Micro-grid linearization model order reduction method, system, device and medium
By employing eigenvalue translation transformation, Krylov subspace algorithm, and improved balance truncation method, the problems of high computational complexity and insufficient accuracy in microgrid model order reduction are solved, achieving efficient order reduction and stability guarantee for unstable systems.
Patent Information
- Application Number
- CN202511800382.0
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-12-02
- Publication Date
- 2026-03-06
AI Technical Summary
Existing methods for reducing the order of microgrid models have high computational complexity when dealing with unstable systems, and existing equilibrium truncation methods are difficult to effectively reduce the order, affecting optimization efficiency and accuracy.
By employing eigenvalue translation transformation, Krylov subspace algorithm, and improved balance truncation method, the order of the microgrid dynamic linearized state-space system is reduced by generating Lyapunov equations to solve for controllable and observable Gram matrices, thus ensuring system stability and accuracy.
It achieves efficient order reduction for unstable microgrid systems, reduces computational complexity, and ensures the accuracy and stability of the reduced-order system, making it suitable for microgrid controller design and dynamic simulation.
Smart Images
Figure CN121614709A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of microgrid stability analysis and control, specifically involving methods, systems, equipment, and media for reducing the order of microgrid linearization models. Background Technology
[0002] From a mathematical representation perspective, a microgrid is a multivariable, strongly coupled nonlinear system. Given the complexity of mathematical analysis for multivariable nonlinear systems, linearization is typically employed to simplify the problem when optimizing microgrid controller parameters. Furthermore, if the components within the microgrid are modeled in detail, the model representing the microgrid's dynamic characteristics will have a high order. In the process of optimizing controller parameters using artificial intelligence algorithms, solving the objective function and verifying constraints involves a large amount of repetitive computation. This results in a massive computational load when calculating based on the original high-order system, thus impacting optimization efficiency. Therefore, reducing the order of the original system becomes a necessary step to improve computational efficiency and ensure the feasibility of the optimization process.
[0003] Current system order reduction methods can be mainly categorized into physical mechanism-based reduction, mathematical model-based reduction, and data-driven reduction. Physical mechanism-based reduction primarily achieves order reduction by analyzing and simplifying the physical model. For example, different control loops of the inverter in a distributed power source can be simplified according to differences in time scales. This method has clear physical meaning but may introduce significant errors. Mathematical model-based reduction methods first mathematically model the system and then use mathematical methods such as balance truncation and Routh approximation to reduce the system order. Furthermore, data-driven reduction methods utilize large amounts of historical or real-time data and employ technologies such as artificial intelligence to reduce the system order. This method does not rely on the system's internal structure and parameters but requires a sufficient dataset of known information. In the optimization design of microgrid de-controllers, since the system structure is usually known, and to ensure the accuracy of the results, mathematical model-based reduction methods are more advantageous. This is because they can utilize known structural information while rigorously controlling errors through mathematical means.
[0004] Among mathematical model-based order reduction methods, the balanced truncation method is one of the most widely used. Its advantage lies in its ability to obtain the error relationship between the original system and the reduced system. Furthermore, it preserves the stability, controllability, and observability of the original system. However, this method requires the use of controllable and observable Gram matrices during the reduction process, which are difficult to solve. Moreover, this method is only applicable to stable systems, but the system before parameter optimization may be unstable. Therefore, the existing balanced truncation method needs improvement to efficiently solve for the Gram matrix and make the method applicable to unstable systems. Summary of the Invention
[0005] The purpose of this invention is to overcome the shortcomings of existing methods in reducing the order of microgrid models and designing controller parameters, and to propose a method, system, device and medium for reducing the order of linearized microgrid models.
[0006] To achieve the above objectives, the present invention adopts the following technical solution: In a first aspect, the present invention provides a method for reducing the order of a microgrid linearization model, comprising the following steps: Based on the distributed power source element model, transmission network element model, and load element model in the microgrid, a set of dynamic nonlinear differential-algebraic equations for the microgrid is established. Solve for the stable equilibrium point of the dynamic nonlinear differential-algebraic equation system of the microgrid, and perform linearization at the stable equilibrium point to obtain the dynamic linearized differential-algebraic equation system of the microgrid; simplify the dynamic linearized differential-algebraic equation system of the microgrid by elimination to obtain the dynamic linearized state-space system of the microgrid. The rightmost eigenvalue of the dynamic linearized state-space system of the microgrid is solved by the eigenvalue algorithm. If the rightmost eigenvalue is located to the right of the imaginary axis on the complex plane, the dynamic linearized state-space system of the microgrid is translated to the left to move the position of the rightmost eigenvalue to the left of the imaginary axis on the complex plane, thus obtaining the translated dynamic linearized state-space system of the microgrid. Lyapunov equations are generated based on the dynamic linearized state-space system of the microgrid after translation transformation, and the Krylov subspace algorithm is used to solve the Lyapunov equations to obtain the controllable Gram matrix and the observable Gram matrix. Based on the controllable Gram matrix and the observable Gram matrix, the balanced truncation method is used to reduce the order of the dynamic linearized state-space system of the microgrid after translation transformation, resulting in the reduced-order dynamic linearized state-space system of the microgrid. The inverse translation transformation is then performed on the reduced-order dynamic linearized state-space system of the microgrid to obtain the linearized reduced-order model of the microgrid, thus completing the reduction of the order of the microgrid linearization model.
