can be diagonalized; then:
I+L＝UΛU-1+ΔL＝U(Λ+U-1ΔLU)U-1formula (14);
based on the definition of the similarity matrix
λ(I+L)＝λ(Λ+U-1ΔLU)＝λ(SM) Formula (15);
wherein
SM＝Λ+U-1Δ LU, equation (16);
the stability margin is equivalent to the minimum eigenvalue for solving the similarity matrix
λ-1(L)＝λmin(SM)-1＝λmin(Λ+U-1Δ LU) -1, formula (17);
the eigenvalues are approximated with minimum modulo diagonal bins to simplify the frequency domain stability margin:
λ-1(L)＝SM11-1, formula (18);
equation (18) is the simplified frequency domain stability margin, where SM11Is the minimum modulo diagonal of the similarity matrix.
The step 5) of analyzing the influence of the prime mover-speed regulator link specifically comprises the following steps:
the transfer matrix component of the prime mover-governor is shown as equation (12).
In the similarity matrix SMThe primary motor-governor having a component of
U-1LGMU＝U-1(jωM+KD)-1GMU, formula (19);
suppose the grid has m units, note that V ═ U-1)TIs a left eigenvector matrix, then
Figure BDA0003117651620000081
Because of SM＝Λ+U-1Δ LU and Λ areA diagonal matrix, focusing on
Figure BDA0003117651620000082
Smallest modular diagonal element in
Figure BDA0003117651620000083
Component, attention to
Figure BDA0003117651620000084
Where u is1,v1Respectively representing the right and left eigenvectors, symbols
Figure BDA0003117651620000086
Representing the Hadamard product of the matrix. The above formula contains one piece of familiar information: the stability margin depends only on the relevant feature vector i.e. (u)1,v1) There is no direct relation to other feature vectors.
Now handle LGMSubstituting the compound into the formula shown in the specification,
Figure BDA0003117651620000085
wherein g isMiRepresenting a diagonal matrix GMThe ith diagonal element of (1)DiRepresenting a diagonal matrix KDThe ith diagonal element of (1)iDenotes the ith diagonal element, u, of the diagonal matrix Mi1、vi1Represents u1、v1The ith element of (1).
At very low frequencies, the matrix v1u1 TOther features are also present. Note K1Close to symmetry and the rows are 0 and the main diagonal is positive, which means that the matrix K is1Like a Laplace matrix. As is well known, the Laplace matrix has one eigenvalue of 0, with the corresponding right eigenvector all positive and each element equal. Obviously, this 0 eigenvalue corresponds to the ultra low frequency mode, u1Is the corresponding feature vector. Due to v1≈Mu1Suppose K D0, then
Figure BDA0003117651620000091
Equation (23) is solved in Matlab, and the influence of the prime mover speed governor is quantitatively analyzed. In the ultra-low mode frequency band, the feature vector of the ultra-low frequency mode generally has u1,v1>0, which means that the stability analysis is very convenient, as long as the phase of each prime mover-speed regulator is concerned. Obviously, if < gM(jω)<P/2, i.e. its lag angle is greater than 90 degrees, the projection of the prime mover-governor component on the imaginary axis is positive, meaning that it contributes negative damping, deteriorating stability.
The step 6) of analyzing the influence of the PSS link of the power system stabilizer specifically comprises the following steps:
Figure BDA0003117651620000092
smallest modular diagonal element in
Figure BDA0003117651620000093
As follows
Figure BDA0003117651620000094
Similar to step 5), we focus on
Figure BDA0003117651620000095
The PSS component in (1) is represented by the formula (2), and the PSS is contained in GQ2In the related item, the transfer matrix component of PSS is as shown in formula (11).
Defining forward channel transfer matrix through which PSS passes
HPVr＝K2[K3+jωT′d0+GEXK6]-1GEXFormula (25);
the transfer matrix component of the PSS may be further written as
Figure BDA0003117651620000096
Let fijRepresentative matrix (j ω M + K)D)-1HPVrAt the element at position (i, j), the contribution of the PSS to the stability margin can be obtained
Figure BDA0003117651620000097
The contribution of PSS to stability margin has the form of a dot product:
Figure BDA0003117651620000098
wherein vector sum (F)uv) Is a matrix FuvColumn (c) and, diag (H)PSS) Are vectors of the same dimension, with the elements representing the PSS frequency response for each unit. Obviously, sum (F) when studying PSS parameter settingsuv) The effect of (1) is similar to the residue, reflecting the angle that the PSS should compensate for. Of course, the feature trajectory stability margin has a more general analytical potential.
