A power system dynamic observability calculation method based on data-driven koopman operator

Abstract

The application discloses a power system dynamic observability calculation method based on a data-driven Koopman operator, belongs to the technical field of power system dynamics and observability analysis, and comprises the following steps: establishing a power system dynamic observability analysis model based on system state variables and measured variables; converting the established dynamic state observability model into an augmented Koopman canonical coordinate system; constructing an observation matrix of the system in the augmented Koopman canonical coordinate system; and calculating the dynamic observability degree of the power system by using a matrix index in the Koopman space. The method of the application uses the data-driven Koopman operator, overcomes the inaccuracy of the model while considering the nonlinear characteristics of the power system dynamic model, effectively solves the low-precision problem of the traditional linear method and the problem of excessive calculation amount of the nonlinear method, and can realize efficient and rapid analysis and calculation of the dynamic observability of the power system.

Description

Technical Field

[0001] This invention belongs to the field of power system dynamics and observability analysis technology, specifically relating to a power system dynamics observability calculation method based on data-driven Koopman operators. Background Technology

[0002] Observability analysis of a system is a prerequisite for its state estimation, as it represents the system's ability to infer its internal state from measurement data. With the increasing application of power system dynamic models (PMUs) in power systems, the observability analysis of nonlinear dynamic models of power systems, which are presented as nonlinear differential-algebraic equations involving time-dependent nonlinear functions, has begun to attract the attention of power researchers.

[0003] As the nonlinearity of the system increases, the accuracy deteriorates accordingly. To address this issue, a Lie-derivative-based nonlinear power system model can be constructed using a classical generator model. However, while this method yields good results in nonlinear dynamic models, it becomes extremely complex and difficult to solve when applied to medium- to large-scale systems. Furthermore, a method based on polynomial chaos can also assess the observability of stochastic nonlinear dynamic models, such as weak observability. However, these observability analysis methods are all model-based, and the errors introduced by them due to differences in power system models and parameter uncertainties are not negligible. Summary of the Invention

[0004] To address the shortcomings of existing technologies, the present invention aims to provide a method for calculating the dynamic observability of power systems based on a data-driven Koopman operator. This method combines the Koopman operator with a nonlinear dynamic state model to obtain an extended model of the Koopman operator in a data-driven manner. This results in a measurement process that is completely derivative-free while maintaining the nonlinearity of the model, thereby facilitating the analysis of the system's observability.

[0005] The objective of this invention can be achieved through the following technical solutions:

[0006] A method for calculating the dynamic observability of a power system based on a data-driven Koopman operator includes the following steps:

[0007] S1. Establish a dynamic observability analysis model for power systems based on system state variables and measurement variables;

[0008] S2. Transform the established dynamic state observability model into the augmented Koopman canonical coordinate system;

[0009] S3. Construct the system's observation matrix in the augmented Koopman canonical coordinate system;

[0010] S4. Calculate the degree of dynamic observability of the power system using matrix indices in Koopman space;

[0011] Furthermore, step S1, establishing a power system dynamic observability analysis model based on system state variables and measurement variables, further includes:

[0012] S11, Assuming that by The autonomous nonlinear system represented evolves on a finite-dimensional manifold M and is governed by f: M→M. A scalar-valued continuous function is defined. As a so-called observable function;

[0013] S12. Given a linear, infinite-dimensional Koopman operator Acting on g:

[0014]

[0015] Where S t Its flow is referred to as its flow, and the calculation formula is as follows:

[0016]

[0017] S13, Koopman eigenvalues ​​λ i and the Koopman characteristic function φ i The following relationship exists between them:

[0018]

[0019] Among them, by selecting a limited observation dataset A subset of Koopman eigenvalues ​​can be estimated. and characteristic function

[0020] S14, due to the n of g d The element is in the Koopman eigenfunction φ. i Within the span of the system, the observation subfunction of the system can be obtained:

[0021]

[0022] in, Indicates the Koopman pattern. These are called Koopman tuples;

[0023] S15. Establish the equations for the system observation subfunctions:

[0024]

[0025] in, This indicates the relevant Koopman operator.