[0007] Furthermore, the system of dynamic nonlinear differential-algebraic equations for the microgrid is shown below:
[0008] in, For state variables, For algebraic variables, For input variables, For output variables; n , m , p and q These represent the number of state variables, algebraic variables, input variables, and output variables, respectively. n , m , p and q All are positive integers;t For time, State variables Regarding time t The first derivative, The state equation function, For functions of algebraic equations, This is the output equation function.
[0009] Furthermore, the system of dynamic linearized differential-algebraic equations for the microgrid is shown below:
[0010] in, For the linearized state variables, For linearized algebraic variables, For the linearized input variables, The output variable after linearization; It is the identity matrix; It is the matrix of partial derivatives of the state equation function with respect to the state variables. It is the matrix of partial derivatives of the state equation function with respect to algebraic variables. It is the matrix of partial derivatives of the state equation function with respect to the input variables. It is the matrix of partial derivatives of the function in the algebraic equation with respect to the state variables. It is the matrix of partial derivatives of the function in an algebraic equation with respect to the algebraic variables. It is the matrix of partial derivatives of the function in the algebraic equation with respect to the input variables. It is the matrix of partial derivatives of the output equation function with respect to the state variables. It is the matrix of partial derivatives of the output equation function with respect to the substituted variables; By eliminating variables and simplifying the system of differential-algebraic equations for the dynamic linearization of the microgrid, the dynamic linearization state-space system of the microgrid is obtained as shown in the following equation:
[0011] in, The state matrix, For the input matrix, For the output matrix, For direct transmission matrix, The linearized state variables Regarding time t The first derivative; The transfer function of the dynamically linearized state-space system of a microgrid is shown in the following equation:
[0012] in, This is the transfer function of the dynamically linearized state-space system of a microgrid.
[0013] Furthermore, the translation distance for the leftward translation transformation is shown in the following formula:
[0014] in, The translation distance is... The real part of the eigenvalue. The rightmost eigenvalue; The dynamically linearized state-space system of the microgrid after translation transformation is shown in the following equation:
[0015] in, This is the state matrix after translation; The state variables of the dynamically linearized state-space system of the microgrid after translation transformation are... The output variables are the dynamic linearized state-space system of the microgrid after translation transformation.
[0016] Furthermore, the Lyapunov equations are generated based on the dynamically linearized state-space system of the microgrid after translation transformation, as shown in the following equation:
[0017] in, For a controllable Gram matrix, For observable Gram matrix, It is the translated state matrix. The transpose of the matrix, It is the input matrix The transpose of the matrix, It is the output matrix The transpose of the matrix; The process of solving the Lyapunov equations using the Krylov subspace algorithm to obtain the controllable Gram matrix and the observable Gram matrix is shown in the following equation:
[0018]
[0019] in, and These are the controllable and observable Krylov subspaces, respectively. l and k , respectively, are the dimensions of the two subspaces; and These are the orthogonal bases of the two subspaces, respectively; and This is a solution to the Lyapunov equation; For a controllable Gram matrix, It is an observable Gram matrix; Let be the projected state matrix on the controllable Krylov subspace. The projected state matrix on the observable Krylov subspace; Let be the solution matrix of the projected controllable Lyapunov equations. This is the solution matrix of the observable Lyapunov equation after projection; The input matrix after projection, This is the output matrix after projection.
[0020] Furthermore, the reduced-order microgrid dynamic linearized state-space system is shown in the following equation:
[0021] in, For the state variables of the reduced-order system, These are the output variables of the reduced-order system; , , and This is the coefficient matrix after order reduction; The linearized order reduction model for microgrids is shown in the following equation:
[0022] in, The state variables are after the inverse translation transformation. The output variable after the inverse translation transformation; The transfer function of the reduced-order microgrid dynamic linearized state-space system is shown in the following equation:
[0023] in, For the reduced-order dynamic linearized state-space system of the microgrid, the transfer function is given. The error between the transfer function of the reduced-order microgrid dynamic linearized state-space system and the transfer function of the microgrid dynamic linearized state-space system satisfies
[0024] Furthermore, based on the controllable Gram matrix and the observable Gram matrix, the order of the dynamically linearized state-space system of the microgrid after translation transformation is reduced using the balanced truncation method. The reduced-order dynamically linearized state-space system of the microgrid includes the following steps: The controllable Gram matrix and the observable Gram matrix are decomposed using Cholesky decomposition, transforming them into matrix operation objects, as shown in the following equation:
[0025] in, For a controllable Gram matrix, The Cholesky factor of the controllable Gram matrix. For observable Gram matrix, The Cholesky factor is the observable Gram matrix; Singular value decomposition is performed on the product of the transpose of the Cholesky factor of the controllable Gram matrix and the Cholesky factor of the observable Gram matrix to extract Hankel singular values, as shown in the following equation:
[0026] in, It is a left singular vector matrix. The columns of the right singular vector matrix; ; These are Hankel singular values; Before keeping r There are 1 Hankel singular value, as shown in the following formula:
[0027] in, To allow for order reduction error; Based on the previous r The left and right projection matrices are constructed using the given Hankel singular values as shown in the following equation:
[0028] in, for The former r column sum for The former r List; ; The left projection matrix, This is the right projection matrix; By using left and right projection matrices, the dynamic linearized state-space system of the microgrid after translation transformation is projected from high dimension to low dimension, resulting in a reduced-order dynamic linearized state-space system of the microgrid.
