Vectorized Jacobian Matrix Construction Method and System Based on Node Voltage Equation

Abstract

The present invention discloses a method and system for constructing a vectorized Jacobian matrix based on a node voltage equation, and relates to the field of power systems. Currently, the construction of the Jacobian matrix requires a large amount of calculation and has poor applicability. The present invention comprises the steps of: establishing a corresponding system model according to the target power system structure, establishing a system admittance matrix and a node voltage equation in combination with system parameters; then, mathematically deducing based on the system node voltage equation to obtain the system flow Jacobian matrix structure, and realizing vectorization during the construction process; finally, calculating the system flow by the Newton-Raphson method, substituting the calculated results, and obtaining the numerical system Jacobian matrix under the corresponding scenario. This technical solution proposes a method for constructing a flow Jacobian matrix that realizes process vectorization through mathematical derivation based on the node voltage equation, thereby providing a new practical method for constructing the flow Jacobian matrix.

Description

Technical Field

[0001] The present invention relates to the field of power systems, and in particular to a method and system for constructing a vectorized Jacobian matrix based on a node voltage equation. Background Art

[0002] The stability, reliability, and economy of power systems are core goals of modern power engineering. Their analysis and assurance rely heavily on the precise modeling and rapid calculation of system operating states. In this process, the power flow Jacobian matrix, a fundamental mathematical tool for power flow calculations, static stability analysis, and solving nonlinear equations, directly impacts the accuracy of system state estimation, the efficiency of optimized scheduling, and the real-time nature of instability risk warnings. Especially in the context of high renewable energy penetration and increasingly complex grid structures, the efficient construction and in-depth analysis of Jacobian matrices have become a key technical bottleneck for achieving the safe and stable operation of large power grids.

[0003] At present, the construction of Jacobian matrix mostly relies on differential transformation method, which calculates the partial derivative of each component with respect to the independent variable one by one and combines them to form the power flow Jacobian matrix. This method is simple and intuitive, but the amount of calculation is large, especially when facing large systems, the calculation efficiency is low. With the development of artificial intelligence technology, a data-driven Jacobian matrix construction method has been proposed. This method does not require line information and is not affected by changes in network topology, but the applicability of this method in dealing with different models is still poor. At the same time,

[0004] The above methods all focus on the numerical generation of the Jacobian matrix, while ignoring the correlation between its intrinsic structural characteristics and physical meaning. For example, the Jacobian matrix constructed by traditional methods is stored in the form of discrete elements, which makes it difficult to directly extract the coupling law between active / reactive sensitivity, and cannot efficiently support deep applications such as matrix eigenvalue analysis and weak node identification, resulting in a waste of resources that are "built but not used". In addition, the existing technology lacks vectorization processing of the matrix construction process, which restricts the application of parallel computing and hardware acceleration technology, making it difficult to meet the real-time requirements of ultra-high voltage AC / DC hybrid power grids.

[0005] To address the above problems, there is an urgent need for a Jacobian matrix construction method that is efficient, scalable, and structurally analyzable. Summary of the Invention

[0006] The technical problem to be solved and the technical task to be addressed by this invention are to improve and enhance existing technical solutions by providing a method and system for constructing a vectorized Jacobian matrix based on node voltage equations, thereby improving computational efficiency and facilitating subsequent analysis through structural features. To this end, this invention adopts the following technical solution.

[0007] A method for constructing a vectorized Jacobian matrix based on a node voltage equation includes the following steps:

[0008] 1) Constructing the target power system model, including eliminating passive interconnection nodes in the system using the Kron order reduction method based on the target power system structure, and using the Thevenin equivalent method to equate the external power grid to an infinite power grid to generate a simplified equivalent circuit model;

[0009] 2) Based on the simplified equivalent circuit model, the system admittance matrix is established, and the node voltage equation is formed based on the product relationship between the node current vector, the admittance matrix, and the node voltage vector;

[0010] 3) Converting the node voltage equation into a node injection complex power equation, and replacing the product of the voltage amplitude and phase angle with the matrix multiplication of the diagonal matrix and the conjugate vector through vectorized matrix operations to obtain a vectorized expression of the complex power;

[0011] 4) Converting the complex power equation into a real power flow equation in a polar coordinate system, calculating partial derivatives of the voltage amplitude vector and the phase angle vector respectively, and constructing a block structure of the Jacobian matrix. The block structure includes the following four sub-matrices: a partial derivative matrix of active power with respect to phase angle, a partial derivative matrix of active power with respect to voltage amplitude, a partial derivative matrix of reactive power with respect to phase angle, and a partial derivative matrix of reactive power with respect to voltage amplitude;

[0012] 5) The Newton-Raphson method is used to iteratively solve the system power flow, and the converged voltage amplitude and phase angle are substituted into the block structure expression of the Jacobian matrix to generate a numerical Jacobian matrix.