[0026] Furthermore, step S14, approximating the Koopman tuple, further includes:

[0027] S141. Establish two data matrices and in

[0028] S142. Establish the vector-valued observation sub-function Observability matrix and in And n d ≥n x ;

[0029] S143. Establish the matrix

[0030]

[0031] in, It is a Moore-Penrose pseudo-reverse;

[0032] S144. Calculate the left eigenvector of K, store it in matrix L, and estimate the Koopman eigenfunction as follows:

[0033] φ(x k )≈Lg(x k )

[0034] in,

[0035] S145, Define the projection matrix Make:

[0036] x k =Pg(x k )

[0037] Therefore, an approximation of the Koopman pattern can be estimated:

[0038] U=PL -1 =x k φ(x k ) -1

[0039] in,

[0040] S146. Represent the evolution of the system state as a finite-dimensional linearized Koopman expansion:

[0041]

[0042] Furthermore, step S2, which transforms the established dynamic state observability model into the augmented Koopman canonical coordinate system, further includes:

[0043] S21. Introduce a set of additional observation sub-functions:

[0044] g y (x(t))=y(t)=h(x(t))

[0045] Among them, choose g y The method of (x) is to transform the measurement function h(x) to the Koopman space, while obtaining the space contained in g(x) = [g x (x) g y (x)] T A larger subset of observations

[0046] S22. Establish the relationship between the system state variable x and the measurement variable y in a higher-dimensional space:

[0047]

[0048] S23. Decompose the augmented Koopman operator K into:

[0049]

[0050] in, The decomposed Koopman operator;

[0051] S24. Map the relationships between different observation subsets using the decomposed Koopman operator, let n a =n d 10 n y Therefore:

[0052]

[0053]

[0054] Furthermore, the decomposition of the augmented Koopman operator in step S24 can map any initial condition to the system measurement at any time, and the specific process is as follows:

[0055] S241, the augmented Koopman operator still satisfies:

[0056] g(x k )=K·g(x k-1 ) = K 2 ·g(x k-2 )=…=Kk ·g(x0)

[0057] S242. Substituting into the decomposed Koopman operator, we get:

[0058] g y (x k+1 ) = K yxy ·K k ·g(x0)

[0059] S243, due to g y (x k+1 )=h(x k+1 Therefore:

[0060]

[0061] Furthermore, step S3, constructing the system's observation matrix in the augmented Koopman canonical coordinate system, further includes:

[0062] S31. Based on the observation function given in the Koopman space, the corresponding state-measurement relationship can be obtained:

[0063]

[0064] S32, the corresponding observability matrix is:

[0065]

[0066] S33. Based on the measurement model stored in the Koopman space, add the measured values ​​as the observation sub-function g. y Therefore, the state-measurement relationship can be expressed as:

[0067]

[0068] S34. Derive the observability O(t) based on Koopman:

[0069]

[0070] The observability of the system can be checked by examining the rank of this matrix.

[0071] Furthermore, step S4, which uses matrix indices in Koopman space to calculate the degree of dynamic observability of the power system, further includes:

[0072] S41. Define the condition number of the system as:

[0073]

[0074] Where, σ min (O) and σmax (O) are the minimum and maximum singular values ​​of the system, respectively;

[0075] S42. Factorize the observability matrix:

[0076] O = U o ∑ o V o T

[0077] in, and U0(V o ) represents orthogonal eigenvectors, and the diagonal matrix ∑ o The elements are eigenvalues ​​relative to U0(V) o The square root in descending order, whose associated singular value is σ of the Koopman observability matrix O. i The square root of (O).