[0029] Secondly, the present invention provides a system for reducing the order of a microgrid linearization model, comprising: The microgrid nonlinear equations module is used to establish a dynamic nonlinear differential-algebraic equation system for a microgrid based on the distributed power source element model, transmission network element model, and load element model in the microgrid. The elimination and simplification module is used to solve for the stable equilibrium point of the dynamic nonlinear differential-algebraic equation system of the microgrid, and to perform linearization at the stable equilibrium point to obtain the dynamic linearized differential-algebraic equation system of the microgrid; the elimination and simplification module of the dynamic linearized differential-algebraic equation system of the microgrid is then used to obtain the dynamic linearized state-space system of the microgrid. The translation transformation module is used to solve for the rightmost eigenvalue of the dynamic linearized state-space system of the microgrid using the eigenvalue algorithm. If the rightmost eigenvalue is located to the right of the imaginary axis on the complex plane, the dynamic linearized state-space system of the microgrid is translated to the left, and the position of the rightmost eigenvalue is moved to the left of the imaginary axis on the complex plane, so as to obtain the translated dynamic linearized state-space system of the microgrid. The controllable and observable Gram matrix modules are obtained and used to generate Lyapunov equations based on the dynamically linearized state-space system of the microgrid after translation transformation. The Krylov subspace algorithm is used to solve the Lyapunov equations to obtain the controllable and observable Gram matrices. The microgrid linearization model order reduction module is used to reduce the order of the microgrid dynamic linearized state-space system after translation transformation based on the controllable Gram matrix and the observable Gram matrix, using the balanced truncation method. The module then performs an inverse translation transformation on the reduced microgrid dynamic linearized state-space system to obtain the microgrid linearized reduced-order model, thus completing the order reduction of the microgrid linearization model.
[0030] Thirdly, the present invention provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement a method for reducing the order of a microgrid linearization model.
[0031] Fourthly, the present invention provides a computer-readable storage medium storing a computer program, which, when executed by a processor, implements a method for reducing the order of a microgrid linearization model.
[0032] Compared with the prior art, the present invention has the following beneficial technical effects: This invention proposes a method for reducing the order of a microgrid linearization model. First, it obtains models of each component in the microgrid and establishes a system of nonlinear differential-algebraic equations characterizing the dynamic characteristics of the microgrid. Then, it solves for the stable equilibrium point and performs linearization to obtain the linearized system of differential-algebraic equations. Furthermore, it uses an eigenvalue algorithm to solve for the rightmost eigenvalue; if the system is unstable, it is shifted to the left to stabilize the system. Next, it generates Lyapunov equations based on the shifted system and uses the Krylov subspace algorithm to solve for the controllable and observable Gram matrices. Finally, it uses the balance truncation method to reduce the order of the shifted system and performs an inverse translation to obtain the final reduced-order model. This invention improves upon existing balance truncation methods, ensuring the accuracy, stability, controllability, and observability of the reduced-order system while efficiently solving the Gram matrix and making the method applicable to unstable systems. The balance truncation method for reducing the order of the microgrid ensures the accuracy, stability, controllability, and observability of the reduced-order system; it also improves upon existing balance truncation methods to efficiently solve the Gram matrix and make the method applicable to unstable systems. The Krylov subspace algorithm is used to solve for controllable and observable Gram matrices, avoiding the direct solution of higher-order Lyapunov equations. A translation transformation is employed to handle unstable systems, enabling the improved equilibrium truncation method to reduce the model order of unstable systems. Attached Figure Description
[0033] The accompanying drawings described herein are for illustrative purposes only and are not intended to limit the scope of the invention in any way. Furthermore, the shapes and proportions of the components in the drawings are merely illustrative to aid in understanding the invention and do not specifically limit the shapes and proportions of the components of the invention.
[0034] In the attached diagram: Figure 1 This is a flowchart of the microgrid linearization model order reduction method of the present invention.
[0035] Figure 2 This is a simplified structural diagram of the reduced-order system of the microgrid linearization model of the present invention.
[0036] Figure 3 This is a schematic diagram of an electronic device for the microgrid linearization model order reduction method of the present invention.
[0037] Figure 4 This is a framework diagram of the microgrid system used in the testing of an embodiment of the present invention.
[0038] Figure 5 This is a flowchart of the method for reducing the order of the microgrid linearization model in an embodiment of the present invention.
[0039] Figure 6 This is a graph showing the eigenvalue distributions of the original system and the reduced-order system.
