Power system online inertia evaluation method based on dynamic mode decomposition

Abstract

The invention discloses a power system online inertia evaluation method based on dynamic mode decomposition, belongs to the technical field of power system operation and maintenance, and deduces a coupling relationship between electromechanical characteristics in a random Koopman space and system inertia by combining a random Koopman theory and a random dynamic system. A Bayesian theory and Gibbs sampling are used to preprocess data, a DMD is used to carry out eigenvalue decomposition and dynamic modal fitting, time domain reconstruction is carried out on a node equivalent swing equation based on extracted eigenvalues and eigenvectors, a linear equation about electromechanical parameters and system features is constructed on this basis, and the dynamic modal fitting is carried out based on the linear equation. And evaluation of inertia and damping coefficient of the generator is realized through linear equation solution. The method is characterized in that power and frequency environment data in a normal operation state of a system are used as input, and a Bayesian-DMD method is used for performing reconstruction calculation on the inertia of the system. The method has the advantages of being scientific, reasonable, high in applicability, high in speed, high in accuracy and good in robustness.

Description

Technical Field

[0001] The present invention relates to the field of power system operation and maintenance technology, and specifically to an online inertia assessment method for power systems based on dynamic mode decomposition, which is suitable for dynamic stability analysis and real-time monitoring of new power systems in the context of large-scale access to clean energy. Background Art

[0002] The accumulation of quantitative changes in power electronic converters in the system will cause a qualitative change effect. Compared with the traditional power system dominated by synchronous power sources, the inertia characteristics of the power system with a high proportion of converters will change significantly. On the one hand, due to the strong randomness, intermittency and volatility of the output of new energy units, the operation mode of the system is complex and changeable, and the system inertia will show a large fluctuation characteristic on the time scale; on the other hand, the low inertia characteristics of the area where a high proportion of converters are connected are significant, which is in sharp contrast to the high inertia situation in the area where synchronous power clusters are distributed, making the system inertia also show certain spatial distribution characteristics. In addition, the inertia response of the system is earlier than the frequency modulation and is also accompanied by inertia response in the early stage of the frequency modulation. Figure 2 As shown in Figure 3, the coupling between the two makes the system modal changes more complex, thus putting forward higher requirements for the accuracy and robustness of inertia evaluation.

[0003] The definition of system node inertia is: "The inherent property of system inertia that hinders node frequency changes during power system energy fluctuations." That is, when the system power is in excess, node inertia prevents the node frequency from rising too quickly; when the system power is insufficient, node inertia prevents the node frequency from falling too quickly. It can be seen that the spatiotemporal distribution characteristics of the dynamic frequency of the power system are the embodiment of the spatiotemporal characteristics of the inertia of the power system. The system inertia is reflected in the inertia of each node, and the inertia of each node constitutes the system inertia. Figure 3 As shown in the figure. Furthermore, in areas where inertia is concentrated, the inertia of each node is relatively large; in areas where inertia is scarce, the inertia of each node is relatively small. The inertia of a power system exhibits spatiotemporal characteristics: the inertia of the same node varies over time, and the inertia of different nodes varies. If the inertia of each node in the system at different time sections can be obtained and the system's inertia distribution diagram can be plotted, the spatiotemporal characteristics of the power system's inertia can be visualized and monitored in real time, providing a reference for controller design, new energy integration, and power system operators. Summary of the Invention

[0004] The purpose of the present invention is to solve the problems mentioned in the above background technology and to propose an online inertia evaluation method for a power system based on dynamic mode decomposition.