[0078] The beneficial effects of this invention are:

[0079] This invention provides an efficient observability analysis tool for dynamic state estimation of power systems and proposes a data-driven Koopman-based method for calculating dynamic observability of power systems. This method preserves the nonlinearity of the model, overcomes computational errors caused by model inaccuracies through a data-driven approach, and enables a completely derivative-free observability calculation process, reducing the computational burden and achieving fast, efficient, convenient, and accurate stochastic dynamic observability analysis of power systems. Furthermore, this observability calculation scheme is helpful in selecting Koopman operator observers. Attached Figure Description

[0080] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, for those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0081] Figure 1 This is a flowchart of a nonlinear dynamic observability analysis method for power systems based on the data-driven Koopman method, according to the present invention.

[0082] Figure 2 This is the power system state prediction comparison result in Embodiment 1 of the present invention. Detailed Implementation

[0083] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0084] A method for calculating the dynamic observability of power systems based on data-driven Koopman operators.

[0085] Example 1

[0086] This embodiment is applied to a simple two-dimensional nonlinear system, whose equations are as follows:

[0087]

[0088] A data-driven Koopman method for analyzing the nonlinear dynamic observability of power systems, such as Figure 1 As shown, it includes the following steps:

[0089] S1. Establish the observation sub-function of the original dynamic system model;

[0090] S11. Initial conditions are given by x1(0) = -1 and x2(0) = 2, and parameters are given by α = -1 and β = -0.05. Different measurement models are chosen to test the observability and related observability of the nonlinear system. Consider five different subsets of Koopman observations, using four different Koopman observations for each subset, g x (x) are respectively a linear Hermite polynomial, a decoupled quadratic polynomial, a quadratic polynomial, and a complex form including a quadratic polynomial.

[0091] S12. Given a linear, infinite-dimensional Koopman operator Acting on g:

[0092]

[0093] Where S t Its flow is referred to as its flow, and the calculation formula is as follows:

[0094]

[0095] S13, Koopman eigenvalues ​​λ i and the Koopman characteristic function φ i The following relationship exists between them:

[0096]

[0097] Among them, by selecting a limited observation dataset A subset of Koopman eigenvalues ​​can be estimated. and characteristic function

[0098] S14, due to the n of g d The element is in the Koopman eigenfunction φ. i Within the span of the system, the observation subfunction of the system can be obtained:

[0099]

[0100] in, Indicates the Koopman pattern. These are called Koopman tuples;

[0101] S15. Establish the equations for the system observation subfunctions:

[0102]

[0103] in, This indicates the relevant Koopman operator.

[0104] S2. Establish augmented Koopman operators to unify the measurement model and system model into Koopman canonical coordinates;

[0105] S21. Introduce a set of additional observation sub-functions:

[0106] g y (x(t))=y(t)=h(x(t))

[0107] Among them, choose g y The method of (x) is to transform the measurement function h(x) to the Koopman space, while obtaining the space contained in g(x) = [g x (x) g y (x)] T A larger subset of observations

[0108] S22. Establish the relationship between the system state variable x and the measurement variable y in a higher-dimensional space:

[0109]

[0110] S3. Decomposition of augmented Koopman operators;

[0111] S31. Decompose the augmented Koopman operator K into:

[0112]

[0113] in, The decomposed Koopman operator;

[0114] S32. Map the relationships between different observation subsets using the decomposed Koopman operator, let n a =n d +n y Therefore:

[0115]

[0116]

[0117] S4. Derive the expression of the observability matrix in Koopman canonical coordinates using the decomposed Koopman operator;

[0118] S41. Based on the observation function given in the Koopman space, the corresponding state-measurement relationship can be obtained:

[0119]

[0120] S42, the corresponding observability matrix is:

[0121]

[0122] S43. Based on the measurement model stored in the Koopman space, add the measured values ​​as the observation sub-function g. y Therefore, the state-measurement relationship can be expressed as:

[0123]

[0124] S44. Derive the Koopman-based observability matrix O(t):

[0125]

[0126] The observability of the system can be checked by examining the rank of this matrix.

[0127] S5. Determine the degree of observability of the system based on the condition number of the observability matrix.

