A method for quantitatively analyzing spatial and temporal distribution characteristics of system frequency based on LQR optimal feedback control

Abstract

A kind of system frequency space-time distribution characteristic quantitative analysis method based on LQR optimal feedback control.This application belongs to the field of power system frequency analysis and control technology, to solve the problem that existing index is difficult to represent dynamic whole process and is not effectively associated with control means, the application defines the integral of frequency difference weighted and square between nodes as objective function, converts it into LQR standard quadratic form, solves optimal feedback control matrix K;The Frobenius norm of matrix K is used as quantitative evaluation index, and its rationality in measuring control potential and frequency distribution significance is proved by matrix inequality;Establish the functional relationship between the index and state coefficient matrix A, input state coefficient matrix B, and calculate the Fréchet derivative to analyze the key factors affecting the frequency space-time distribution.This application starts from the control level, considers the frequency dynamic whole process, and can quantitatively analyze the influence of system structure and disturbance on frequency difference.

Description

Technical Field

[0001] This invention belongs to the field of power system frequency analysis and control technology, specifically relating to a quantitative analysis method for the spatiotemporal distribution characteristics of system frequency based on LQR optimal feedback control. Background Technology

[0002] With the large-scale integration of new energy sources into the power grid, traditional power systems are gradually transforming, and the spatiotemporal distribution differences of system frequencies are intensifying. The traditional research concept of a uniform power system frequency is outdated. Currently, there is a lack of clear analytical indicators and methods for the significance of the spatiotemporal distribution of frequencies at various nodes in the power system, making it difficult to conduct refined studies on the spatiotemporal distribution characteristics of frequencies and further increasing the difficulty of frequency control. Therefore, how to establish quantitative evaluation indicators for the spatiotemporal distribution characteristics of system frequencies from the control level, and analyze the key factors affecting the spatiotemporal distribution characteristics based on these indicators to reduce system frequency differences, has become one of the key issues that urgently need to be addressed to improve the frequency stability of power systems.

[0003] Currently, various solutions exist for analyzing the spatiotemporal distribution characteristics of system frequencies. However, current research on this topic still has certain limitations. Most existing quantitative analysis indicators are based on inherent physical quantities of the system itself, only describing instantaneous differences after disturbances and failing to reflect the dynamic process of the system frequency throughout. Furthermore, the initial goal of analyzing the spatiotemporal distribution characteristics of frequencies is to reduce the spatiotemporal variability of system frequencies through relevant control measures, but existing methods cannot analyze the spatiotemporal distribution of system frequencies at the control level. Therefore, there is an urgent need for an analytical method that can fully consider the entire dynamic process of system frequencies and analyze the spatiotemporal distribution characteristics of system frequencies at the control level, in order to better reduce the spatiotemporal variability of system frequencies and improve system frequency stability. Summary of the Invention

[0004] To address the shortcomings of current quantitative indices for frequency spatiotemporal distribution characteristics in representing dynamic frequency processes and the lack of an effective correlation between system frequency spatiotemporal distribution characteristics and control, this invention proposes a quantitative analysis method for system frequency spatiotemporal distribution characteristics based on LQR optimal feedback control. Specifically, it includes the following steps:

[0005] S1. Define the integral of the weighted sum of squares of frequency differences between nodes as the objective function of frequency spatiotemporal distribution differences. Based on the proposed objective function, solve for the optimal feedback control matrix of LQR.

[0006] S2. Based on the obtained optimal feedback control matrix, determine the quantitative evaluation index of the system's frequency spatiotemporal distribution characteristics;

[0007] S3. Based on the proposed quantitative indicators from the control perspective, establish functional relationships to analyze the influencing factors of the system's frequency spatiotemporal distribution characteristics from the control perspective.