[0013] Starting from the system node voltage equations, this technical solution achieves vectorized reconstruction of the matrix construction process through mathematical derivation, overcoming the efficiency bottleneck of traditional methods while preserving the physical characteristics of the matrix, providing direct support for subsequent stability analysis and optimal control. Through model order reduction, vectorized operations, and block matrix construction, the computational efficiency and structural analyzability of the Jacobian matrix are significantly improved while maintaining accuracy. Combining engineering practicality with theoretical rigor, this method provides an efficient and reliable foundational tool for real-time analysis, optimal control, and AI-integrated power system applications. Specifically, by replacing the element-by-element product of voltage amplitude and phase angle with matrix multiplication of a diagonal matrix and a conjugate vector, the redundant element-by-element calculations in traditional methods are avoided. Vectorized processing fully exploits the parallelism of matrix operations, effectively improving computational speed, especially in large-scale power systems. Kron order reduction eliminates passive tie nodes and the Thevenin equivalent external grid, significantly simplifying the system model, reducing the admittance matrix dimensionality, avoiding redundant calculations, and preserving the electrical characteristics of key nodes. This approach is suitable for a variety of scenarios, such as power flow calculations and short-circuit analysis. Vectorized matrix operations replace traditional element-by-element partial derivative calculations, leveraging the parallel nature of matrices to reduce computational complexity. Block-based Jacobian matrix construction (the submatrix of the partial derivatives of active / reactive power with respect to voltage amplitude and phase angle) preserves the matrix's physical coupling characteristics, facilitating direct extraction of sensitivity coefficients or analysis of matrix eigenvalues, providing structured support for stability assessment. The Jacobian matrix is generated through a block structure, decomposing the matrix into four submatrices, reducing the dimensionality of each matrix, memory usage, and computational complexity. The mathematical derivation of the node voltage equation ensures that the matrix elements strictly correspond to the grid topology and parameters, avoiding model inaccuracy caused by training data bias in data-driven methods and truncation errors introduced by traditional numerical difference methods, thereby improving computational accuracy. The Newton-Raphson iterative solution, combined with the vectorized Jacobian matrix, leverages the symmetry of the matrix block structure to accelerate iterative convergence, avoiding oscillation or non-convergence caused by matrix ill-conditioning in traditional methods. By constructing the Jacobian matrix from the real-valued power flow equation in polar coordinates, the accumulated errors of complex operations are avoided, resulting in numerical stability superior to methods that directly process complex equations. This avoids the manual derivation of partial derivatives required in traditional differential transformation methods, reducing code implementation complexity and the risk of human error. Furthermore, the block matrix structure facilitates subsequent functional expansion, such as evaluating voltage stability margins through the minimum singular value of the Jacobian matrix; extracting the sensitivity coefficients of active and reactive power to voltage to guide generator voltage regulation or capacitor switching; and using the Jacobian matrix eigenvalues as input to deep learning models to predict system stability boundaries.

[0014] As a preferred technical means: the system admittance matrix in step 2) is constructed as follows: Assuming that the system has n nodes, the admittance matrix Y The dimension is n×n, construct the system admittance matrix Yas follows:

[0015] (1)

[0016] Where, For nodes i With node j The admittance value between When , it represents the mutual admittance between nodes; When , it indicates the self-admittance of the node;

[0017] Then the node voltage equation of the system is obtained as:

[0018] (2)

[0019] Where, is the node current vector, is the node voltage vector.

[0020] This technical solution achieves precise mathematical representation of grid topology and parameters through the systematic construction of an admittance matrix. It combines physical intuitiveness, computational efficiency, and engineering practicality, providing a reliable basic model for subsequent power flow calculations, stability analysis, and optimization control. Its advantages are particularly significant in large-scale and complex power grid scenarios. Specifically, by constructing an admittance matrix Y with a dimension of n×n, the connection relationship of all nodes in the system is fully covered, ensuring a comprehensive mapping of the grid topology and parameters and avoiding the omission of key node or branch information. The admittance matrix Y serves as the basic input for power flow calculations, short-circuit analysis, and stability assessment, ensuring the mathematical consistency of the entire analysis process and reducing intermediate conversion errors.

[0021] As a preferred technical means: the complex power equation in step 3) is constructed as follows: multiply the two ends of the node voltage equation by the diagonal matrix , and the complex power equation of each node is obtained:

[0022] (3)

[0023] In the formula, the symbol It represents conjugate transpose for matrices and conjugate for vectors; S Inject complex power column vector into the node.