Now see the contribution of PSS to individual mode damping. Since the contributions of the individual PSS to the stability margin are simply added, it is sufficient to look at the contribution of one PSS. Assuming that the grid has n units, then
Figure BDA0003117651620000099
According to the above formula, the parameters of the PSS should satisfy the following phase condition, so that the minimum feature trajectory "moves down vertically", away from the-1 point on the complex plane:
Figure BDA0003117651620000101
in practical production, the ideal phase frequency characteristic method isAccording to HPVr(j ω) tuning the PSS in terms of the diagonal phase, i.e. tuning the PSS parameters according to the following phase conditions:
arg[f11h11]0, formula (31);
and (4) optimally designing the PSS parameters by taking the formula (30) as a constraint condition.
And in the step 7), selecting the controller parameters according to the frequency domain stability margin, namely using the formula (23) obtained in the step 5) and the formula (31) obtained in the step 6), and designing the controller parameters by using a Matlab tool.
Examples of the applications
In an application example of the present invention, a method for analyzing electromechanical transient small interference stability of a multiple-input multiple-output system based on a minimum feature trajectory method is provided, as shown in fig. 1, the method includes:
step 101: selecting a rotor loop as a forward channel link to obtain a compact Heffron-Phillips model;
step 102: applying a characteristic trajectory method to give a frequency domain stability margin of the system;
step 103: writing the frequency domain stability margin into a loop addition form;
step 104: simplifying the frequency domain stability margin by using a similar matrix method;
step 105: analyzing the influence of a prime motor-speed regulator link;
step 106: analyzing the influence of a PSS link of a power system stabilizer;
step 107: and selecting the parameters of the controller according to the frequency domain stability margin.
In specific implementation, in a Heffron-Phillips model of a conventional small interference stability analysis as shown in FIG. 2, a rotor loop (sM + K) is selectedD)-1As a forward path transfer matrix, to a compact form as shown in fig. 3.
In specific implementation, a frequency domain stability margin of the system is given by applying a characteristic trajectory method:
vis (sM + K)D)-1A forward channel transfer matrix is adopted, and the other links are feedback channels, so that a return difference matrix I + L of the closed-loop system is obtained
Figure BDA0003117651620000102
According to the nature of the matrix eigenvalues:
λ(I+L)＝I+λ(L)
following the convention of single input-single output (SISO) system stability analysis, we take the feature trajectory λmin(I + L) distance | λ from-1 Point-1(L) | is the margin of stability.
In specific implementation, the frequency domain stability margin is written into a loop addition form, and the rotor loop open-loop transfer function is written into a sum form as follows:
Figure BDA0003117651620000111
wherein the definition:
Figure BDA0003117651620000112
LGQ1＝(jωM+KD)-1GQ1ω0/(jω)
LGQ2＝(jωM+KD)-1GQ2ω0/(jω)
LGM＝(jωM+KD)-1GM
LKDrepresents a damping component, LGQ1Representing a field component, LGQ2Represents the PSS component, LGMRepresenting the prime mover-governor component.
In specific implementation, a similar matrix method is used for simplifying the frequency domain stability margin:
in the formula (4), the characteristic decomposition is carried out on the related item of K1
Figure BDA0003117651620000113
Then
I+L＝UΛU-1+ΔL＝U(Λ+U-1ΔLU)U-1
The stability margin is equivalent to the minimum eigenvalue for solving the similarity matrix
λ-1(L)＝λmin(SM)-1＝λmin(Λ+U-1ΔLU)-1
The minimum mode diagonal element in the above similar matrix is used to approximate the characteristic value as the simplified stability margin
λ-1(L)＝SM11-1
Wherein S isM11Is the minimum modulo diagonal of the similarity matrix.
In specific implementation, the influence of a prime mover-speed regulator link is analyzed by using the following formula:
Figure BDA0003117651620000114
at very low frequencies, the influence of the prime mover-speed regulator can be further simplified according to the matrix properties:
Figure BDA0003117651620000115
in specific implementation, the influence of the PSS link of the power system stabilizer is analyzed by using the following formula:
Figure BDA0003117651620000121
actually, the phase parameter of the PSS is set by using the following formula when setting the PSS link:
Figure BDA0003117651620000122
therefore, the problem of small interference stability of electromechanical transient of the multi-input and multi-output system can be completely analyzed.
For example, in a 6 machine 11 node system, we use the method described in the patent to analyze the effect of the prime mover-governor, the power system stabilizer PSS on the stability of small disturbances.
We plot vector diagrams of prime mover-governor components for thermal power generating units and hydroelectric generating units, respectively, according to equation (23), as shown in fig. 4. To ensure that the system is stable, we need the prime mover-governor link to provide positive damping to the system, i.e. the vector of its components is located on the lower side of the vector diagram.
From fig. 4 we find that the effect of the hydro-electric machine prime mover-governor loop is negative due to the inherent non-minimum phase characteristics of the hydro-electric machine (i.e. the frequency response characteristics are substantially lagging). In contrast, the prime mover-governor circuit of a thermal power unit contributes some positive damping.