[0008] Specifically, the method for establishing the objective function for the frequency spatiotemporal distribution difference in step S1 is as follows:

[0009] Assuming the system has N nodes, the objective function for defining the frequency difference between the inertia centers of the system during the full dynamic process is the frequency difference between the nodes. This index represents the significance of the spatiotemporal distribution of the system in the full dynamic process. The larger the function value, the more significant the frequency spatiotemporal distribution characteristics in the dynamic process of the system, and vice versa. Specifically, it is expressed as follows:

[0010]

[0011] In the formula, a ij A positive number represents the weight of the frequency difference between node i and node j. and These are the frequency deviations of nodes i and j, respectively.

[0012] Specifically, the method for solving the LQR optimal feedback control matrix in step S1 based on the proposed objective function is as follows:

[0013] The objective function for the frequency difference between nodes is transformed into a standard quadratic objective function for LQR control.

[0014]

[0015] In the formula, J(x) is the control target. and Let these represent the state vector and input vector at time t, respectively. Including disturbance inputs and control inputs; and Let Q and R represent the transposes of the two matrices, respectively. Q and R are the state penalty matrix and the control penalty matrix, respectively, representing the accumulation of state deviations in the objective function and the cost of control. Matrix R is not specifically defined and is taken as the identity matrix. Matrix Q should satisfy the following form:

[0016]

[0017] In the formula, Represents the diagonal elements of the state penalty matrix. This represents the off-diagonal elements of the state penalty matrix.

[0018] Specifically, applying the standard quadratic objective function to the state equation of the following linear control system, a given linear control system can be characterized by the following state equation:

[0019]

[0020] In the formula, Let be the derivative of the state vector, x be the state vector, u be the input vector, and y be the output vector; A, B, C, and D are the state coefficient matrix, input-state coefficient matrix, state-output coefficient matrix, and input-output coefficient matrix, respectively; A represents the system structure, B represents the influence of the system input on the state, C represents the influence of the state on the output, and D represents the direct influence of the input on the output; where matrix B can be derived from the perturbation matrix Bt. d and control matrix B c composition:

[0021]

[0022] Based on equation (4), since the linear feedback control law introduces the state variables into the input by linear superposition, the output u(t) at time t can be expressed as:

[0023]

[0024] In the formula, Let t be the input vector at time t, and K be the feedback control matrix. When the controlled objective function is at its minimum, K is the optimal feedback control matrix.

[0025] Specifically, in the process of solving the optimal feedback control matrix of LQR, the optimal feedback control matrix is ​​calculated through the Riccati equation. Based on equations (2) to (5), assuming that matrices A and B are controllable and matrices R and Q are positive definite, then when the objective function J(x) is at its minimum, there exists a unique optimal feedback control matrix K in the system, which can be calculated by the Riccati equation, as follows:

[0026]

[0027] In the formula, P is the matrix solution obtained from the Riccati equation. For the Riccati equation, and Let A and B represent the transposes of matrices A and B, respectively; A is the state coefficient matrix, representing the structure of the system; B is the input state coefficient matrix, representing the influence of the system input on the state; matrix R is not specifically defined and is taken as the identity matrix. This represents the inverse of matrix R.

[0028] Specifically, in step S2, the quantitative evaluation index is the Frobenius norm of the optimal feedback control matrix K. The optimal feedback control matrix K It can be derived from equation (4). For the convenience of subsequent derivation, we take its square, specifically:

[0029]

[0030] In the formula, P is the matrix solution obtained from the Riccati equation, B is the input state coefficient matrix, which characterizes the influence of the system input on the state; matrix R is not specifically defined and is taken as the identity matrix. This represents the square of the inverse of matrix R; This indicates finding the trace of a matrix.