[0024] Using diagonal matrix multiplication , replacing element-by-element scalar operations, reduces the complexity of complex power calculations and is particularly suitable for rapid modeling of large-scale power grids. The matrix form of the complex power equation naturally supports parallel computing and can fully utilize GPU or distributed computing resources to accelerate the solution. Directly generating the complex power equation through matrix operations avoids the tedious derivation of the node power equation manually expanded in traditional methods and reduces the risk of human error. Complex conjugate operation and The data is processed uniformly in matrix form to avoid rounding errors in element-by-element operations.

[0025] As a preferred technical means: the block matrix construction in step 4) specifically includes:

[0026] Convert the complex node voltage vector into polar coordinates to get the real Jacobian matrix of the power flow equation. Let:

[0027] (4)

[0028] Where, U 、 θ yes step voltage magnitude and phase angle vector, represents the exponential operation of the phase angle vector θ element by element, where j is the imaginary unit; The vector product of represents the Hadamard product between vectors;

[0029] The partial derivatives of the voltage amplitude vector and phase angle vector are obtained for the complex power equation.

[0030] (5)

[0031] Where the derivatives of a complex voltage with respect to its phase and amplitude are defined as follows:

[0032] (6)

[0033] Where: j is the imaginary unit;

[0034] Substituting equations (2) and (6) into equation (5), we can obtain:

[0035] (7)

[0036] In which, let the coupling matrix ;

[0037] According to formula (7), we can get the real number Jacobian matrix J :

[0038] (8)

[0039] In the formula, Re is the real part of each number, and Im is the imaginary part of each number;

[0040] Restoring power S = P +j Q , then the Jacobian matrix J Re-expressed as:

[0041] (9)

[0042] Where, P Inject active power column vector into the node, Q Inject reactive power column vector into the node,

[0043] Thus, the power flow Jacobian matrix for realizing process vectorization is constructed.

[0044] The block matrix construction separates the sensitivity of active power to phase angle and reactive power to voltage amplitude, intuitively reflecting the power-voltage coupling characteristics and providing a direct basis for stability analysis. Through Hadamard product and diagonal matrix operations, the partial derivative calculation is upgraded from element-by-element scalar operation to matrix-level vector operation, which improves the computational efficiency several times and supports sparse matrix optimization. The coupling matrix Γ is introduced to convert the network parameters ( Y S ) and running status ( U , θ ) decoupling, avoiding repeated calculations of the admittance matrix and reducing memory usage. The real Jacobian matrix in polar coordinates avoids numerical oscillation issues in complex operations and improves the convergence of the Newton-Raphson method, making it particularly suitable for scenarios with high impedance ratios or heavily loaded grids.

[0045] As a preferred technical means: after generating the numerical Jacobian matrix, further include any of the following:

[0046] 6) Performing a static stability analysis of the power system based on the numerical Jacobian matrix, including: calculating the minimum singular value or eigenvalue of the Jacobian matrix and evaluating the voltage stability margin of the current operating section; locating the system instability mode based on the change in the sign of the eigenvalue and identifying weak nodes or critical transmission lines;

[0047] 6) Constructing a sensitivity matrix based on the numerical Jacobian matrix for optimizing scheduling decisions, including: extracting the sensitivity coefficients of node active / reactive power to voltage amplitude and phase in the Jacobian matrix; generating a preventive control strategy based on the optimization objective function to adjust generator output or reactive compensation equipment;

[0048] 6) Inputting the numerical Jacobian matrix into a deep learning model to train the mapping relationship between the grid operation state and the stability boundary, including: using the Jacobian matrix eigenvalues, node voltage and power data as input features; and outputting the load growth limit or fault tolerance capacity prediction result under the critical stability state of the system.

[0049] This technical solution can upgrade the Jacobian matrix from a basic mathematical tool to a core engine for intelligent analysis, thereby improving the grid's proactive defense capabilities, economic operation level, and digital transformation speed, becoming a key technical support for building a new power system.

[0050] Another object of the present invention is to provide a vectorized Jacobian matrix construction system based on node voltage equations, the system comprising:

[0051] The target power system model construction module is used to eliminate the passive connection nodes in the system through the Kron order reduction method according to the target power system structure, and use the Thevenin equivalent method to equate the external power grid to an infinite power grid to generate a simplified equivalent circuit model;

[0052] an admittance matrix generation module, connected to the target power system model construction module, for constructing a system admittance matrix based on the simplified equivalent circuit model, and forming a node voltage equation based on the product relationship between the node current vector, the admittance matrix, and the node voltage vector;

[0053] a complex power conversion module, connected to the admittance matrix generation module, for converting the node voltage equation into a node injection complex power equation, and replacing the product term of the voltage amplitude and the phase angle with the matrix multiplication of the diagonal matrix and the conjugate vector through vectorized matrix operation to obtain a vectorized expression of the complex power;