In addition, the compensation angle for solving the PSS according to equation (30) is shown in fig. 5.
To verify the feasibility of the method, the method of the invention is compared with the current mainstream ideal phase frequency characteristic method and the residue method. The compensation angle of the PSS obtained by the ideal phase-frequency characteristic method is shown in fig. 6, and the compensation angle of the PSS obtained by the residue method is shown in fig. 7. We see that the method of the invention leads to conclusions consistent with both mainstream methods. Compared with the two methods, the method of the invention has more rigorous mathematical basis.
According to the above example, it can be seen that the stability margin obtained by the present invention well analyzes the electromechanical transient small interference stability characteristics of the multiple input-multiple output system.
It will be apparent to those skilled in the art that the modules or steps of the embodiments of the invention described above may be implemented by a general purpose computing device, they may be centralized on a single computing device or distributed across a network of multiple computing devices, and alternatively, they may be implemented by program code executable by a computing device, such that they may be stored in a storage device and executed by a computing device, and in some cases, the steps shown or described may be performed in an order different than that described herein, or they may be separately fabricated into individual integrated circuit modules, or multiple ones of them may be fabricated into a single integrated circuit module. Thus, embodiments of the invention are not limited to any specific combination of hardware and software.
The above description is only a preferred embodiment of the present invention, and is not intended to limit the present invention, and various modifications and changes may be made to the embodiment of the present invention by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (8)

1. The electromechanical transient small interference stability analysis method based on the minimum characteristic trajectory method is characterized by comprising the following steps of:
1) selecting a rotor loop as a forward channel link to obtain a compact Heffron-Phillips model;
2) applying a characteristic trajectory method to give a frequency domain stability margin of the system;
3) writing the frequency domain stability margin into a loop addition form;
4) simplifying the frequency domain stability margin by using a similar matrix method;
5) analyzing the influence of a prime motor-speed regulator link;
6) analyzing the influence of a PSS link of a power system stabilizer;
7) and selecting the parameters of the controller according to the frequency domain stability margin.
2. The electromechanical transient small interference stability analysis method based on the minimum feature trajectory method according to claim 1, wherein the specific method of the step 1) is as follows:
extracting a rotor loop in a Heffron-Phillips model, and enabling a rotor channel (sM + K)D)-1The other links are combined together to be used as a feedback channel to obtain a compact Heffron-Phillips model;
defined in the compact form of the Heffron-Phillips model described above:
GQ(s)＝GQ1(s)+GQ2(s), formula (3);
wherein G isQ1(s)＝-K2[[(K3+sT′d0)+GEX(s)K6]]-1(GEX(s)K5+K4) Formula (1);
Figure FDA0003117651610000011
in the formula, matrix K2、K3、K4、K5、K6Is a linearized model coefficient matrix; diagonal matrix T'd0The transient time constant of the d axis of each generator is contained; the diagonal matrix M contains a generator rotor motion inertia constant; diagonal matrix KDThe damping coefficient of the rotor motion is contained; diagonal matrix GEXIs an excitation system transfer function matrix; diagonal matrix HPSS(s) is the PSS transfer function matrix; omega0For system synchronous speed, s is the generalized frequency.
3. The method for analyzing electromechanical transient small interference stability based on the minimum feature trajectory method according to claim 1 or 2, wherein the applying the feature trajectory method in step 2) gives a frequency domain stability margin of the system, specifically:
vis (sM + K)D)-1And (3) transmitting a matrix for a forward channel, and taking other links as feedback channels to obtain a return difference matrix I + L of the closed-loop system:
Figure FDA0003117651610000012
wherein, the matrix K1Is a linearized model coefficient matrix; diagonal matrix GMIs a governor-prime mover system transfer function matrix;
according to the nature of the matrix eigenvalues:
λ (I + L) ═ I + λ (L), formula (5);
taking a characteristic track lambdamin(I + L) distance | λ from-1 Point-1(L) | is the margin of stability.
4. The method for analyzing electromechanical transient small disturbance stability based on the minimum feature trajectory method according to claim 3, wherein the step 3) writes the frequency domain stability margin into a loop addition form, specifically:
substituting s ═ j ω into the expression for L according to equation (4)
Figure FDA0003117651610000013
Wherein ω represents an angular frequency;
in formula (6), the matrices M and KDAre all diagonal matrices, therefore
Figure FDA0003117651610000024
Bringing formula (7) into formula (6)
Figure FDA0003117651610000021
The following matrix is introduced:
Figure FDA0003117651610000022
LGQ1＝(jωM+KD)-1GQ1ω0/(j ω), formula (10);
LGQ2＝(jωM+KD)-1GQ2ω0/(j ω), formula (11);
LGM＝(jωM+KD)-1GMformula (12);
equation (9) represents the damping component, equation (10) represents the excitation component, equation (11) represents the PSS component, and equation (12) represents the prime mover-governor component.