[0031] Specifically, in step S2, the rationality of the frequency spatiotemporal distribution characteristics of the quantization evaluation system is proven using matrix inequalities, and the specific constraints include:

[0032] Suppose there are two distinct active disturbances, with corresponding input state coefficient matrices B1 and B2, respectively. The following inequality holds:

[0033]

[0034] In the formula, The symbol is used to determine the size of the Loewner partial order of matrices B1 and B2;

[0035] For the optimal feedback problem of the quadratic LQR control law in a linear control system, the following conclusion holds:

[0036]

[0037] In the formula, P1 and P2 are the matrix solutions obtained from the Riccati equations corresponding to B1 and B2, respectively;

[0038] Based on the monotonicity of the matrix trace, the inequality holds, specifically:

[0039]

[0040] In the formula, and Each is a matrix The minimum and maximum eigenvalues, This represents the square of the inverse of matrix R;

[0041] P and The relationship between them is:

[0042]

[0043] In the formula, This indicates finding the trace of a matrix. This indicates that the value has increased;

[0044] at the same time:

[0045]

[0046] In the formula, The output average energy after optimal feedback control with unit state deviation is represented by x, where x is the state vector and n is the dimension of the state variables.

[0047] Combining equations (7) to (12), we can conclude that the Frobenius norm of the proposed optimal feedback matrix K is... It possesses the ability to measure the spatiotemporal significance of power system attenuation frequencies under different active power disturbances, meaning it can measure control-level potential. Specifically, When the value increases, it means that in order to suppress the same degree of frequency spatiotemporal distribution differences, the system needs to exert a greater active power regulation, thus reflecting an increase in the difficulty of system control and a decrease in control potential. Therefore, It can be used as a quantitative evaluation index of the spatiotemporal distribution characteristics of system frequency from a control perspective.

[0048] Specifically, the functional relation established in step S3 is:

[0049]

[0050] In the formula, Let K(A,B) be the functional relation between matrices A and B, and let A be the optimal feedback control matrix obtained by combining matrices A and B. Let A be the state coefficient matrix, which represents the structure of the system; and let B be the input state coefficient matrix, which represents the influence of the system input on the state.

[0051] Specifically, due to the functional First, determined by the implicit function of the Riccati equation, we first differentiate the implicit function with respect to matrices P, A, and B. In the process of analyzing the influencing factors of the system's frequency spatiotemporal distribution characteristics from a control perspective, based on functional... Pairs of Fréchet derivatives with respect to matrices A and B An analysis is conducted to determine the factors influencing the spatiotemporal distribution characteristics of the system's frequency. This involves examining the functional... Fréchet's process of finding the derivatives of matrices A and B includes:

[0052] Taking the Fréchet derivative with respect to P yields the defined operator. :

[0053]

[0054] Operator Taking the derivative with respect to P, we get:

[0055]

[0056] In the formula, Operator The derivative of P, ΔP, is a small change in P, where P is the matrix solution obtained from the Riccati equation;

[0057] Operator Taking the derivatives with respect to A and B respectively, we get:

[0058]

[0059] In the formula, and Representing operators respectively The derivatives with respect to A and B, ΔA and ΔB are the small changes in A and B, respectively;

[0060] By using a linear approximation, its Fréchet differential is approximated by a small change ΔK in matrix K:

[0061]

[0062] By the definition of the Frobenius norm, for the perturbation direction (ΔA, ΔB), The Fréchet derivative with respect to K is:

[0063]

[0064] In the formula, Let K be the Frobenius inner product of ΔK;

[0065] Substituting equation (17) into equation (18), we get The Fréchet derivative with respect to B is:

[0066]

[0067] Similarly, substituting equation (17) into equation (18) again, we get... The Fréchet derivative with respect to A is:

[0068] .

[0069] Specifically, based on functional Pairs of Fréchet derivatives with respect to matrices A and B The process of analyzing the factors influencing the spatiotemporal distribution characteristics of system frequency includes:

[0070] Based on equations (19) and (20), we can analyze from a mathematical perspective: matrix A pairs with... The impact is indirect and requires intervention. This will affect or in turn, affect ; Matrix B There are both through The direct impact, and also through The indirect impact, its influence on The influence of matrix B is direct and dominant. In other words, from a control perspective, the control difficulty of the system's frequency spatiotemporal distribution is mainly affected by matrix B, while the influence of matrix A is relatively minor. The control difficulty and frequency spatiotemporal distribution characteristics can be reduced and improved mainly by optimizing matrix B (such as adjusting the position of the control input and enhancing the influence of the control input on the state variables).