[0005] The object of the present invention can be achieved by the following technical solution: A method for online inertia evaluation of a power system based on dynamic mode decomposition, comprising the following steps:

[0006] Step 1: Dynamic data input and preprocessing: At time intervals Δt, random response data of each node in the power system under normal operation is collected, including node angular velocity Δω(t), angular frequency Δf(t), and active power deviation ΔPe(t);

[0007] Step 2: Sort the sampled data in the order of each node number, introduce the Bayesian probability model, assume that the data follows the distribution X~p(X / Φ), parameter Φ=(θ1, θ2); resample the initial data set through Gibbs posterior sampling to generate a new data matrix Δx d ';Use this matrix as the input matrix to perform system characteristic dynamic mode decomposition DMD calculation to obtain the eigenvalue matrix λ and its corresponding eigenvector matrix V;

[0008] Step 3: By combining the random Koopman theory with the random dynamic system, the coupling relationship between the electromechanical characteristics and the system inertia in the random Koopman space is derived. Then, the equivalent inertia value of each node is calculated based on the eigenvalue matrix λ and the corresponding eigenvectors obtained from the dynamic modal decomposition in step 2 and the coupling model of electromechanical characteristics and inertia derived above.

[0009] As a preferred embodiment of the present invention, step 1 measures the environmental data of each power grid node through the PMU, including the input matrices X0 and X1 composed of the changes in the active power and frequency of the nodes.

[0010] As a preferred embodiment of the present invention, step 2 performs Gibbs posterior sampling, assuming that the data obeys the distribution X~p(X / Φ), parameter Φ=(θ1, θ2), and the prior distribution of Φ is p(θ1, θ2); {X1, ..., X n} is the set of samples drawn from this distribution; the posterior distribution is calculated using Bayes' theorem:

[0011]

[0012] Initialization parameters From the conditional distribution and The samples are extracted from the Gibbs sampling to generate a Markov chain {Φ (1) ,...,Φ (N)}; Then, the denoised data set is extracted through the stationary distribution of the Markov chain to form a new input matrix Δx d ′=[Δx1′,Δx2′,...,Δx m ′], its data matrix is:

[0013]

[0014] P in the matrixz is the data matrix composed of the first m-1 data sequences, F z The data matrix composed of the last m-1 data sequences; construct the state matrix and A d =F z V -1 ∑ -1 U * Perform singular value decomposition on it: A d v j =μ j v j ; U and V are unitary matrices, ∑ is a singular value diagonal matrix;

[0015] Solve A d The eigenvalue μ j and the eigenvector v j , and is converted into a continuous eigenvalue by λ=ln(μ) / Δt.

[0016] As a preferred embodiment of the present invention, step three derives the coupling relationship between the electromechanical characteristics and the system inertia in the random Koopman space, including:

[0017] First, the electromechanical response process of the power system stimulated by the environment is described as:

[0018]

[0019] Where M, D, and K represent the diagonalizable matrices consisting of the inertia time constant, damping coefficient, and synchronization coefficient, respectively; Δδ(t) represents the rotor angular deviation near the equilibrium point at time t; and F(t) represents the matrix composed of the natural excitation vector at time t.

[0020] The deduction of each state variable in the random Koopman space is expressed as:

[0021]

[0022] K Ω is the random Koopman operator; F(0) represents the excitation at the initial moment; Δδ(0) represents the rotor angle deviation near the equilibrium point at the initial moment;

[0023] In random Koopman space: According to the rotor angle active power change ΔP e The angular velocity change Δω satisfies ΔP e =KΔδ and , then:

[0024]

[0025] Where: and are the modal vibration shapes extracted from the angular velocity and active power changes respectively; b(0) is the initial input value; since t is arbitrary, the intermediate variables are eliminated. After that,

[0026]

[0027] According to the characteristics of electromechanical dynamics, the eigenvalues containing frequency and damping ratio information and the corresponding system modal vibration shapes are all in complex form. The projection transformation is used to decompose the equation into a complex plane based on the real axis and imaginary axis to expand the dimension of the above equation:

[0028]

[0029] M is the inertia matrix, D is the damping matrix, are the modal vibration shapes of angular velocity and electromagnetic power respectively, b is the initial eigenfunction, and the equivalent inertia of each node is calculated based on the eigenvalue and eigenvector in step 2.