[0054] A Jacobian matrix construction module is connected to the complex power conversion module and is used to convert the complex power equation into a real power flow equation in a polar coordinate system, obtain partial derivatives of the voltage amplitude vector and the phase angle vector respectively, and construct a block structure of the Jacobian matrix, wherein the block structure includes the following four sub-matrices: a partial derivative matrix of active power with respect to phase angle, a partial derivative matrix of active power with respect to voltage amplitude, a partial derivative matrix of reactive power with respect to phase angle, and a partial derivative matrix of reactive power with respect to voltage amplitude;

[0055] The power flow solving module is connected to the Jacobian matrix construction module and is used to iteratively solve the system power flow using the Newton-Raphson method, substitute the converged voltage amplitude and phase angle into the block structure expression of the Jacobian matrix, and generate a numerical Jacobian matrix.

[0056] Through modular architecture, vectorized computing and physical model drive, it achieves high efficiency, high precision and strong scalability in Jacobian matrix construction, while providing full-scenario coverage from basic analysis to intelligent decision-making. It is a core technology platform that supports real-time monitoring, optimized operation and security defense of new power systems.

[0057] As a preferred technical means: the admittance matrix generation module is specifically configured as follows: assuming that the system has n nodes and the dimension of the admittance matrix Y is n×n, the system admittance matrix Y is constructed as follows:

[0058] (1)

[0059] Where, For nodes iWith node j The admittance value between When , it represents the mutual admittance between nodes; When , it indicates the self-admittance of the node;

[0060] The admittance matrix generation module further generates the node voltage equation:

[0061] (2)

[0062] Where, is the node current vector, is the node voltage vector.

[0063] The admittance matrix generation module achieves efficient and high-precision representation of power grid topology through systematic modeling and sparse optimization. It combines engineering practicality and scalability, providing a reliable foundation for power flow calculation, stability analysis and intelligent decision-making, and is a core technical component of the digitalization of power systems.

[0064] As an optimal technical means: the complex power conversion module is specifically configured as follows: multiplying the two ends of the node voltage equation by the diagonal matrix , and the complex power equation of each node is obtained:

[0065] (3)

[0066] In the formula, the symbol It represents the conjugate transpose for matrices and the conjugate for vectors; S is the column vector of the complex power injected into the node.

[0067] As an optimal technical means: the Jacobian matrix construction module includes the following submodules:

[0068] The polar coordinate conversion submodule is used to convert the complex node voltage vector into a polar coordinate system representation. Let:

[0069] (4)

[0070] Where, U 、 θ yes step voltage magnitude and phase angle vector, represents the exponential operation of the phase angle vector θ element by element, where j is the imaginary unit; The vector product of represents the Hadamard product between vectors;

[0071] The partial derivative calculation submodule is used to obtain the partial derivatives of the voltage amplitude vector and phase angle vector for the complex power equation:

[0072] (5)

[0073] Where the derivative of a complex voltage with respect to its phase and amplitude is defined as:

[0074] (6)

[0075] Where: j is the imaginary unit;

[0076] The Jacobian matrix generation submodule is used to substitute equations (2) and (6) into equation (5) to obtain:

[0077] (7)

[0078] Among them, let the symbol ;

[0079] According to formula (7), we can get the real number Jacobian matrix J :

[0080] (8)

[0081] In the formula, Re is the real part of each number, and Im is the imaginary part of each number;

[0082] Restoring power S = P +j Q , then the Jacobian matrix J Re-expressed as:

[0083] (9)

[0084] Where, P Inject active power column vector into the node, Q Inject reactive power column vector into the node,

[0085] Thus, the power flow Jacobian matrix for realizing process vectorization is constructed.

[0086] This technical solution, through vectorized complex power conversion and block Jacobian matrix construction, achieves a step-change improvement in computational efficiency and enhanced numerical stability while maintaining physical interpretability. It provides an efficient and reliable foundational tool and represents a core technological breakthrough supporting the safe and economic operation of new power systems.

[0087] As a preferred technical means: also include any of the following functional modules:

[0088] A static stability analysis module is used to calculate the minimum singular value or eigenvalue of the numerical Jacobian matrix based on the matrix, evaluate the voltage stability margin of the current operating section, locate the system instability mode according to the change of the eigenvalue sign, and identify weak nodes or key transmission lines;

[0089] An optimization scheduling decision module is used to extract the sensitivity coefficient of node active / reactive power to voltage amplitude and phase based on the numerical Jacobian matrix, generate a preventive control strategy based on the optimization objective function, and adjust the generator output or reactive compensation equipment;

[0090] A deep learning prediction module is used to input the numerical Jacobian matrix into a deep learning model, train the mapping relationship between the power grid operation state and the stability boundary, use the Jacobian matrix eigenvalues, node voltage and power data as input features, and output the load growth limit or fault tolerance capacity prediction result under the critical stability state of the system.