[0071] In summary, for a linear system that can be described by equation (4), the relative influence of matrices A and B on the spatiotemporal distribution of system frequency can be quantified based on equations (19) and (20). This is the proposed method for quantifying the spatiotemporal distribution characteristics of system frequency based on LQR optimal feedback control.

[0072] It is worth noting that the above derivation of the indicators The Fréchet derivative is used to build a rigorous mathematical foundation, but in the subsequent examples, a different approach is adopted. The derivative is quantized. This is because for nonnegative scalar functions, Its square They possess the same monotonicity and extreme points on their domains. Therefore, the analysis... The derivative not only maintains consistency with the aforementioned derivation, but its value also more intuitively reflects the change in control difficulty caused by a unit disturbance, making it easier for engineers to interpret directly.

[0073] The beneficial effects of this invention are as follows:

[0074] To address the shortcomings of current quantitative indicators for frequency spatiotemporal distribution characteristics in representing the dynamic process of frequency and in linking system frequency spatiotemporal distribution characteristics with control, this invention, based on LQR optimal feedback control, establishes an index for quantitatively representing the system's frequency spatiotemporal distribution characteristics at the control level and proposes a quantitative analysis method for system frequency spatiotemporal distribution characteristics based on LQR optimal feedback control. This method considers the complete dynamic characteristics of the system frequency, introduces a control theory perspective, and uses the Frobenius norm of the optimal feedback matrix to quantify the control potential for frequency spatiotemporal distribution differences, further realizing the quantitative analysis of system frequency spatiotemporal distribution characteristics and providing indicative significance for subsequent control to mitigate frequency spatiotemporal differences. This method can provide technical support for frequency regulation in new power systems. Attached Figure Description

[0075] Figure 1 This is a schematic diagram illustrating the steps for solving the optimal feedback control matrix in this invention;

[0076] Figure 2 A schematic diagram illustrating the steps for establishing quantitative evaluation indicators for the spatiotemporal distribution characteristics of the system frequency of the present invention;

[0077] Figure 3 This is a schematic diagram illustrating the analysis process of the system frequency spatiotemporal distribution characteristics of the present invention;

[0078] Figure 4 For matrix B c Fréchet derivative heatmap;

[0079] Figure 5 For matrix B d Fréchet derivative heatmap. Detailed Implementation Specific Implementation Example 1:

[0081] This implementation takes an actual ACTIVSg500 system with 500 bus nodes as an example. The system state matrix A and disturbance matrix B can be directly obtained through parameter identification. d Control matrix B c Among them, B d and B c Together, they constitute the input state coefficient matrix B. The following is a quantitative analysis method for the spatiotemporal distribution characteristics of a system based on LQR optimal feedback control, focusing on the influence of matrix B on the system's frequency spatiotemporal distribution characteristics. The method includes the following steps:

[0082] S1. Define the integral of the weighted sum of squares of frequency differences between nodes as the objective function of frequency spatiotemporal distribution differences. Based on the proposed objective function, solve for the optimal feedback control matrix of LQR.

[0083] Combination Figure 1The steps for solving the LQR optimal feedback control matrix are as follows:

[0084] A1. Establish an objective function to characterize the spatiotemporal distribution differences of frequency;

[0085] Assuming the system has N nodes, the objective function for defining the frequency difference between the inertia centers of the system during the full dynamic process is the frequency difference between the nodes. This index represents the significance of the spatiotemporal distribution of the system in a fully dynamic process. A larger function value indicates a more significant spatiotemporal distribution of frequency during the system's dynamic process, and vice versa. Specifically, it is expressed as:

[0086]

[0087] In the formula, a ij A positive number represents the weight of the frequency difference between node i and node j. and These are the frequency deviations of nodes i and j, respectively.