[0128] S51. Define the condition number of the system as:

[0129]

[0130] Where, σ min (O) and σ max (O) are the minimum and maximum singular values ​​of the system, respectively;

[0131] S52. Factorize the observability matrix:

[0132] O = U o ∑ o V o T

[0133] in, and U0(V o ) represents orthogonal eigenvectors, and the diagonal matrix ∑ o The elements are eigenvalues ​​relative to U0(V) o The square root in descending order, whose associated singular value is σ of the Koopman observability matrix O. i The square root of (O).

[0134] The proposed methods are evaluated for different measurement configurations and different subsets of Koopman observations. It is evident that choosing different Koopman observations leads to varying degrees of observability. Typically, the observation vector g with more elements... x (x) can easily lead to a relatively small observability of the system.

[0135] Example 2

[0136] The power system in this embodiment is an IEEE 9-bus system using a classic generator model. The simulation duration is set to 20 seconds, and three different Koopman observation sets are used: a quadratic decoupled form, a quadratic form, and a complex Hermite polynomial form, which includes an additional kernel function for each state variable. Based on this, V, θ, and P are measured for different types of PMUs. e and Q e The Koopman observability was also analyzed, with a frame rate of 60 frames per second.

[0137] The evaluation of different Koopman observation subsets of the system according to the steps of the present invention shows that different sets of observables lead to different values ​​of system observability. The observability of the measured voltage amplitude V on each generator bus is the lowest. Furthermore, adding a measurement with very low observability will reduce the overall observability to some extent. Only by adding a measurement with relatively strong observability will the overall observability, quantified by conditions, increase. Finally, the more Koopman observations selected, the lower the observability tends to be.

[0138] Example 3

[0139] The power system in this embodiment is the New England 39 bus system using a classic generator model. A set of PMUs is placed at the generator terminals to provide V, θ, P. e and Qe The measured values. To analyze the dynamics of the power system under disturbance, the transmission line between bus 15 and bus 16 was opened, and the observability level was tested by the condition number c(O), using the Hermite polynomial in the decoupled quadratic form, quadratic form, and the complex form with an additional kernel function, while other settings remained unchanged.

[0140] The evaluation of different Koopman observation subsets of the system according to the steps of the present invention shows that using more measurement devices, while using the same Koopman observations, we can obtain better observability. Furthermore, with more Koopman observations, it is more likely to obtain an unobservable system in the Koopman space. Secondly, due to the increase in system dimension, the dimension of the enhanced Koopman space also increases accordingly. Finally, although the addition of a large number of Koopman observations generally better characterizes the nonlinearity of the system, it may jeopardize the observability of the system to some extent. This can be improved by removing negligible Koopman observations that contribute little to the Koopman operator, thus ensuring the observability matrix is ​​of full rank.

[0141] Example 4

[0142] The power system in this embodiment is a New England 39-bus system using a detailed generator model, specifically a 9th-order synchronous generator with an IEEE-DC1A exciter and a TGOV1 turbine governor. The PMU is placed only on the PV bus to estimate the system state. A decoupled quadratic form of Hermite polynomials is used, which contains far fewer observations.

[0143] Evaluation of different Koopman observation subsets of the system according to the steps of the present invention reveals that, despite identical measurement locations, observability significantly decreases for measurement locations using detailed generator models, consistent with the fundamental principles of our proposed method. Secondly, by placing PMUs only on the PV bus, adjustments to the Koopman observations are necessary as the system state dimension increases. Finally, the proposed Koopman observability method is based on the linear Koopman operator, which yields very high computational efficiency. For example, for this power system, simulation can be completed in seconds. This computational speed will be highly desirable when performing observability calculations for dynamic state estimation.