[0091] This technical solution can upgrade the Jacobian matrix from a basic mathematical tool to the core of intelligent power grid analysis, effectively improving the stability, economy and digitalization level of the system, and is a key technical pillar for building a new power system.

[0092] Beneficial effects:

[0093] This technical solution simplifies the power grid model through the Kron reduction method and the Thevenin equivalent, significantly reducing the computational complexity; uses vectorized matrix operations instead of element-by-element operations to improve computational efficiency; constructs the Jacobian matrix in blocks through the polar coordinate system to clearly separate the sensitivity of active / reactive power to voltage amplitude and phase angle. It has clear physical meaning and supports parallel computing, has strong numerical stability, and combines the Newton-Raphson method to iteratively accelerate convergence. At the same time, it provides a high-precision and highly compatible mathematical foundation for static stability assessment, optimized scheduling, and AI prediction, significantly improving the real-time, economical, and intelligent level of power grid analysis.

[0094] This technical solution starts from the system node voltage equation and realizes vectorized reconstruction of the matrix construction process through mathematical derivation, breaking through the efficiency bottleneck of traditional methods while retaining the physical characteristic information of the matrix, providing direct support for subsequent stability analysis and optimization control. BRIEF DESCRIPTION OF THE DRAWINGS

[0095] Figure 1 It is a partial schematic diagram of the AC power grid.

[0096] Figure 2 It is the power grid equivalent circuit diagram of the present invention.

[0097] Figure 3 It is a flow chart of the present invention. DETAILED DESCRIPTION

[0098] The technical solution of the present invention is further described in detail below with reference to the accompanying drawings.

[0099] Example 1:

[0100] like Figure 3 As shown, this embodiment includes the following steps:

[0101] 1. Build the target power system model

[0102] by Figure 1 Take the partial diagram of AC power grid as an example to explain. Figure 1 It includes local AC network, new energy stations and synchronous generators, among which the AC network contains loads, passive connection nodes, etc. When analyzing the system power flow or the system static voltage stability, it is necessary to construct the system Jacobian matrix. In order to reduce the complexity of the calculation, the system structure is simplified first. The Kron reduction method can be used to eliminate the passive connection nodes in the system, and the Thevenin equivalent is used to equate the external power grid to an infinite power grid. The equivalent circuit of the simplified system is as follows Figure 2 shown.

[0103] Based on the above system equivalent circuit model structure and parameters, assuming that the system has n nodes, construct the system admittance matrix Y as follows:

[0104] (1)

[0105] Where, is the admittance value between corresponding nodes, i, j is the node number, When , it represents the mutual admittance between nodes; When , it indicates the self-admittance of the node.

[0106] Then the system node voltage equation is obtained as:

[0107] (2)

[0108] Where, I S The current of each node in the system n ×1-order matrix, U S The voltage of each node in the system n ×1-order matrix.

[0109] 2. Vectorized Jacobian Matrix Construction

[0110] Next, we construct the power flow equation of the local power grid based on the equivalent circuit. Multiply both ends of the equivalent circuit equation (2) by , the complex power of each node can be obtained, as shown in formula (3).

[0111] (3)

[0112] In the formula, the symbol For matrices, it represents conjugate transpose, for vectors, it represents conjugate, and the symbol diag represents taking a diagonal matrix,S A column vector consisting of complex power values injected into each node.

[0113] By changing the rectangular coordinate system in equation (3) to the polar coordinate system, we can obtain the traditional real-number Jacobian matrix of the power flow equation, and let:

[0114] (4)

[0115] Where, U 、 θ yes step voltage magnitude and phase angle vector, The vector product of represents the Hadamard product between vectors, and the Hadamard product between vectors represents the item-by-item product of corresponding elements.

[0116] Let (3) be respectively and Vector partial derivative,

[0117] (5)

[0118] Where the derivatives of a complex voltage with respect to its phase and amplitude are defined as follows (j is an imaginary unit):

[0119] (6)

[0120] Substituting equations (2) and (6) into equation (5), we can obtain:

[0121] (7)

[0122] Among them, let the symbol .

[0123] According to formula (7), we can get the real number Jacobian matrix J :

[0124] (8)

[0125] In the formula, Re is the real part of each number, and Im is the imaginary part of each number.