[0088] A2. Transform equation (2) into a standard quadratic objective function for LQR control, as shown below:

[0089]

[0090] In the formula, J(x) is the control target. and Let these represent the state vector and input vector at time t, respectively. and Let Q and R represent the transposes of the two matrices, respectively. Q and R are the state penalty matrix and the control penalty matrix, respectively, representing the accumulation of state deviations in the objective function and the cost of control. Matrix R is not specifically defined and is taken as the identity matrix. Matrix Q should satisfy the following form:

[0091]

[0092] In the formula, Represents the diagonal elements of the state penalty matrix. This represents the off-diagonal elements of the state penalty matrix.

[0093] A3. Solve for the optimal feedback control matrix K:

[0094] For a given linear control system, it can be characterized by the following state equation:

[0095]

[0096] In the formula, Let be the derivative of the state vector, where x is the state vector, u is the input vector, and y is the output vector; A, B, C, and D are the state coefficient matrix, input-state coefficient matrix, state-output coefficient matrix, and input-output coefficient matrix, respectively; A represents the system structure, B represents the influence of the system input on the state, C represents the influence of the state on the output, and D represents the direct influence of the input on the output. Matrix B is derived from the perturbation matrix Bt. d and control matrix B c composition:

[0097]

[0098] Based on equation (25), since the linear feedback control law introduces the state variables into the input by linear superposition, the output u(t) at time t can be expressed by the following equation:

[0099]

[0100] In the formula, Let t be the input vector at time t, and K be the feedback control matrix. When the controlled objective function is at its minimum, K is the optimal feedback control matrix.

[0101] Based on equations (23) to (26), assuming that matrices A and B are controllable and matrices R and Q are positive definite, then when the objective function J(x) is at its minimum, there exists a unique optimal feedback control matrix K, which can be calculated from the Riccati equation:

[0102]

[0103] In the formula, P is the matrix solution obtained from the Riccati equation. For the Riccati equation, and Let A and B represent the transposes of matrices A and B, respectively. This represents the inverse of matrix R.

[0104] Combination Figure 2 The steps for establishing the quantitative evaluation index of the system's frequency spatiotemporal distribution characteristics are as follows:

[0105] S2. Based on the obtained optimal feedback control matrix, and combining the ideas of matrix inequalities and norms, a quantitative evaluation index for the spatiotemporal distribution characteristics of system frequency is proposed from the perspective of control.

[0106] B1. Solve for the Frobenius norm of the optimal feedback control matrix K;

[0107] The optimal feedback control matrix K It can be derived from equation (25). For the convenience of subsequent derivation, we take its square, as shown in the following equation:

[0108]

[0109] In the formula, This indicates finding the trace of a matrix.

[0110] B2. Use inequalities to prove the rationality of this index in quantitatively evaluating the spatiotemporal distribution characteristics of the system frequency from a control perspective;

[0111] Based on equation (27), assuming there are two different active disturbances with corresponding input state coefficient matrices B1 and B2, the following equation holds:

[0112]

[0113] In the formula, The symbol is used to determine the Loewner partial order size of matrices B1 and B2.

[0114] Based on equation (29), for the optimal feedback problem of the quadratic LQR control law of a linear control system, the following conclusions are drawn:

[0115]

[0116] In the formula, P1 and P2 are the matrix solutions obtained by the Riccati equations corresponding to matrices B1 and B2, respectively;

[0117] Based on the monotonicity of the matrix trace, the following inequality holds:

[0118]

[0119] In the formula, and Each is a matrix The minimum and maximum eigenvalues, This represents the inverse of matrix R, squared.

[0120] Combining equations (28) to (31), we can obtain P and Relationship between them:

[0121]

[0122] Meanwhile, considering the average energy amplification effect of linear transformation, the following equation also holds:

[0123]

[0124] In the formula, Let x represent the average output energy after optimal feedback control with unit state deviation, where x is the state vector and n is the dimension of the state variables.