[0144] Therefore, this method proposes a Koopman-based observability analysis framework for faster and more accurate computation of the observability of stochastic nonlinear dynamic models of power systems. We formulate an augmented Koopman operator that reflects the relationship between system state and measurements in a data-driven manner, through... Figure 2 The observation results show that different subsets of observations result in different observability of the system. When the condition number of the observability matrix is ​​smaller, the system exhibits stronger observability. The proposed method not only considers the nonlinearity of the system model but also achieves a completely derivative-free calculation process, improving the convenience and feasibility of the method. Ultimately, it overcomes the inaccuracy of the model through a data-driven approach, achieving efficient and accurate dynamic observability calculation of the power system. Furthermore, the proposed observability calculation scheme provides valuable guidance for the selection of Koopman operator observables, a major challenge in applying the Koopman operator. In the description of this specification, references to terms such as "an embodiment," "example," "specific example," etc., indicate that the specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials, or characteristics described can be combined in any suitable manner in one or more embodiments or examples.

[0145] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the claimed invention.

Claims

1. A method for calculating the dynamic observability of a power system based on a data-driven Koopman operator, characterized in that, Includes the following steps: Establish a dynamic observability analysis model for power systems based on system state variables and measurement variables; The established dynamic observability analysis model is transformed into the augmented Koopman canonical coordinate system; Construct the system's observation matrix in the augmented Koopman canonical coordinate system; The degree of dynamic observability of the power system is calculated using the observation matrix index in Koopman space. The transformation of the established dynamic observability analysis model to the augmented Koopman canonical coordinate system further includes: Establish system model equations: in, and These are the vector-valued system function and the observation function, respectively. Introducing a set of additional observation subfunctions , to measure function Convert to Koopman space; Get included A larger subset of observations; Establish system state variables based on the observation subset and measurement variables The connection between them: in, To augment the Koopman operator; Establish connections between different subsets of Koopman observations; The observation matrix of the system constructed in the augmented Koopman canonical coordinate system further includes: Construct the corresponding state-measurement relationship for the system: Constructing the observability matrix : Will Substituting into the observability matrix, the state-measurement relationship is reformulated into a form containing decomposition and reinforcement of the Koopman operator based on the saved measurement model. The observability of the system is then detected by calculating the rank of this matrix.

2. The method for calculating the dynamic observability of a power system based on the data-driven Koopman operator according to claim 1, characterized in that, The establishment of a power system dynamic observability analysis model based on system state variables and measurement variables further includes: Assume that the autonomous nonlinear system is in a finite-dimensional manifold Evolution, by Control, define a scalar continuous function As an observation subfunction, given a linear infinite-dimensional Koopman operator Acting on ,Right now: in, For its traffic, and has; in, for The state vector of the system at any given time; The Koopman feature function is represented as: Selecting a limited number of observations Estimate a subset of Koopman eigenvalues and characteristic function : For the Koopman mode, middle The elements can be obtained within the spanned space of the Koopman feature function. , For the relevant Koopman operators.

3. The method for calculating the dynamic observability of a power system based on the data-driven Koopman operator according to claim 2, characterized in that, Estimating Koopman eigenvalues ​​and eigenfunctions further includes: Establish two data matrices and Vector-valued observation subfunction Observable matrix and ; Establish matrix ,satisfy: in This is a Moore-Penrose pseudo-inverse; Establish the relationship between characteristic functions, eigenvalues, and observation subsets; Define the projection matrix and establish the relationship between the state variables and the observation sub-functions; The evolution of the system state is characterized as a finite-dimensional linear Koopman expansion.

4. The method for calculating the dynamic observability of a power system based on the data-driven Koopman operator according to claim 1, characterized in that, The calculation of the dynamic observability of the power system using the observation matrix index in Koopman space further includes: Define the condition number as an index characterizing the invertibility of the observation matrix: in, and These are the maximum and minimum singular values ​​of the observability matrix, respectively; Will It can be represented in the following form: in, , , The column vectors are orthogonal eigenvectors. The diagonal elements are the eigenvalues ​​relative to The square root of the system, arranged in descending order, is represented by the square root of the singular value of the Koopman observability matrix. This transformation yields the maximum and minimum singular values ​​of the observability matrix, thus providing the degree of observability of the system.

5. The method for calculating the dynamic observability of a power system based on the data-driven Koopman operator according to claim 1, characterized in that, The integration process of the observed subset needs to be fully preserved. But from Remove observations that are negligible and contribute little to the Koopman operator.

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