[0040] Figure 7 The graphs show the frequency response curves of the original system and the reduced-order system. Detailed Implementation
[0041] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. 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 should fall within the scope of protection of the present invention.
[0042] Example 1 See Figure 1 The method for reducing the order of a microgrid linearization model includes the following steps: Based on the distributed power source element model, transmission network element model, and load element model in the microgrid, a set of dynamic nonlinear differential-algebraic equations for the microgrid is established. Solve for the stable equilibrium point of the dynamic nonlinear differential-algebraic equation system of the microgrid, and perform linearization at the stable equilibrium point to obtain the dynamic linearized differential-algebraic equation system of the microgrid; simplify the dynamic linearized differential-algebraic equation system of the microgrid by elimination to obtain the dynamic linearized state-space system of the microgrid. The rightmost eigenvalue of the dynamic linearized state-space system of the microgrid is solved by the eigenvalue algorithm. If the rightmost eigenvalue is located to the right of the imaginary axis on the complex plane, the dynamic linearized state-space system of the microgrid is translated to the left to move the position of the rightmost eigenvalue to the left of the imaginary axis on the complex plane, thus obtaining the translated dynamic linearized state-space system of the microgrid. Lyapunov equations are generated based on the dynamic linearized state-space system of the microgrid after translation transformation, and the Krylov subspace algorithm is used to solve the Lyapunov equations to obtain the controllable Gram matrix and the observable Gram matrix. Based on the controllable Gram matrix and the observable Gram matrix, the balanced truncation method is used to reduce the order of the dynamic linearized state-space system of the microgrid after translation transformation, resulting in the reduced-order dynamic linearized state-space system of the microgrid. The reduced-order dynamic linearized state-space system of the microgrid is then subjected to an inverse translation transformation to obtain the linearized reduced-order model of the microgrid, thus completing the reduction of the order of the microgrid linearized model.
[0043] This embodiment constructs a system of equations based on core component models such as distributed power sources, transmission networks, and loads. The linearization process anchors a stable equilibrium point, fully preserving the key dynamic coupling relationships of the microgrid and avoiding distortion of core characteristics. By employing a leftward translation transformation, the unstable system is converted into a stable system, overcoming the limitations of classical order reduction methods and handling unstable linearized systems under scenarios such as microgrid load fluctuations and power output adjustments. The Krylov subspace algorithm is used to solve the Lyapunov equations, significantly reducing the computational complexity of high-dimensional Gram matrices. The balance truncation method uses Hankel singular values to screen core states, ensuring accuracy through error bounds while compressing dimensions (e.g., reducing from tens to more than ten dimensions). The reduced-order model balances real-time performance and accuracy, and can be quickly used for microgrid controller design, dynamic simulation, and stability analysis, addressing the pain points of large computational load and slow real-time response of the original high-dimensional model, providing an efficient and reliable mathematical tool for engineering implementation.
[0044] Example 2 See Figure 2 The microgrid linearization model order reduction system includes: The microgrid nonlinear equations module is used to establish a dynamic nonlinear differential-algebraic equation system for a microgrid based on the distributed power source element model, transmission network element model, and load element model in the microgrid. The elimination and simplification module is used to solve for the stable equilibrium point of the dynamic nonlinear differential-algebraic equation system of the microgrid, and to perform linearization at the stable equilibrium point to obtain the dynamic linearized differential-algebraic equation system of the microgrid; the elimination and simplification module of the dynamic linearized differential-algebraic equation system of the microgrid is then used to obtain the dynamic linearized state-space system of the microgrid. The translation transformation module is used to solve for the rightmost eigenvalue of the dynamic linearized state-space system of the microgrid using the eigenvalue algorithm. If the rightmost eigenvalue is located to the right of the imaginary axis on the complex plane, the dynamic linearized state-space system of the microgrid is translated to the left, and the position of the rightmost eigenvalue is moved to the left of the imaginary axis on the complex plane, so as to obtain the dynamic linearized state-space system of the microgrid after translation transformation. The controllable and observable Gram matrix modules are obtained and used to generate Lyapunov equations based on the dynamically linearized state-space system of the microgrid after translation transformation. The Krylov subspace algorithm is used to solve the Lyapunov equations to obtain the controllable and observable Gram matrices. The microgrid linearization model order reduction module is used to reduce the order of the microgrid dynamic linearized state-space system after translation transformation based on the controllable Gram matrix and the observable Gram matrix, using the balanced truncation method. The module then performs an inverse translation transformation on the reduced microgrid dynamic linearized state-space system to obtain the microgrid linearized reduced-order model, thus completing the order reduction of the microgrid linearization model.