[0030] Compared with the prior art, the present invention has the following beneficial effects:

[0031] 1. This paper uses Bayesian theory and Gibbs sampling to preprocess data, treating the linear modal parameters (such as eigenvalues and eigenvectors) of the Koopman operator as random variables and estimating their posterior distribution. This quantifies the uncertainty of the modal parameters (such as the confidence interval of the frequency estimate), thereby improving the robustness of the inertia estimation.

[0032] 2. The linear representation of the Koopman operator in this invention allows system dynamics to be reconstructed using fewer data snapshots, while the Bayesian approach further constrains the solution space through prior knowledge (such as historical data on inertia distribution). Therefore, in real-time power system monitoring, even with missing data or limited sampling rate, high-precision inertia estimation can be achieved through Koopman-Bayesian DMD, adapting to scenarios with random fluctuations in renewable energy. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] To facilitate understanding by those skilled in the art, the present invention is further described below with reference to the accompanying drawings.

[0034] Figure 1 This is a flowchart of online evaluation of power system node inertia based on random data driven Bayesian-DMD.

[0035] Figure 2 It is the time series diagram of the inertia response and frequency response of the power system

[0036] Figure 3 It is a diagram showing the relationship between the overall power system and the node inertia. DETAILED DESCRIPTION

[0037] The technical solutions of the present invention will be clearly and completely described below with reference to the embodiments. Obviously, the embodiments described are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making any creative efforts shall fall within the scope of protection of the present invention.

[0038] See also Figure 1-Figure 3 As shown in the figure, a method for online inertia assessment of power systems based on dynamic mode decomposition is proposed. By combining the Bayesian probability framework, dynamic mode decomposition (DMD) and stochastic Koopman theory, it can achieve rapid inertia assessment without the need for an accurate model. The core steps are as follows:

[0039] Step 1: Data acquisition and processing: Dynamic data input and preprocessing. At time intervals Δt, random response data of each node under normal power system operation is collected, including node angular velocity change Δω(t), angular frequency change Δf(t), and active power deviation ΔPe(t).

[0040] Step 2: DMD modal identification: Sort the sampled data in the order of each node number, introduce the Bayesian probability model, assume that the data follows the distribution X~p(X / Φ), and the parameter Φ=(θ1,θ2). Resample the initial data set through Gibbs posterior sampling to generate a new data matrix Δx d ';Use this matrix as the input matrix to perform system characteristic dynamic mode decomposition DMD calculation to obtain the eigenvalue matrix λ and its corresponding eigenvector matrix V;

[0041] Step 3: Node Inertia Calculation: By combining random Koopman theory with random dynamic systems, we derive the coupling relationship between electromechanical characteristics and system inertia in random Koopman space. We then calculate the equivalent inertia of each node based on the eigenvalue matrix λ and corresponding eigenvectors obtained from the dynamic modal decomposition (DMD) in Step 2 and the previously derived coupling model between electromechanical characteristics and inertia.

[0042] The first step is to measure the environmental data of each power grid node through the PMU, including the input matrices X0 and X1 composed of the changes in the active power and frequency of the nodes.

[0043] The second step is to perform Gibbs posterior sampling on it; first assume that the data obeys the distribution X~p(X / Φ), parameter Φ=(θ1,θ2); the prior distribution of Φ is p(θ1,θ2); the sample set {X1,...,X n}.

[0044] The posterior distribution can be calculated using Bayes' theorem:

[0045]

[0046] Initialization parameters From the conditional distribution and It can be seen that in the sampling process, each sample Φ i are affected by the previous sample Φ i-1 impact.

[0047] Therefore, the samples obtained by Gibbs sampling can generate a Markov chain {Φ (1) ,...,Φ (N) Then, the denoised data set is extracted through the stationary distribution of the Markov chain to form a new input matrix Δx d '=[Δx1',Δx2',...,Δx m '].

[0048] Its data matrix is P in the matrix z is the data matrix composed of the first m-1 data sequences, F z The data matrix consists of the last m-1 data sequences.