[0125] Combining equations (28) to (33), we can conclude that the Frobenius norm of the proposed optimal feedback matrix K is... It can measure the power system's ability to mitigate the spatiotemporal distribution of frequency under different active power disturbances, that is, it can measure the potential at the control level. Specifically, When the value increases, it means that in order to suppress the same degree of frequency spatiotemporal distribution differences, the system needs to exert a greater active power regulation, thus reflecting an increase in the difficulty of system control and a decrease in control potential. Therefore, It can be used as a quantitative evaluation index of the spatiotemporal distribution characteristics of system frequency from a control perspective.

[0126] S3. Based on the proposed quantitative indicators from the control perspective, establish functional relationships to analyze the influencing factors of the system's frequency spatiotemporal distribution characteristics from the control perspective.

[0127] Combination Figure 3 The analysis process of the system's frequency spatiotemporal distribution characteristics is as follows:

[0128] C1. Establish the functional relationship between matrix K and A, B:

[0129] Based on equation (25) described in step S1, the functional relation is established as follows:

[0130]

[0131] In the formula, Let K(A,B) be the functional relation between matrices A and B, and let K(A,B) represent the optimal feedback matrix obtained by combining matrices A and B. A is the state coefficient matrix, which characterizes the structure of the system; B is the input state coefficient matrix, which characterizes the influence of the system input on the state.

[0132] C2. Find the Fréchet derivatives of the functional with respect to matrices A and B:

[0133] Due to functional First, determined by the implicit function of the Riccati equation, we first differentiate the implicit function with respect to matrices P, A, and B, as follows:

[0134] Taking the Fréchet derivative with respect to P yields the defined operator. :

[0135]

[0136] Operator Taking the derivative with respect to P, we get:

[0137]

[0138] In the formula, Operator The derivative with respect to A and B, ΔP, is a small change in P.

[0139] Operator Taking the derivatives with respect to A and B respectively, we get:

[0140]

[0141] In the formula, and Representing operators respectively The derivatives with respect to A and B, ΔA and ΔB are the small changes in A and B, respectively.

[0142] By using a linear approximation, its Fréchet differential is approximated by a small change ΔK in matrix K:

[0143]

[0144] By the definition of the Frobenius norm, for the perturbation direction (ΔA, ΔB), The Fréchet derivative with respect to K is:

[0145]

[0146] In the formula, Let K be the Frobenius inner product of K and ΔK.

[0147] Substituting equation (38) into equation (39), we obtain the results respectively. Regarding control matrix B c and perturbation matrix B d The Fréchet derivative, combined with Figure 4 , Figure 5 Explanation of matrix B c Fréchet derivative heatmap and B d Fréchet derivative heatmap:

[0148]

[0149]

[0150] Similarly, substituting equation (38) into equation (39) again, we get... Fréchet derivative with respect to A:

[0151]

[0152] C3. Analysis of the proposed quantitative indicators Key influencing factors;

[0153] Based on equations (40) to (42) obtained from C2, we can analyze from a mathematical perspective: matrix A pairs The impact is indirect and requires intervention. This will affect or in turn, affect ; Matrix B There are both through The direct impact, and also through The indirect impact, its influence on The influence of matrix B is direct and dominant. In other words, from a control perspective, the control difficulty of the system's frequency spatiotemporal distribution is mainly affected by matrix B, while the influence of matrix A is relatively minor. The control difficulty and frequency spatiotemporal distribution characteristics can be reduced and improved mainly by optimizing matrix B (such as adjusting the position of the control input and enhancing the influence of the control input on the state variables).

[0154] In summary, for a linear system that can be described by equation (25), the relative influence of matrices A and B on the spatiotemporal distribution of system frequency can be quantitatively analyzed based on equations (40) and (41). This is the proposed quantitative analysis method for the spatiotemporal distribution characteristics of system frequency based on LQR optimal feedback control.