[0045] Example 3 See Figure 3 An electronic device includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements a method for reducing the order of a microgrid linearization model: Based on models of distributed power source components, transmission network components, and load components in the microgrid, a system of dynamic nonlinear differential-algebraic equations for the microgrid is established; the stable equilibrium points of the system of dynamic nonlinear differential-algebraic equations are solved, and linearization is performed at these points to obtain a system of dynamically linearized differential-algebraic equations for the microgrid; the system of dynamically linearized differential-algebraic equations for the microgrid is simplified by elimination to obtain a dynamically linearized state-space system of the microgrid; the rightmost eigenvalue of the dynamically linearized state-space system of the microgrid is solved using an eigenvalue algorithm. If the rightmost eigenvalue is located to the right of the imaginary axis on the complex plane, then... A leftward translation transformation is performed on the dynamic linearized state-space system of the microgrid, shifting the position of the rightmost eigenvalue to the left of the imaginary axis in the complex plane, resulting in the translated dynamic linearized state-space system of the microgrid. Lyapunov equations are generated based on the translated dynamic linearized state-space system, and the Krylov subspace algorithm is used to solve the Lyapunov equations, yielding the controllable Gram matrix and the observable Gram matrix. Based on the controllable Gram matrix and the observable Gram matrix, the order of the translated dynamic linearized state-space system of the microgrid is reduced using the balanced truncation method, resulting in the reduced-order dynamic linearized state-space system of the microgrid. An inverse translation transformation is then performed on the reduced-order dynamic linearized state-space system of the microgrid to obtain the reduced-order linearized model of the microgrid, thus completing the order reduction of the microgrid linearization model.
[0046] Example 4 A computer-readable storage medium stores a computer program that, when executed by a processor, implements a method for reducing the order of a microgrid linearization model: Based on models of distributed power source components, transmission network components, and load components in the microgrid, a system of dynamic nonlinear differential-algebraic equations for the microgrid is established; the stable equilibrium points of the system of dynamic nonlinear differential-algebraic equations are solved, and linearization is performed at these points to obtain a dynamically linearized system of differential-algebraic equations for the microgrid; the system of dynamic linearized differential-algebraic equations for the microgrid is simplified by elimination to obtain a dynamically linearized state-space system of the microgrid; the rightmost eigenvalue of the dynamically linearized state-space system of the microgrid is solved using an eigenvalue algorithm. If the rightmost eigenvalue is located to the right of the imaginary axis in the complex plane, then the dynamic linearization of the microgrid is reduced. The linearized state-space system is transformed by leftward translation, shifting the position of the rightmost eigenvalue to the left of the imaginary axis in the complex plane, resulting in the transformed dynamic linearized state-space system of the microgrid. Lyapunov equations are generated based on this transformed system, and the Krylov subspace algorithm is used to solve them, yielding the controllable Gram matrix and the observable Gram matrix. Based on these matrices, the order of the transformed dynamic linearized state-space system is reduced using the balanced truncation method, resulting in the reduced-order dynamic linearized state-space system. Finally, an inverse translation transformation is performed on this reduced-order system to obtain the linearized reduced-order model of the microgrid, thus completing the order reduction of the microgrid linearization model.
[0047] Example 5 See Figure 5 The method for reducing the order of a microgrid linearization model includes the following steps: 1) Obtain models of components such as distributed power sources, transmission networks, and loads in the microgrid, and establish a set of nonlinear differential-algebraic equations characterizing the dynamic characteristics of the microgrid; 2) Solve for the stable equilibrium points of the nonlinear differential-algebraic equation system, and perform linearization at the equilibrium points to obtain the linearized differential-algebraic equation system. 3) Use the eigenvalue algorithm to solve for the rightmost eigenvalue of the system. If the system is unstable, shift the system to the left so that the rightmost eigenvalue is located to the left of the imaginary axis in the complex plane. 4) Generate the Lyapunov equations based on the system after translation transformation, and solve the equations using the Krylov subspace algorithm to obtain the controllable and observable Gram matrix; 5) The system after translation transformation is reduced in order by using the balanced truncation method, and then the reduced system is subjected to inverse translation transformation to obtain the final reduced-order model.
[0048] In step 1), the specific form of the nonlinear differential-algebraic equation system characterizing the dynamic characteristics of the microgrid is as follows: (1) In the formula: For state variables, For algebraic variables, For input variables, For output variables; n , m , p and q These represent the number of state variables, algebraic variables, input variables, and output variables, respectively. n , m , p and q All are positive integers; t For time, State variables Regarding time t The first derivative, The state equation function, For functions of algebraic equations, This is the output equation function.
[0049] In step 2), the specific form of the system of differential-algebraic equations obtained by linearization at the equilibrium point is as follows: (2) In the formula: , , and The variables are linearized; It is the identity matrix; It is the matrix of partial derivatives of the state equation function with respect to the state variables. It is the matrix of partial derivatives of the state equation function with respect to algebraic variables. It is the matrix of partial derivatives of the state equation function with respect to the input variables. It is the matrix of partial derivatives of the function in the algebraic equation with respect to the state variables. It is the matrix of partial derivatives of the function in an algebraic equation with respect to the algebraic variables. It is the matrix of partial derivatives of the function in the algebraic equation with respect to the input variables. It is the matrix of partial derivatives of the output equation function with respect to the state variables. It is the matrix of partial derivatives of the output equation function with respect to the substituted variables; , , , , , , and The linearized matrix has the following specific form: (3) After eliminating the algebraic variables in equation (2), we can obtain: (4) In the formula: The state matrix, For the input matrix, For the output matrix, For direct transmission matrix The linearized state variables Regarding time t The first derivative; , and The linearized matrix has the following specific form: (5) From equation (4), the transfer function of the original system can be obtained as: (6) In step 3), the eigenvalue algorithm is used to solve the problem. rightmost eigenvalue If the eigenvalue If the eigenvalue is unstable, then the system is shifted to the left, where the shift distance is... Set to: (7) In the formula: The real part of the eigenvalue; This represents the translation distance.