[0126] From Equations (7) and (8), we can see that the four block matrices in the Jacobian matrix are correlated. To further analyze the structural characteristics of the Jacobian matrix, let the complex power S = P +j Q , then the Jacobian matrix J can be re-expressed as:

[0127] (9)

[0128] In the formula PInject active power column vector into the node, Q Inject reactive power column vector into the node,

[0129] Thus, the power flow Jacobian matrix construction for realizing process vectorization is obtained.

[0130] Finally, the Newton-Raphson method is used to solve the system power flow, and the parameter results are substituted into formula (9) to obtain the corresponding Jacobian matrix.

[0131] Specific implementation examples

[0132] The present invention uses a three-machine system for illustration. The system nodes are numbered, with the three generator nodes designated as nodes 1, 2, and 3, and the remaining nodes numbered sequentially. The impedance parameters of each line in the system are shown in Table 1.

[0133] Table 1 Impedance parameters of each line

[0134]

[0135] Based on the line parameter data, the system Jacobian matrix is constructed according to the method proposed in this invention as follows:

[0136] (10)

[0137] To verify the effectiveness of the method proposed in this invention, the Jacobian matrix is calculated by comparing it with the traditional method. The traditional method is as follows:

[0138] (11)

[0139] Where, H 、 N 、 M 、 L are the sub-matrix blocks that constitute the Jacobian matrix, represents the Jacobian matrix constructed using traditional methods.

[0140] 1) When When , the calculation method of each element is as follows:

[0141]

[0142]

[0143]

[0144] (12)

[0145] Where, P i is the active power of each node, Q iReactive power of each node, represents the conductance value of the corresponding circuit, represents the susceptance value of the corresponding line, For nodes i The voltage amplitude, For nodes j The voltage amplitude, For nodes i, j The phase angle difference between them.

[0146] 2) When , the calculation method of each element is as follows:

[0147]

[0148]

[0149]

[0150] (13)

[0151] Where Σ is the summation symbol.

[0152] Based on the same line model as above, the Jacobian matrix obtained using the traditional method is:

[0153] (14)

[0154] By comparing the matrix obtained by the method of the present invention with that obtained by the traditional method, it can be seen that the Jacobian matrix obtained by the method of the present invention is the same as that obtained by the traditional method, which proves the accuracy of the method proposed by the present invention. However, the method proposed by the present invention realizes the vectorization of the construction process, which provides a basis for further deconstructing the Jacobian matrix or further transformation.

[0155] Example 2:

[0156] A vectorized Jacobian matrix construction system based on node voltage equations includes:

[0157] The target power system model construction module is used to eliminate the passive connection nodes in the system through the Kron order reduction method according to the target power system structure, and use the Thevenin equivalent method to equate the external power grid to an infinite power grid to generate a simplified equivalent circuit model;

[0158] an admittance matrix generation module, connected to the target power system model construction module, for constructing a system admittance matrix based on the simplified equivalent circuit model, and forming a node voltage equation based on the product relationship between the node current vector, the admittance matrix, and the node voltage vector;

[0159] a complex power conversion module, connected to the admittance matrix generation module, for converting the node voltage equation into a node injection complex power equation, and replacing the product term of the voltage amplitude and the phase angle with the matrix multiplication of the diagonal matrix and the conjugate vector through vectorized matrix operation to obtain a vectorized expression of the complex power;

[0160] A Jacobian matrix construction module is connected to the complex power conversion module and is used to convert the complex power equation into a real power flow equation in a polar coordinate system, obtain partial derivatives of the voltage amplitude vector and the phase angle vector respectively, and construct a block structure of the Jacobian matrix, wherein the block structure includes the following four sub-matrices: a partial derivative matrix of active power with respect to phase angle, a partial derivative matrix of active power with respect to voltage amplitude, a partial derivative matrix of reactive power with respect to phase angle, and a partial derivative matrix of reactive power with respect to voltage amplitude;

[0161] The power flow solving module is connected to the Jacobian matrix construction module and is used to iteratively solve the system power flow using the Newton-Raphson method, substitute the converged voltage amplitude and phase angle into the block structure expression of the Jacobian matrix, and generate a numerical Jacobian matrix.

[0162] This embodiment achieves high efficiency, high precision, and strong scalability in Jacobian matrix construction through modular architecture, vectorized computing, and physical model drive. It also provides full-scenario coverage from basic analysis to intelligent decision-making, and is a core technology platform supporting real-time monitoring, optimized operation, and security defense of new power systems.

[0163] It is understandable that the detailed functional implementation of the above modules can be found in the introduction of the aforementioned method embodiment, and no further details are given here.

[0164] The above-mentioned method and system for constructing a vectorized Jacobian matrix based on the node voltage equation is a specific embodiment of the present invention, which has reflected the essential characteristics and progress of the present invention. According to actual use needs and under the guidance of the present invention, equivalent modifications in shape, structure, etc. can be made to it, which are all within the scope of protection of this scheme.