[0155] The above examples of this invention are merely illustrative of the computational model and process of this invention, and are not intended to limit the implementation of this invention. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is impossible to exhaustively list all possible implementations here. Any obvious variations or modifications derived from the technical solutions of this invention are still within the scope of protection of this invention.

[0156] The scope of this invention is not limited to the above-described embodiments; a combination of one or more specific embodiments can also achieve the purpose of the invention.

Claims

1. A method for quantitative analysis of the spatiotemporal distribution characteristics of system frequency based on LQR optimal feedback control, characterized in that, Includes the following steps: S1. Define the integral of the weighted sum of squares of frequency differences between nodes as the objective function of frequency spatiotemporal distribution differences. Based on the proposed objective function, solve for the optimal feedback control matrix of LQR. S2. Based on the obtained optimal feedback control matrix, determine the quantitative evaluation index of the system's frequency spatiotemporal distribution characteristics; S3. Based on the proposed quantitative indicators from the control perspective, establish functional relationships to analyze the influencing factors of the system's frequency spatiotemporal distribution characteristics from a control perspective.

2. The method for quantitative analysis of the spatiotemporal distribution characteristics of a system based on LQR optimal feedback control according to claim 1, characterized in that, The specific method for establishing the objective function for the frequency spatiotemporal distribution difference in step S1 is as follows: Assuming the system has N nodes, the objective function for defining the frequency difference between the inertia centers of the system during the full dynamic process is the frequency difference between the nodes. : In the formula, a ij A positive number represents the weight of the frequency difference between node i and node j. and These are the frequency deviations of nodes i and j, respectively.

3. The method for quantitative analysis of the spatiotemporal distribution characteristics of a system based on LQR optimal feedback control according to claim 1, characterized in that, In step S1, the process of solving the optimal feedback control matrix of LQR based on the proposed objective function includes: The objective function for frequency differences between nodes is transformed into a standard quadratic objective function for LQR control. In the formula, J(x) is the control target. and Let these represent the state vector and input vector at time t, respectively. Including disturbance inputs and control inputs; and Let Q and R represent the transposes of the two matrices, respectively. Q and R are the state penalty matrix and the control penalty matrix, respectively, representing the accumulation of state deviations in the objective function and the cost of control. Matrix R is not specifically defined and is taken as the identity matrix. Matrix Q should satisfy the following form: In the formula, Represents the diagonal elements of the state penalty matrix. This represents the off-diagonal elements of the state penalty matrix.

4. The method for quantitative analysis of the spatiotemporal distribution characteristics of a system based on LQR optimal feedback control according to claim 3, characterized in that, The standard quadratic objective function is applied to the state equation of the following linear control system: In the formula, Let be the derivative of the state vector, x be the state vector, u be the input vector, and y be the output vector; A, B, C, and D are the state coefficient matrix, input-state coefficient matrix, state-output coefficient matrix, and input-output coefficient matrix, respectively; A represents the system structure, B represents the influence of the system input on the state, C represents the influence of the state on the output, and D represents the direct influence of the input on the output; where matrix B can be derived from the perturbation matrix Bt. d and control matrix B c composition: The input vector u(t) at time t can be expressed as: In the formula, Let t be the state vector at time t, and K be the feedback control matrix. When the controlled objective function is at its minimum, K is the optimal feedback control matrix.

5. The method for quantitative analysis of the spatiotemporal distribution characteristics of a system based on LQR optimal feedback control according to claim 4, characterized in that, In the process of solving for the optimal feedback control matrix of LQR, the optimal feedback control matrix is ​​calculated through the Riccati equation, specifically: In the formula, P is the matrix solution obtained from the Riccati equation. For the Riccati equation, and Let A and B represent the transposes of matrices A and B, respectively. A is the state coefficient matrix, which represents the structure of the system; B is the input state coefficient matrix, which represents the influence of the system input on the state. Unless otherwise specified, matrix R is taken as the identity matrix. This represents the inverse of matrix R.