[0050] The specific form of the system after translation transformation is as follows: (8) In the formula: This is the state matrix after translation; and These represent the state and output variables of the system after translation.
[0051] In step 4), the specific form of the Lyapunov equations generated from the translated system is as follows: (9) In the formula: and These are controllable and observable Gram matrices, respectively. It is the translated state matrix. The transpose of the matrix, It is the input matrix The transpose of the matrix, It is the output matrix The transpose of .
[0052] The Krylov subspace is then generated in the following form: (10) In the formula: and For Krylov subspace; l and k These are the dimensions of the two subspaces, which are much smaller than the number of state variables. n .
[0053] Projecting equation (9) using Krylov subspaces yields: (11) In the formula: ; ; ; ; and These are the orthogonal bases of the two subspaces, respectively; and The solution to equation (11); It is the projected state matrix on the controllable Krylov subspace. It is the solution matrix of the projected controllable Lyapunov equations. It is the input matrix after projection. It is the projected state matrix on the observable Krylov subspace. It is the solution matrix of the observable Lyapunov equations after projection. It is the output matrix after projection.
[0054] The expressions for the controllable and observable Gram matrices are: (12) In step 5), the specific process of reducing the order of the system after translation transformation using the balanced truncation method is as follows: first, ... and Perform Cholesky decomposition. (13) In the formula: For a controllable Gram matrix, The Cholesky factor of the controllable Gram matrix. For observable Gram matrix, Let be the Cholesky factor of the observable Gram matrix. Then, consider the product... Perform singular value decomposition: (14) In the formula: and It is a column orthogonal matrix; It is a left singular vector matrix. The columns of the right singular vector matrix; ; These are Hankel singular values. (Preserve the original value.) r There are n Hankel singular values such that: (15) In the formula: This is the tolerance for order reduction.
[0055] Then the projection matrix is constructed as follows: (16) In the formula: and They are respectively and The former r List; ; and These are the left and right projection matrices, respectively.
[0056] The reduced matrix is then obtained as follows: (17) In the formula: and These are the state and output variables of the system after order reduction, respectively; , , and The coefficient matrix after order reduction has the following specific form: (18) The final reduced-order model is obtained by performing an inverse translation transformation on the reduced-order system. (19) In the formula: and These are the state and output variables after the inverse translation transformation, respectively; .
[0057] The final transfer function of the reduced-order system is: (20) In the formula: It is the transfer function; its relationship with the original system transfer function The error between them satisfies: (twenty one) To verify the proposed method for reducing the order of a microgrid linearization model based on the improved balanced truncation method, this embodiment validates the effectiveness of the method on a two-region interconnected microgrid system containing four diesel generators. The microgrid system framework diagram is shown below. Figure 4 As shown, the flowchart of the invention is as follows: Figure 5 As shown. The corresponding model parameters are shown in Table 1, where... R 0, X L0 and X C0 These represent the resistance, inductive reactance, and capacitive reactance per unit length of the line, respectively. l ij Represents a node i and nodes j The length of the line between them; P Gi and Q Gi It is the first i The active and reactive power output of the diesel generator; P Li and Q Li It is the first i The active and reactive power of each load; Q Ci Indicates the first i Reactive power compensation capacity at each load node. All diesel generators are equipped with excitation and speed control systems. Area 1 is connected via tie lines. l 67 400kW of active power is delivered to Region 2.
[0058] Table 1 Microgrid Model Parameters
[0059] Next, models of the diesel generators, transmission network, loads, and parallel reactive power compensation capacitors in the microgrid are obtained, and a set of nonlinear differential-algebraic equations characterizing the dynamic characteristics of the microgrid is established. The generators are modeled as four-winding thyristor excitation systems with transient gain attenuation without PSS (Power Supply System), and the load model is a constant impedance model. Then, the stable equilibrium point of the system is determined through power flow calculations, resulting in a linearized set of differential-algebraic equations. There are 72 state variables and 64 algebraic variables. The rightmost eigenvalue of the system is then solved using the eigenvalue algorithm, which is 0.048 ± j3.930. This eigenvalue characterizes the oscillation between the synchronous generators in the two regions. To suppress this oscillation mode, a controller is added to the excitation system of diesel generator No. 1, with its input signal being the speed deviation between diesel generators No. 1 and No. 3. Conversely, for the microgrid system without the added controller, its input signal comes from the excitation system, and its output signal is the speed deviation between diesel generators No. 1 and No. 3.