Claims

1. A method for constructing a vectorized Jacobian matrix based on a node voltage equation, characterized in that: The following steps are involved: 1) Constructing the target power system model, including eliminating passive interconnection nodes in the system using the Kron order reduction method based on the target power system structure, and using the Thevenin equivalent method to equate the external power grid to an infinite power grid to generate a simplified equivalent circuit model; 2) Based on the simplified equivalent circuit model, the system admittance matrix is established, and the node voltage equation is formed based on the product relationship between the node current vector, the admittance matrix, and the node voltage vector; 3) Converting the node voltage equation into a node injection complex power equation, and replacing the product of the voltage amplitude and phase angle with the matrix multiplication of the diagonal matrix and the conjugate vector through vectorized matrix operations to obtain a vectorized expression of the complex power; 4) Converting the complex power equation into a real power flow equation in a polar coordinate system, calculating partial derivatives of the voltage amplitude vector and the phase angle vector respectively, and constructing a block structure of the Jacobian matrix. The block structure includes the following four sub-matrices: a partial derivative matrix of active power with respect to phase angle, a partial derivative matrix of active power with respect to voltage amplitude, a partial derivative matrix of reactive power with respect to phase angle, and a partial derivative matrix of reactive power with respect to voltage amplitude; 5) The Newton-Raphson method is used to iteratively solve the system power flow, and the converged voltage amplitude and phase angle are substituted into the block structure expression of the Jacobian matrix to generate a numerical Jacobian matrix.

2. The method for constructing a vectorized Jacobian matrix based on a node voltage equation according to claim 1, wherein: The system admittance matrix in step 2) is constructed as follows: Assuming that the system has n nodes, the admittance matrix Y The dimension is n×n, construct the system admittance matrix Y as follows: （1） Where, For nodes i With node j The admittance value between When , it represents the mutual admittance between nodes; When , it indicates the self-admittance of the node; Then the node voltage equation of the system is obtained as: （2） Where, is the node current vector, is the node voltage vector.

3. The method for constructing a vectorized Jacobian matrix based on a node voltage equation according to claim 2, wherein: The complex power equation in step 3) is constructed as follows: multiply both ends of the node voltage equation by the diagonal matrix , and the complex power equation of each node is obtained: （3） In the formula, the symbol It represents conjugate transpose for matrices and conjugate for vectors; S Inject complex power column vector into the node.

4. The method for constructing a vectorized Jacobian matrix based on a node voltage equation according to claim 3, wherein: The block matrix construction in step 4) specifically includes: Convert the complex node voltage vector into polar coordinates to get the real Jacobian matrix of the power flow equation. Let: （4） Where, U 、 θ yes step voltage magnitude and phase angle vector, represents the exponential operation of the phase angle vector θ element by element, where j is the imaginary unit; The vector product of represents the Hadamard product between vectors; The partial derivatives of the voltage amplitude vector and phase angle vector are obtained for the complex power equation. （5） Where the derivatives of a complex voltage with respect to its phase and amplitude are defined as follows: （6） Where: j is the imaginary unit; Substituting equations (2) and (6) into equation (5), we can obtain: （7） In which, let the coupling matrix ; According to formula (7), we can get the real number Jacobian matrix J : （8） In the formula, Re is the real part of each number, and Im is the imaginary part of each number; Restoring power S = P +j Q , then the Jacobian matrix J Re-expressed as: （9） Where, P Inject active power column vector into the node, Q Inject reactive power column vector into the node, Thus, the power flow Jacobian matrix for realizing process vectorization is constructed.

5. The method for constructing a vectorized Jacobian matrix based on a node voltage equation according to claim 1, wherein: After generating the numerical Jacobian matrix, further include any of the following: 6) Performing a static stability analysis of the power system based on the numerical Jacobian matrix, including: calculating the minimum singular value or eigenvalue of the Jacobian matrix and evaluating the voltage stability margin of the current operating section; locating the system instability mode based on the change in the sign of the eigenvalue and identifying weak nodes or critical transmission lines; 6) Constructing a sensitivity matrix based on the numerical Jacobian matrix for optimizing scheduling decisions, including extracting sensitivity coefficients of node active and reactive power to voltage amplitude and phase in the Jacobian matrix; generating a preventive control strategy based on the optimization objective function to adjust generator output or reactive compensation equipment; 6) Inputting the numerical Jacobian matrix into a deep learning model to train the mapping relationship between the grid operation state and the stability boundary, including: using the Jacobian matrix eigenvalues, node voltage and power data as input features; and outputting the load growth limit or fault tolerance capacity prediction result under the critical stability state of the system.