[0049] Construct the state matrix and A d =F z V -1 ∑ -1 U * Perform singular value decomposition on it:

[0050] A d v j =μ j v j ;

[0051] U and V are unitary matrices, Σ is a singular value diagonal matrix, solve A d The eigenvalue μ j and the eigenvector v j , and is converted into a continuous eigenvalue by λ=ln(μ) / Δt.

[0052] The third step deduces the coupling relationship between electromechanical characteristics and system inertia in the random Koopman space. First, the electromechanical response process of the power system stimulated by the environment can be described as:

[0053]

[0054] M, D, and K represent the diagonalizable matrices consisting of the inertia time constant, damping coefficient, and synchronization coefficient, respectively; Δδ(t) represents the rotor angular deviation near the equilibrium point at time t; and F(t) represents the matrix composed of the natural excitation vector at time t.

[0055] The deduction form of each state variable in the random Koopman space can be expressed as:

[0056]

[0057] Where KΩ is the random Koopman operator; F(0) represents the excitation at the initial moment; Δδ(0) represents the rotor angle deviation near the equilibrium point at the initial moment.

[0058] In random Koopman space:

[0059] Due to the rotor angle active power change ΔP e The angular velocity change Δω satisfies ΔP e =KΔδ and The above formula can be rewritten as:

[0060]

[0061] Where: and are the modal vibration shapes extracted from the angular velocity and active power changes respectively; b(0) is the initial input value. Since t is arbitrary, the intermediate variables are eliminated. The above formula can be rewritten as:

[0062]

[0063] According to the characteristics of electromechanical dynamics, the eigenvalues containing frequency and damping ratio information and the corresponding system modal vibration shapes are all in complex form. The dimension of the above equation can be expanded by using projection transformation to decompose the equation into a complex plane based on the real axis and imaginary axis:

[0064]

[0065] M is the inertia matrix, D is the damping matrix, are the mode shapes of angular velocity and electromagnetic power, respectively, and b is the initial eigenfunction. Based on the eigenvalues and eigenvectors in step 2, the equivalent inertia of each node can be calculated.

[0066] When in use, the present invention collects random response data of each node in the normal operation of the power system through the PMU at time intervals Δt, including node angular velocity change Δω(t), frequency change Δf(t) and active power deviation ΔPe(t); sorts the sampled data in the order of each node number, introduces a Bayesian probability model, and assumes that the data follows the distribution X~p(X / Φ), with parameters Φ=(θ1, θ2). Bayes' theorem calculates the posterior distribution:

[0067]

[0068] Resample the initial data set through Gibbs sampling to generate a new data matrix Δx d '; Using this matrix as the input matrix, the system characteristic dynamic mode decomposition (DMD) calculation is performed to obtain the eigenvalue matrix λ and its corresponding eigenvector matrix V; By combining the random Koopman theory with the random dynamic system, the coupling relationship between the electromechanical characteristics and the system inertia in the random Koopman space is derived. Then, based on the eigenvalue matrix λ and the corresponding eigenvectors obtained from the dynamic mode decomposition (DMD) in step 2 and the coupling model of electromechanical characteristics and inertia derived above, the equivalent inertia value of each node is calculated. Finally, the inertia distribution heat map of the power system is output based on the geographical distribution and inertia of each node.

[0069] The preferred embodiments of the present invention disclosed above are intended only to help illustrate the present invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the present invention to specific embodiments. Obviously, many modifications and variations are possible based on the contents of this specification. These embodiments are selected and described in detail in this specification to better explain the principles and practical applications of the present invention, thereby enabling those skilled in the art to better understand and utilize the present invention. The present invention is limited only by the claims and their full scope and equivalents.