5. The method for analyzing electromechanical transient small interference stability based on the minimum feature trajectory method according to claim 4, wherein the step 4) of simplifying the frequency domain stability margin by using a similarity matrix method specifically comprises:
in equation (4), feature decomposition is performed on the related term of K1:
Figure FDA0003117651610000023
wherein Λ represents a characteristic value matrix, and U represents a characteristic vector matrix; the above equation holds if the matrix K1 can be diagonalized; then:
I+L＝UΛU-1+ΔL＝U(Λ+U-1ΔLU)U-1formula (14);
based on the definition of the similarity matrix
λ(I+L)＝λ(Λ+U-1ΔLU)＝λ(SM) Formula (15);
wherein
SM＝Λ+U-1Δ LU, equation (16);
the stability margin is equivalent to the minimum eigenvalue for solving the similarity matrix
λ-1(L)＝λmin(SM)-1＝λmin(Λ+U-1Δ LU) -1, formula (17);
the eigenvalues are approximated with minimum modulo diagonal bins to simplify the frequency domain stability margin:
λ-1(L)＝SM11-1, formula (18);
equation (18) is the simplified frequency domain stability margin, where SM11Is the minimum modulo diagonal of the similarity matrix.
6. The electromechanical transient small disturbance stability analysis method based on the minimum feature trajectory method according to claim 5, wherein the step 5) analyzes the influence of the prime mover-speed regulator link, specifically:
since the transfer matrix component of the prime mover-governor is as shown in equation (12), the similarity matrix S isMThe primary motor-governor having a component of
U-1LGMU＝U-1(jωM+KD)-1GMU, formula (19);
suppose that the power grid has m units, V ═ U-1)TIs a left eigenvector matrix, then
Figure FDA0003117651610000031
Due to SM＝Λ+U-1Δ LU, Λ is a diagonal matrix, consider
Figure FDA0003117651610000032
Smallest modular diagonal element in
Figure FDA0003117651610000033
Component(s) of
Figure FDA0003117651610000034
Wherein u is1,v1Respectively representing the right and left eigenvectors, symbols
Figure FDA0003117651610000035
Represents the Hadamard product of the matrix;
handle LGMFormula (21):
Figure FDA0003117651610000036
wherein, gMiRepresenting a diagonal matrix GMThe ith diagonal element of (1)DiRepresenting a diagonal matrix KDThe ith diagonal element of (1)iDenotes the ith diagonal element, u, of the diagonal matrix Mi1、vi1Represents u1、v1The ith element of (1);
in the ultra-low mode frequency band, due to v1≈Mu1Right eigenvectoru1Each element is equal, assuming KD0, then
Figure FDA0003117651610000037
7. The electromechanical transient small-disturbance stability analysis method based on the minimum feature trajectory method according to claim 6, wherein the step 6) is used for analyzing the influence of a Power System Stabilizer (PSS) link, and specifically comprises the following steps:
Figure FDA0003117651610000038
smallest modular diagonal element in
Figure FDA0003117651610000039
As follows
Figure FDA00031176516100000310
The transfer matrix component of the PSS is shown in formula (11);
defining forward channel transfer matrix through which PSS passes
HPVr＝K2[K3+jωT′d0+GEXK6]-1GEXFormula (25);
the transfer matrix component of the PSS may be further written as
Figure FDA0003117651610000041
Let fijRepresentative matrix (j ω M + K)D)-1HPVrElement at position (i, j), the contribution of PSS to stability margin is obtained:
Figure FDA0003117651610000042
the contribution of PSS to stability margin has the form of a dot product:
Figure FDA0003117651610000043
wherein, vector sum (F)uv) Is a matrix FuvColumn (c) and, diag (H)PSS) Are vectors of the same dimension, where the elements represent the PSS frequency response for each unit;
assuming that the grid has n units, then
Figure FDA0003117651610000044
According to the above formula, the parameter of the PSS should satisfy the following phase condition, so that the minimum feature trajectory moves vertically downward and is away from the-1 point on the complex plane, that is, the phase parameter of the PSS is set by using the following formula:
Figure FDA0003117651610000045
8. the method for analyzing the electromechanical transient small disturbance stability based on the minimum feature trajectory method according to claim 7, wherein the step 7) selects the controller parameters according to the frequency domain stability margin, and the specific method is as follows: selecting PID parameters of the speed regulator according to the formula (23) obtained in the step 5), and selecting parameters of the power system stabilizer PSS according to the formula (30) obtained in the step 6).
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