6. A vectorized Jacobian matrix construction system based on node voltage equations, characterized in that: include: The target power system model construction module is used to eliminate the passive connection nodes in the system through the Kron order reduction method according to the target power system structure, and use the Thevenin equivalent method to equate the external power grid to an infinite power grid to generate a simplified equivalent circuit model; an admittance matrix generation module, connected to the target power system model construction module, for constructing a system admittance matrix based on the simplified equivalent circuit model, and forming a node voltage equation based on the product relationship between the node current vector, the admittance matrix, and the node voltage vector; a complex power conversion module, connected to the admittance matrix generation module, for converting the node voltage equation into a node injection complex power equation, and replacing the product term of the voltage amplitude and the phase angle with the matrix multiplication of the diagonal matrix and the conjugate vector through vectorized matrix operation to obtain a vectorized expression of the complex power; A Jacobian matrix construction module is connected to the complex power conversion module and is used to convert the complex power equation into a real power flow equation in a polar coordinate system, obtain partial derivatives of the voltage amplitude vector and the phase angle vector respectively, and construct a block structure of the Jacobian matrix, wherein the block structure includes the following four sub-matrices: a partial derivative matrix of active power with respect to phase angle, a partial derivative matrix of active power with respect to voltage amplitude, a partial derivative matrix of reactive power with respect to phase angle, and a partial derivative matrix of reactive power with respect to voltage amplitude; The power flow solving module is connected to the Jacobian matrix construction module and is used to iteratively solve the system power flow using the Newton-Raphson method, substitute the converged voltage amplitude and phase angle into the block structure expression of the Jacobian matrix, and generate a numerical Jacobian matrix.

7. The vectorized Jacobian matrix construction system based on node voltage equations according to claim 6, characterized in that: The specific configuration of the admittance matrix generation module is as follows: assuming that the system has n nodes and the dimension of the admittance matrix Y is n×n, the system admittance matrix Y is constructed as follows: （1） Where, For nodes i With node j The admittance value between When , it represents the mutual admittance between nodes; When , it indicates the self-admittance of the node; The admittance matrix generation module further generates the node voltage equation: （2） Where, is the node current vector, is the node voltage vector.

8. The vectorized Jacobian matrix construction system based on node voltage equations according to claim 7, characterized in that: The complex power conversion module is specifically configured as follows: multiplying the two ends of the node voltage equation by the diagonal matrix , and the complex power equation of each node is obtained: （3） In the formula, the symbol It represents conjugate transpose for matrices and conjugate for vectors; S is the column vector of complex power injected into the node.

9. The vectorized Jacobian matrix construction system based on node voltage equations according to claim 8, characterized in that: The Jacobian matrix construction module includes the following submodules: The polar coordinate conversion submodule is used to convert the complex node voltage vector into a polar coordinate system representation. Let: （4） Where, U 、 θ yes step voltage magnitude and phase angle vector, represents the exponential operation of the phase angle vector θ element by element, where j is the imaginary unit; The vector product of represents the Hadamard product between vectors; The partial derivative calculation submodule is used to obtain the partial derivatives of the voltage amplitude vector and phase angle vector for the complex power equation: （5） Where the derivative of a complex voltage with respect to its phase and amplitude is defined as: （6） Where: j is the imaginary unit; The Jacobian matrix generation submodule is used to substitute equations (2) and (6) into equation (5) to obtain: （7） In which, let the coupling matrix ; According to formula (7), we can get the real number Jacobian matrix J : （8） In the formula, Re is the real part of each number, and Im is the imaginary part of each number; Restoring power S = P +j Q , then the Jacobian matrix J Re-expressed as: （9） Where, P Inject active power column vector into the node, Q Inject reactive power column vector into the node, Thus, the power flow Jacobian matrix for realizing process vectorization is constructed.

10. A vectorized Jacobian matrix construction system based on node voltage equations according to claim 9, characterized in that: Also includes any of the following functional modules: A static stability analysis module is used to calculate the minimum singular value or eigenvalue of the numerical Jacobian matrix based on the matrix, evaluate the voltage stability margin of the current operating section, locate the system instability mode according to the change of the eigenvalue sign, and identify weak nodes or key transmission lines; An optimization scheduling decision module is used to extract the sensitivity coefficients of node active and reactive power to voltage amplitude and phase based on the numerical Jacobian matrix, generate a preventive control strategy based on the optimization objective function, and adjust the generator output or reactive compensation equipment; A deep learning prediction module is used to input the numerical Jacobian matrix into a deep learning model, train the mapping relationship between the power grid operation state and the stability boundary, use the Jacobian matrix eigenvalues, node voltage and power data as input features, and output the load growth limit or fault tolerance capacity prediction result under the critical stability state of the system.

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