Claims

1. A method for online inertia assessment of power systems based on dynamic mode decomposition, characterized in that: The following steps are involved: Step 1: Dynamic data input and preprocessing: At time intervals Δt, random response data of each node in the power system under normal operation is collected, including node angular velocity Δω(t), angular frequency Δf(t), and active power deviation ΔPe(t); Step 2: Sort the sampled data in the order of each node number, introduce the Bayesian probability model, assume that the data follows the distribution X~p(X / Φ), parameter Φ=(θ1, θ2); resample the initial data set through Gibbs posterior sampling to generate a new data matrix Δx d ';Use this matrix as the input matrix to perform system characteristic dynamic mode decomposition DMD calculation to obtain the eigenvalue matrix λ and its corresponding eigenvector matrix V; Step 3: By combining the random Koopman theory with the random dynamic system, the coupling relationship between the electromechanical characteristics and the system inertia in the random Koopman space is derived. Then, the equivalent inertia value of each node is calculated based on the eigenvalue matrix λ and the corresponding eigenvectors obtained from the dynamic modal decomposition in step 2 and the coupling model of electromechanical characteristics and inertia derived above.

2. The method for online inertia assessment of a power system based on dynamic mode decomposition according to claim 1, characterized in that: In step 1, the environmental data of each power grid node is measured by the PMU, including the input matrices X0 and X1 composed of the changes in the node's active power and frequency.

3. The method for online inertia assessment of a power system based on dynamic mode decomposition according to claim 1, characterized in that: Step 2: Perform Gibbs posterior sampling. First, assume that the data follows the distribution X~p(X / Φ), parameter Φ=(θ1,θ2), and the prior distribution of Φ is p(θ1,θ2); {X1,……,X n } is the set of samples drawn from this distribution; the posterior distribution is calculated using Bayes' theorem: Initialization parameters From the conditional distribution and The samples are extracted from the Gibbs sampling to generate a Markov chain {Φ (1) ,...,Φ (N) }; Then, the denoised data set is extracted through the stationary distribution of the Markov chain to form a new input matrix Δx d ′=[Δx1′,Δx2′,...,Δx m ′], its data matrix is: P in the matrix z is the data matrix composed of the first m-1 data sequences, F z The data matrix composed of the last m-1 data sequences; construct the state matrix and A d =F z V -1 ∑ -1 U * Perform singular value decomposition on it: A d v j =μ j v j ; U and V are unitary matrices, ∑ is a singular value diagonal matrix; Solve A d The eigenvalue μ j and the eigenvector v j , and is converted into a continuous eigenvalue by λ=ln(μ) / Δt.

4. The method for online inertia assessment of a power system based on dynamic mode decomposition according to claim 1, characterized in that: Step 3 derives the coupling relationship between the electromechanical characteristics and the system inertia in the random Koopman space, including: First, the electromechanical response process of the power system stimulated by the environment is described as: Where M, D, and K represent the diagonalizable matrices consisting of the inertia time constant, damping coefficient, and synchronization coefficient, respectively; Δδ(t) represents the rotor angular deviation near the equilibrium point at time t; and F(t) represents the matrix composed of the natural excitation vector at time t. The deduction of each state variable in the random Koopman space is expressed as: K Ω is the random Koopman operator; F(0) represents the excitation at the initial moment; Δδ(0) represents the rotor angle deviation near the equilibrium point at the initial moment; In random Koopman space: According to the rotor angle active power change ΔP e The angular velocity change Δω satisfies ΔP e =KΔδ and , then: Where: and are the modal vibration shapes extracted from the angular velocity and active power changes respectively; b(0) is the initial input value; since t is arbitrary, the intermediate variables are eliminated. After that, According to the characteristics of electromechanical dynamics, the eigenvalues containing frequency and damping ratio information and the corresponding system modal vibration shapes are all in complex form. The projection transformation is used to decompose the equation into a complex plane based on the real axis and imaginary axis to expand the dimension of the above equation: M is the inertia matrix, D is the damping matrix, are the modal vibration shapes of angular velocity and electromagnetic power respectively, b is the initial eigenfunction, and the equivalent inertia of each node is calculated based on the eigenvalue and eigenvector in step 2.

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