[0060] Note that the system is unstable, and according to equation (7), it needs to be shifted to the left by 0.058. Then, the Lyapunov equation is generated based on the translated system, and the controllable and observable Gram matrices are obtained by solving. Next, the equilibrium truncation method is used to reduce the order of the translated system, and the inverse translation transformation is performed on the reduced system to obtain the final reduced-order model. The order reduction tolerance is set to 10. -5 The reduced system contains only 11 state variables. The eigenvalue distributions of the original system and the reduced system are as follows: Figure 6 As shown in the figure. The original unstable eigenvalues of the system are preserved, and the reduced-order system does not add any additional unstable eigenvalues. Furthermore, the frequency response curves of the original system and the reduced-order system are shown in the figure. Figure 7 As shown in the figure, the error between the frequency response of the reduced-order system and the original system is less than 1% of its amplitude. This invention achieves high order reduction accuracy.
[0061] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.
[0062] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.
[0063] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.
[0064] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.
[0065] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the specific implementation of the present invention. Any modifications or equivalent substitutions that do not depart from the spirit and scope of the present invention should be covered within the protection scope of the claims of the present invention.
Claims
1. A microgrid linearization model order reduction method characterized by, The method comprises the following steps: Based on the distributed power supply element model, the power transmission network element model and the load element model in the micro-grid, a dynamic nonlinear differential algebraic equation set of the micro-grid is established; A stable equilibrium point of the dynamic nonlinear differential algebraic equation set of the micro-grid is solved, and linearization processing is performed at the stable equilibrium point to obtain a dynamic linearized differential algebraic equation set of the micro-grid; the dynamic linearized differential algebraic equation set of the micro-grid is simplified by elimination to obtain a dynamic linearized state space system of the micro-grid; An eigenvalue algorithm is used to solve the rightmost eigenvalue of the dynamic linearized state space system of the micro-grid, and if the rightmost eigenvalue is located on the right side of the imaginary axis on a complex plane, a leftward translation transformation is performed on the dynamic linearized state space system of the micro-grid to move the position of the rightmost eigenvalue to the left side of the imaginary axis on the complex plane to obtain a dynamic linearized state space system of the micro-grid after the translation transformation; A Lyapunov equation is generated according to the dynamic linearized state space system of the micro-grid after the translation transformation, and a Krylov subspace algorithm is used to solve the Lyapunov equation to obtain a controllable Gram matrix and an observable Gram matrix; Based on the controllable Gram matrix and the observable Gram matrix, a balanced truncation method is used to reduce the order of the dynamic linearized state space system of the micro-grid after the translation transformation to obtain a reduced-order dynamic linearized state space system of the micro-grid, and an inverse translation transformation is performed on the reduced-order dynamic linearized state space system of the micro-grid to obtain a linearized reduced-order model of the micro-grid, thereby completing the order reduction of the linearized model of the micro-grid.
2. The microgrid linearization model reduction method of claim 1, wherein, The dynamic nonlinear differential algebraic equation set of the micro-grid is shown in the following formula: in, For state variables, For algebraic variables, For input variables, For output variables; n , m , p and q These represent the number of state variables, algebraic variables, input variables, and output variables, respectively. n , m , p and q All are positive integers; t For time, State variables Regarding time t The first derivative; The state equation function, For functions of algebraic equations, This is the output equation function.
3. The microgrid linearization model reduction method of claim 1, wherein, The dynamic linearized differential algebraic equation set of the micro-grid is shown in the following formula: wherein is a linearized state variable, is a linearized algebraic variable, is a linearized input variable, is a linearized output variable; is an identity matrix; is a matrix of partial derivatives of the state equation function with respect to the state variable, is a matrix of partial derivatives of the state equation function with respect to the algebraic variable, is a matrix of partial derivatives of the state equation function with respect to the input variable, is a matrix of partial derivatives of the algebraic equation function with respect to the state variable, is a matrix of partial derivatives of the algebraic equation function with respect to the algebraic variable, is a matrix of partial derivatives of the algebraic equation function with respect to the input variable, is a matrix of partial derivatives of the output equation function with respect to the state variable, is a matrix of partial derivatives of the output equation function with respect to the algebraic variable; The dynamic linearized state space system of the micro-grid obtained by simplifying the dynamic linearized differential algebraic equation set of the micro-grid by elimination is shown in the following formula: wherein, is a state matrix, is an input matrix, is an output matrix, is a direct transmission matrix, is a linearized state variable first derivative with respect to time t t. The transfer function of the dynamic linearized state space system of the micro-grid is shown in the following formula: wherein is the transfer function of the microgrid dynamic linearized state-space system.
4. The microgrid linearization model reduction method of claim 1, wherein, The translation distance of the leftward translation transformation is shown in the following formula: wherein is the translation distance, is the real part of the eigenvalue, is the rightmost eigenvalue; The dynamic linearized state space system of the micro-grid after the translation transformation is shown in the following formula: wherein, is the state matrix after translation; is the state variable of the microgrid dynamic linearized state space system after translation transformation, is the output variable of the microgrid dynamic linearized state space system after translation transformation.