6. The method for quantitative analysis of the spatiotemporal distribution characteristics of a system based on LQR optimal feedback control according to claim 1, characterized in that, In step S2, the quantitative evaluation index is the Frobenius norm of the optimal feedback control matrix K. .

7. The method for quantitative analysis of the spatiotemporal distribution characteristics of a system based on LQR optimal feedback control according to claim 1, characterized in that, The rationality of the quantitative evaluation index of the system frequency spatiotemporal distribution characteristics mentioned in step S2 is determined in the following way: Suppose there are two distinct active power disturbances, with corresponding input state coefficient matrices B1 and B2, respectively. The following inequality holds: In the formula, The symbol is used to determine the size of the Loewner partial order of matrices B1 and B2; For the optimal feedback problem of the quadratic LQR control law of a linear control system, we have The equation holds true, where P1 and P2 are the matrix solutions obtained from the Riccati equations corresponding to B1 and B2, respectively. Based on the monotonicity of the matrix trace, the inequality holds, specifically: In the formula, and Each is a matrix The minimum and maximum eigenvalues, This represents the square of the inverse of matrix R; P and The relationship between them is: In the formula, This indicates finding the trace of a matrix. This indicates that the value has increased; at the same time: In the formula, The output average energy after optimal feedback control with unit state deviation is represented by x, where x is the state vector and n is the dimension of the state variables. Therefore, the Frobenius norm of the proposed optimal feedback matrix K It can measure the ability of a power system to reduce the spatiotemporal distribution of frequency under different active power disturbances, that is, its potential to measure the control level.

8. The method for quantitative analysis of the spatiotemporal distribution characteristics of a system based on LQR optimal feedback control according to claim 1, characterized in that, The functional relation established in step S3 is In the formula, Let K(A,B) be the functional relation between matrices A and B, and let A be the optimal feedback control matrix obtained by combining matrices A and B. Let A be the state coefficient matrix, which represents the structure of the system; and let B be the input state coefficient matrix, which represents the influence of the system input on the state.

9. The method for quantitative analysis of the spatiotemporal distribution characteristics of a system based on LQR optimal feedback control according to claim 8, characterized in that, In the process of analyzing the influencing factors of the system's frequency spatiotemporal distribution characteristics from a control perspective, based on functional analysis... Pairs of Fréchet derivatives with respect to matrices A and B Analysis is conducted to analyze the factors influencing the spatiotemporal distribution characteristics of the system's frequency; the functional is determined. The process of obtaining the Fréchet derivatives of matrices A and B includes: Taking the Fréchet derivative with respect to P yields the defined operator. : Operator Taking the derivative with respect to P, we get: In the formula, Operator The derivative of P, ΔP, is a small change in P, where P is the matrix solution obtained from the Riccati equation; Operator Taking the derivatives with respect to A and B respectively, we get: In the formula, and Representing operators respectively The derivatives with respect to A and B, ΔA and ΔB are the small changes in A and B, respectively; By using a linear approximation, its Fréchet differential is approximated by a small change ΔK in matrix K: By the definition of the Frobenius norm, for the perturbation direction (ΔA, ΔB), The Fréchet derivative with respect to K is: In the formula, Let K be the Frobenius inner product of ΔK; The Fréchet derivative with respect to B is: The Fréchet derivative with respect to A is: 。 10. The method for quantitative analysis of the spatiotemporal distribution characteristics of a system based on LQR optimal feedback control according to claim 9, characterized in that, Based on functional Pairs of Fréchet derivatives with respect to matrices A and B The process of analyzing the factors influencing the spatiotemporal distribution characteristics of system frequency includes: Matrix A pairs The impact is indirect and requires intervention. This will affect This will affect ; Matrix B There are both through The direct impact, and also through The indirect impact, its influence on The influence of matrix B is direct and dominant; that is, from a control perspective, the difficulty of controlling the spatiotemporal distribution of system frequency is mainly affected by matrix B, while the influence of matrix A is relatively minor.

Images