5. The microgrid linearization model reduction method of claim 1, wherein, The Lyapunov equation generated according to the dynamic linearized state space system of the micro-grid after the translation transformation is shown in the following formula: wherein is a controllable Gram matrix, is an observable Gram matrix, is the translated state matrix is the transpose matrix of is the transpose matrix of is the transpose matrix of is the transpose matrix of is the transpose matrix of The process of solving the Lyapunov equation by using the Krylov subspace algorithm to obtain the controllable Gram matrix and the observable Gram matrix is shown in the following formula: where and are controllable and observable Krylov subspaces, respectively; l and k are the dimensions of the two subspaces, respectively; and are orthonormal bases of the two subspaces, respectively; and are the solutions of the Lyapunov equations; is the controllability Gramian, is the observability Gramian; is the projected state matrix on the controllable Krylov subspace, is the projected state matrix on the observable Krylov subspace; is the solution matrix of the projected controllable Lyapunov equation, is the solution matrix of the projected observable Lyapunov equation; is the projected input matrix, is the projected output matrix.
6. The microgrid linearization model reduction method of claim 1, wherein, The reduced-order dynamic linearized state space system of the micro-grid is shown in the following formula: wherein, is a state variable of the reduced system, is an output variable of the reduced system; , , and is a coefficient matrix of the reduced system; The linearized reduced-order model of the micro-grid is shown in the following formula: wherein is the state variable after inverse translation, is the output variable after inverse translation; The transfer function of the reduced-order dynamic linearized state space system of the micro-grid is shown in the following formula: wherein, is the transfer function of the reduced-order microgrid dynamic linearized state-space system; The error between the transfer function of the reduced-order dynamic linearized state space system of the micro-grid and the transfer function of the dynamic linearized state space system of the micro-grid satisfies 。 7. The microgrid linearization model reduction method of claim 1, wherein, The process of reducing the order of the dynamic linearized state space system of the micro-grid after the translation transformation by using the balanced truncation method based on the controllable Gram matrix and the observable Gram matrix to obtain the reduced-order dynamic linearized state space system of the micro-grid comprises the following steps: The Cholesky decomposition is performed on the controllable Gram matrix and the observable Gram matrix, and the matrix operation object is converted, as shown in the following formula: wherein, is a controllable Gram matrix, is a Cholesky factor of the controllable Gram matrix, is an observable Gram matrix, is a Cholesky factor of the observable Gram matrix; The singular value decomposition is performed on the product of the transpose of the Cholesky factor of the controllable Gram matrix and the Cholesky factor of the observable Gram matrix, and the Hankel singular value is extracted, as shown in the following formula: wherein is a left-singular vector matrix, is a right-singular vector matrix array; ; is a Hankel singular value; Before reservation r Hankel singular values, as follows: wherein is the reduced order admissible error; Based on the Hankel singular values of the previous r The left and right projection matrices are constructed as follows: wherein, is the preceding r column and is the preceding r column; ; is a left projection matrix, is a right projection matrix; The micro-grid dynamic linearization state space system after the translation transformation is projected from high dimension to low dimension by using the left and right projection matrices, to obtain the reduced micro-grid dynamic linearization state space system.
8. A reduced order system of a microgrid linearized model characterized by, The micro-grid nonlinear equation set module is configured to establish a micro-grid dynamic nonlinear differential algebraic equation set based on a distributed power supply element model, a power transmission network element model and a load element model in the micro-grid. The elimination and simplification module is configured to solve a stable equilibrium point of the micro-grid dynamic nonlinear differential algebraic equation set, linearize the stable equilibrium point, obtain a micro-grid dynamic linearization differential algebraic equation set, and eliminate and simplify the micro-grid dynamic linearization differential algebraic equation set to obtain a micro-grid dynamic linearization state space system. The translation transformation module is configured to solve a rightmost eigenvalue of the micro-grid dynamic linearization state space system by using an eigenvalue algorithm, and perform a left translation transformation on the micro-grid dynamic linearization state space system if the rightmost eigenvalue is located on the right side of the imaginary axis in a complex plane, so as to move the position of the rightmost eigenvalue to the left side of the imaginary axis in the complex plane, and obtain a micro-grid dynamic linearization state space system after the translation transformation. The controllable and observable Gram matrix module is configured to generate a Lyapunov equation based on the micro-grid dynamic linearization state space system after the translation transformation, and solve the Lyapunov equation by using a Krylov subspace algorithm to obtain a controllable Gram matrix and an observable Gram matrix. The micro-grid linearization model reduction module is configured to reduce the micro-grid dynamic linearization state space system after the translation transformation by using a balanced truncation method based on the controllable Gram matrix and the observable Gram matrix, obtain a reduced micro-grid dynamic linearization state space system, perform an inverse translation transformation on the reduced micro-grid dynamic linearization state space system to obtain a micro-grid linearization reduced model, and complete the micro-grid linearization model reduction.
9. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the micro-grid linearization model reduction method in any one of claims 1-7 when executing the computer program.
10. A computer readable storage medium storing a computer program, wherein the computer program implements the micro-grid linearization model reduction method in any one of claims 1-7 when executed by a processor.